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54    \fancyhead[OC]{\textsc{Jonathan W. Tooker}}
55    \fancyhead[EC]{\textsc{Fractional Distance: The Topology of the Real Number Line}}
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98\title{{ \textbf{Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis}}}
99 
100\author{{\textsc{Jonathan W. Tooker}}}
101\begin{document}
102   
103 
104   
105\begin{minipage}{\textwidth}
106    \maketitle
107    \thispagestyle{empty}
108    \begin{abstract}
109        Recent analysis has uncovered a broad swath of rarely considered real numbers called real numbers in the neighborhood of infinity.  Here we extend the catalog of the rudimentary analytical properties of all real numbers by defining a set of fractional distance functions on the real number line and studying their behavior.  The main results of this paper are (1) to prove with modest axioms that some real numbers are greater than any natural number, (2) to develop a technique for taking a limit at infinity via the ordinary Cauchy definition reliant on the classical epsilon-delta formalism, and (3) to demonstrate an infinite number of non-trivial zeros of the Riemann zeta function in the neighborhood of infinity.  We define numbers in the neighborhood of infinity as Cartesian products of Cauchy equivalence classes of rationals.  We axiomatize the arithmetic of such numbers, prove all the operations are well-defined, and then make comparisons to the similar axioms of a complete ordered field.  After developing the many underling foundations, we present a basis for a topology.
110    \end{abstract}
111\end{minipage}
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115 
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127   
128   
129 
130\section{Introduction}
131 
132The original Euclidean definition of a real number \cite{EE} has given way over time to newer constructive definitions such as the Cauchy equivalence class suggested by Cantor \cite{CANTOR} and the Dedekind cut \cite{DEDE}, and also axiomatic definitions, the most popular of which are the axioms of a complete ordered field based in Hilbert's axioms of geometry \cite{HILB}.  The main purpose of this paper is to compare and contrast geometric and algebraic constructions of the real numbers, and then to give a hybrid constructive-axiomatic definition which increases the mutual complements among the two notions of geometry and algebra.  Throughout most of the history of mathematics, it was sufficient to give the Euclidean geometric conception of numbers as cuts in an infinite line, or ``magnitudes'' as Euclid is usually translated \cite{EE}.  The Euclid definition of $\mathbb{R}$ has its foundation in physical measurement.  In modernity, the preoccupation of mathematics with algebra more so than geometric measurement has stimulated the development of alternatives which are said to be ``more rigorous.''  The main development of this paper is to present an alternative set of constructive and algebraic axioms which more thoroughly preserve the underlying geometric notion that a number is a cut in an infinite line.  We will show that the Cauchy definition leaves something to be desired with respect to the underlying conception of $\mathbb{R}$ as an open-ended infinite line $(-\infty,\infty)$.  This something relates to the notion that the reals are the ``completion'' of the rationals.''
133 
134The equivalence class construction of $\mathbb{R}$ based on an assumed set of rational numbers $\mathbb{Q}$ precludes the existence of a neighborhood of infinity distinct from any neighborhood of the origin, as does the similar Dedekind cut.  For a finite interval $x'\in[0,\frac{\pi}{2})$, we can use $x=\tan(x')$ to construct the interval $x\in[0,\infty)$ wherein everything is usually considered to be a real number.  In this paper, we will develop the notion of fractional distance to prove that if there exists a number at the Euclidean midpoint $x'=\frac{\pi}{4}$ of $[0,\frac{\pi}{2})$, then the bijectivity of compositions of bijective functions, in this case the identity function $f(x)=x$ and the tangent function $f(x)=\tan(x)$ on $[0,\frac{\pi}{2})$, should require a real number at the Euclidean midpoint of $[0,\infty)$.  A proof that there must exist such a number is the linchpin of everything in this paper.  Such a number is said to be a number in the neighborhood of infinity because it has non-zero ``fractional distance'' with respect to infinity.  We will show that this number is required to preserve Euclid's conception of a number as a cut in an infinite line and we will argue that a construction which preserves it is necessarily better.  Indeed, since Euler used this number in his own proofs \cite{EULER1748,EULER1988,OLDINF}, the fractional distance approach to $\mathbb{R}$ presented here should be considered \textbf{\textit{a return to the old rather than a proposition for something new}}.
135 
136 
137Treatment of the neighborhood of infinity as a distinct numerical mode with separate behavior from the neighborhood of the origin is the direct motivator for everything new reported in this paper.  We will posit one very modest change to the Cauchy construction such that it will more fully preserve the favorable notion that $\mathbb{R}=(-\infty,\infty)$ which equivalent to the assumption that $\mathbb{R}$ has the usual topology.  The modified equivalence class construction will give formal constructions for real numbers in the neighborhood of infinity rather than preclude their existence.  With our new constructions and axioms given, we will present an analysis of $\mathbb{R}$ yielding unexpected properties which are non-trivial and exciting, and then we will give the formal topological basis.  
138 
139In previous work \cite{RINF,ZEROSZZ}, we have demonstrated the existence of a broad class of real numbers: those in the neighborhood of infinity.  In the present paper, we will again demonstrate the existence of real numbers in the neighborhood of infinity.  Then we will construct such numbers more or less directly from $\mathbb{Q}$, and then we will axiomatize the arithmetic of such numbers and study the consequences which follow.
140 
141The paper is structured as follows.
142\begin{itemize}
143    \item Section Two: We give a simple Euclidean definition for real numbers.  These geometric considerations set the stage for the algebraic considerations which follow.
144    \item Section Three: We define and analyze a set of functions called fractional distance functions.  These functions constitute the kernel of the analytical direction of this paper.
145    \item Section Four: We give the properties of real numbers in the neighborhood of infinity.  The \textbf{\textit{formal algebraic construction}} of such numbers by Cauchy sequences is given therein.
146    \item  Section Five: We axiomatize a set of arithmetic operations for $\mathbb{R}$ and make a comparison with the similar field axioms.  We find they are mostly the same, but slightly different.
147    \item Section Six: We prove some results with the present arithmetic axioms.  Interestingly, we develop a technique by which it is possible to take a limit at infinity with the ordinary Cauchy prescription for limits: something that has been considered heretofore impossible.
148    \item Section Seven: This section is dedicated most specifically to the topological and generally set theoretical properties of the real number line.  The main thrust is to define a Cantor-like set on $\mathbb{R}$ and then to examine its consequences for the least upper bound property of connected sets.
149   
150    \item Section Eight: We apply the notions and consequences of fractional distance to the Riemann hypothesis.  We show that the Riemann $\zeta$ function \textit{does} have non-trivial zeros off the critical line.
151\end{itemize}
152 
153\section{Mathematical Preliminary}\label{sec:RN1ddd}
154   
155 
156\subsection{Real Numbers}\label{sec:RN1}
157 
158 
159In this section, the reader is invited to recall the distinction between the real numbers $\mathbb{R}$ and the real number field $\mathcal{R}=\{\mathbb{R},+,\times,\leq\}$.  Real numbers exist independently of their operations.  Here, we define real numbers as cuts in the real number line pending a more formal, complementary definition by Cauchy sequences in Section \ref{sec:rg8hr8y8hh}, and by Dedekind cuts in Section \ref{sec:topoR}.  By defining a line, giving it a label ``real,'' defining cuts in a line, and then defining real numbers as cuts in the real number line, we make a rigorous definition of real numbers sufficient for applications at any level of rigor.  Specifically, the definition given in this section underpins the Cauchy and Dedekind definitions given later.  
160 
161Generally, the definition of real numbers given in the present section is totally equivalent to the Euclidean magnitude defined in Euclid's \textit{Elements}.  Fitzpatrick, the translator of Euclid's original Greek in Reference \cite{EE}, points out that Euclid's analysis was deliberately restricted to that which may be measured with a physical compass and straight edge: what are called the constructible numbers.  Euclid surely was well aware, however, that the real number line is of immeasurable, non-constructible length, and that non-constructible numbers exist.  The main motivator for the presentation of new formalism in this paper is that we would like to consider both measurable and immeasurable magnitudes, or constructible and non-constructible numbers, which exceed those that can be defined in the canonical Cauchy and Dedekind approaches \cite{CANTOR,DEDE}.
162 
163 
164\begin{defin}\label{def:lineXXX}
165    A line is a 1D Hausdorff space extending infinitely far in both directions.  The interval representation of a line is $(-\infty,\infty)$.  In other words, the connected interval $(-\infty,\infty)$ is an infinite line.
166\end{defin}
167 
168\begin{defin}\label{def:metspa}
169    A number line is a line equipped with a chart $x$ and the Euclidean metric
170    \begin{equation}
171    d(x,y)=
172    \big|y-x\big|~~.\nonumber
173    \end{equation}
174\end{defin}
175 
176\begin{defin}\label{def:2412b24}
177    The real number line is a unique number line given the label ``real.''
178\end{defin}
179 
180\begin{defin}\label{def:cuts}
181    If $x$ is a cut in a line, then
182    \begin{equation}
183    (-\infty,\infty)=(-\infty,x]\cup(x,\infty)~~.\nonumber
184    \end{equation}
185\end{defin}
186 
187 
188\begin{defin}\label{def:real}
189    A real number $x\in\mathbb{R}$ is a cut in the real number line.
190\end{defin}
191 
192\begin{axio}\label{ax:97977080}
193    Real numbers are such that
194    \begin{equation*}
195    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x\neq y\quad\exists n\in\mathbb{N}\quad\text{s.t.}\quad \big|y-x\big|>\frac{1}{n}~~.
196    \end{equation*}
197   
198    \noindent In other words, neither infinitesimals nor numbers having infinitesimal parts are real numbers.  
199\end{axio}
200 
201 
202 
203\begin{axio} \label{ax:mainR}
204    Real numbers are represented in algebraic interval notation as
205    \begin{equation}
206    \mathbb{R}=(-\infty,\infty)~~.\nonumber
207    \end{equation}
208   
209    \noindent In other words, $x\in\mathbb{R}$ if $x$ is both less than infinity and greater than minus infinity.  The connectedness of $\mathbb{R}$ is explicit in the interval notation.
210\end{axio}
211 
212\begin{rem}
213    In Section \ref{sec:fconst}, we will supplement Axiom \ref{ax:mainR} by giving a definition in terms of Cauchy equivalence classes.  Axiom \ref{ax:mainR} is often considered as lacking sufficient rigor but the Cauchy definition will remedy any so-called insufficiencies of the broad generality of Axiom \ref{ax:mainR}.  
214\end{rem}
215 
216\begin{defin}\label{def:orig}
217    $\mathbb{R}_0$ is a subset of all real numbers
218    \begin{equation}
219    \mathbb{R}_0=\left\{  x\in\mathbb{R}~\big|~(  \exists n\in\mathbb{N})[ -n<x<n ]  \right\}     ~~.\nonumber
220    \end{equation}
221   
222    \noindent Here we define $\mathbb{R}_0$ as the set of all $x\in\mathbb{R}$ such that there exists an $n\in\mathbb{N}$ allowing us to write $-n<x<n$.  We call this the set of real numbers less than some natural number (where absolute value is implied.)  These numbers are said to lie within the neighborhood of the origin.
223\end{defin}
224 
225 
226\begin{defin}\label{def:Rinf}
227    $\mathbb{R}_\infty$ is a subset of all real numbers with the property
228    \begin{equation}
229    \mathbb{R}_\infty=\mathbb{R}\,\backslash\,\mathbb{R}_0~~.\nonumber
230    \end{equation}
231\end{defin}
232 
233 
234\subsection{Affinely Extended Real Numbers}\label{sec:33333}
235 
236 
237 
238To prove in Section \ref{sec:RN} that $\mathbb{R}_\infty$ is not the empty set, namely that there are real numbers larger than every natural number, we will make reference to ``line segments'' beyond the simpler construction called ``a line.''  Most generally, a line with two different endpoints $A$ and $B$ is a called a line segment $AB$.  We will use notation such that $AB\equiv[a,b]$ where $[a,b]$ is an interval of numbers.  Nowhere will we require that the endpoints must be real numbers so the interval $[a,b]=[-\infty,\infty]$ will conform to the definition of a line segment.  The real line $\mathbb{R}$ together with two endpoints $\{\pm\infty\}$ is called the affinely extended real number line $\overline{\mathbb{R}}=[-\infty,\infty]$.  The present section lays the foundation for an analysis of general line segments in Section \ref{sec:LineSegs} by first giving some properties of $\overline{\mathbb{R}}$.  
239 
240 
241 
242\begin{defin}  \label{def:RRRinf9999}
243    For $x\in\mathbb{R}$ and $n,k\in\mathbb{N}$, we have the properties
244    \begin{equation}
245    \lim\limits_{x\to0^\pm}\dfrac{1}{x}=\text{diverges}~~,\quad\text{and}\qquad \lim\limits_{n\to\infty}\sum_{k=1}^{n}k=\text{diverges}~~. \nonumber
246    \end{equation}
247\end{defin}
248 
249 
250 
251\begin{defin} \label{def:RRRinf}
252    Define two affinely extended real numbers $\pm\infty$ such that for $x\in\mathbb{R}$ and $n,k\in\mathbb{N}$, we have the properties
253    \begin{equation}
254    \lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\infty~~, \quad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
255    \end{equation}
256   
257    \noindent The limit as $x$ approaches zero shall be referred to as the limit definition of infinity.
258\end{defin}
259 
260 
261\begin{axio} \label{ax:undefin7666y}
262    The infinite element $\infty$ is such that
263    \begin{align}
264    \infty-\infty=\text{undefined}~~,\quad\text{and}\qquad\frac{\infty}{\infty}=\text{undefined}~~.\nonumber
265    \end{align}
266\end{axio}
267 
268 
269\begin{defin} \label{def:overlineR}
270    The set of all affinely extended real numbers is
271    \begin{equation}
272    \overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}~~.\nonumber
273    \end{equation}
274 
275    \noindent This set is defined in interval notation as
276    \begin{equation}
277    \overline{\mathbb{R}}=[-\infty,\infty]~~.\nonumber
278    \end{equation}
279\end{defin}
280 
281 
282\begin{rem}\label{def:12ccc2}
283    If $x_n>0$ with $\{x_n\}$ being a monotonic sequence, the $\infty$ symbol is such that if $x_n\in\mathbb{R}$, and if
284    \begin{equation*}
285    \lim\limits_{n\to\infty}x_n=\text{diverges}~~,
286    \end{equation*}
287   
288    \noindent then for the same $x_n\in\overline{\mathbb{R}}$ we have
289    \begin{equation}
290    \lim\limits_{n\to\infty}x_n=\infty~~.\nonumber
291    \end{equation}
292\end{rem}
293 
294 
295 
296\begin{defin}\label{def:AFF2real}
297    An affinely extended real number $x\in\overline{\mathbb{R}}$ is $\pm\infty$ or it is a cut in the affinely extended real number line:
298    \begin{equation}
299    [-\infty,\infty]=[-\infty,x]\cup(x,\infty]~~.\nonumber
300    \end{equation}
301\end{defin}
302 
303 
304 
305 
306\begin{thm} \label{thm:RnotR}
307    If $x\in\overline{\mathbb{R}}$ and $x\neq\pm\infty$, then $x\in\mathbb{R}$.
308\end{thm}
309 
310 
311 
312\begin{proof}
313    Proof follows from Definition \ref{def:overlineR}.
314\end{proof}
315 
316 
317\subsection{Line Segments}\label{sec:LineSegs}
318 
319 
320In this section, we review what is commonly understood regarding Euclidean line segments \cite{EE}.  We begin to develop the relationship between points in a line segment and cuts in a line.  During the analyses which follow in the remainder of this paper, we will closely examine the differences between cuts and points as a proxy for the fundamental relationship between algebra and geometry.  Section \ref{sec:3322442} is dedicated specifically to these distinctions though they are treated throughout this text.  The general principle of the distinction between cuts and points is the following.  If $x$ is a cut in a line, then
321\begin{equation}
322(-\infty,\infty)=(-\infty,x]\cup(x,\infty)~~.\nonumber
323\end{equation}
324 
325\noindent If $x$ is a point in a line segment, then we have a tentative, preliminary understanding that
326\begin{equation}
327[a,b]=[a,x)\cup\{x\}\cup(x,b]~~.\nonumber
328\end{equation}
329 
330\begin{defin}\label{def:lineseg}
331    A line segment $AB$ is a line together with two different endpoints $A\neq B$.  
332\end{defin}
333 
334 
335\begin{defin}\label{def:lineseg2}
336    $AB$ is a real line segment if and only if the endpoints $A$ and $B$ bound some subset of the real line $\mathbb{R}=(-\infty,\infty)$.
337\end{defin}
338 
339 
340\begin{defin}
341    Much of the analysis in this paper will depend on relationships between geometric and algebraic expressions.  The $\equiv$ symbol will be used to denote symbolic equality between geometric and algebraic expressions.
342\end{defin}
343 
344 
345 
346\begin{defin}\label{def:li5t35y5y5y2}
347    A real line segment $AB$ is represented in interval notation as $AB\equiv[a,b]$ where $a$ and $b$ are any two affinely extended real numbers $a,b\in\overline{\mathbb{R}}$ such that $a<b$.
348\end{defin}
349 
350\begin{defin}\label{def:eucdef9}
351    The Euclidean notation $AB$ is called the geometric representation of a line segment.  The interval notation $[a,b]$ is called the algebraic representation of a line segment.
352\end{defin}
353 
354\begin{axio}\label{def:2points}
355Line segments have the property that
356\begin{equation*}
357AB=AC\quad\iff\quad B=C~~.
358\end{equation*}
359\end{axio}
360 
361 
362\begin{axio}\label{def:li5t3f5y5y5y2}
363    Two line segments $AB$ and $CD$ are equal, meaning $AB=CD$, if and only if
364    \begin{equation}
365    \cfrac{AB}{CD}=\cfrac{CD}{AB}=1~~.\nonumber
366    \end{equation}
367\end{axio}
368 
369 
370\begin{defin}
371    $\mathbf{AB}$ is a special label given to the unique real line segment $AB\equiv[0,\infty]$.  We have
372    \begin{equation}
373    AB=\mathbf{AB}\quad\iff\quad AB\equiv[0,\infty]~~.\nonumber
374    \end{equation}
375\end{defin}
376 
377 
378 
379 
380\begin{defin}
381    $X$ is an interior point of $AB$ if and only if
382    \begin{equation}
383    X\neq A~~,~~X\neq B~~,\quad\text{and}\qquad X\in AB  ~~.\nonumber
384    \end{equation}
385\end{defin}
386 
387\begin{axio}\label{def:5y5dddy2}
388    If $X$ is an interior point of $AB$, then
389    \begin{equation}
390    AB=AX+XB~~.\nonumber
391    \end{equation}
392\end{axio}
393 
394 
395\begin{axio}\label{ax:779hzz}
396    Every geometric point $X$ along a real line segment $AB$ has one and only one algebraic interval representation $\mathscr{X}$.  If $\mathscr{X}$ is the algebraic representation of $X$, then $X\equiv\mathscr{X}$ and $\mathscr{X}$ is a unique subset of $[a,b]\equiv AB$.
397\end{axio}
398 
399\begin{defin}
400    The formal meaning of the relation $AB\equiv[a,b]$ is that $a$ is the least number in the algebraic representation of $A$, $b$ is the greatest number in the algebraic representation of $B$, and that every other number $x$ in the algebraic representation of any point in $AB$ has the property $a<x<b$.
401\end{defin}
402 
403 
404\begin{thm}
405    If $X$ is an interior point of a real line segment $AB$, then $X$ has an algebraic interval representation as one or more real numbers.
406\end{thm}
407 
408\begin{proof}
409    $X$ is an interior point of $AB$ so, by Axiom \ref{def:5y5dddy2}, we have
410    \begin{equation}
411    AB=AX+XB~~.\nonumber
412    \end{equation}
413   
414    \noindent Since $AB\equiv[a,b]$ and $(a,b)\subset\mathbb{R}$, it follows that the algebraic representation $\mathscr{X}$ of an interior point $X$ is such that
415    \begin{equation}
416    x\in\mathscr{X}\quad\implies\quad a<x<b~~.\nonumber
417    \end{equation}
418   
419    \noindent For $(a,b)\subset\mathbb{R}$, this inequality is only satisfied by $x\in\mathbb{R}$.  The theorem is proven.
420\end{proof}
421 
422\begin{rem}
423    It will be a main result of this paper to show that the infinite length of a line segment such as $\mathbf{AB}\equiv[0,\infty]$ will allow us to put more than one number into the algebraic representation $\mathscr{X}$ of a geometric point $X$.  If a line segment has finite length $L\in\mathbb{R}_0$, we will show that there is at most one real number in the algebraic representation of one its interior points.  However, this constraint will vanish in certain cases of $\text{len}(AB)$.
424\end{rem}
425 
426 
427\begin{defin}\label{def:XrepR}
428    The algebraic representation $\mathscr{X}$ of a geometric point $X$ lying along a real line segment $AB$ is
429    \begin{equation}
430    \mathscr{X}=[x_1,x_2]~~,\quad\text{where}\qquad x_1,x_2\in\overline{\mathbb{R}}~~. \nonumber
431    \end{equation}
432   
433    \noindent  The special (intuitive) case of $x_1=x_2=x$ gives
434    \begin{equation}
435    \mathscr{X}=[x,x]=\{x\}=x~~. \nonumber
436    \end{equation}
437   
438    \noindent Here, we have expressed $\mathscr{X}$ with included endpoints $x_1$ and $x_2$.  Most generally, however, an algebraic representation of a geometric point is a single number or it is some interval of numbers, \textit{i.e.}: all variations of $(x_1,x_2)$, $(x_1,x_2]$, and $[x_1,x_2)$ are allowable algebraic representations of $X$.  We do not require that $x_1\neq x_2$ in all cases.
439\end{defin}
440 
441 
442\begin{rem}
443    A point in a line segment has a representation as a set of numbers, possibly only one number, and it remains to identify the exact relationship between numbers (cuts) and geometric points.  The key feature of Definition \ref{def:XrepR} is that it allows, provisionally, a many-to-one relationship between cuts in lines (algebraic) and points in line segments (geometric.)  In Section \ref{sec:3322442}, we will strictly prove that which has been suggested: the algebraic representation of $X\in AB$ is only constrained to be a unique real number for certain cases of $AB$ with finite length.
444\end{rem}
445 
446 
447\begin{defin}
448    If $X\equiv\mathscr{X}=[x_1,x_2]$ with $x_1\neq x_2$, and if $x\in[x_1,x_2]$, then $x$ is said to be a \textit{possible} algebraic representation of $X$.  If $x_1=x_2=x$, then $x$ is said to be \textit{the} algebraic representation of $X$.  If $x$ is the algebraic representation of $X$, then $x\equiv X$.  If $x$ is a possible representation of $X$, then $x\in X$, \textit{i.e.}: if $x$ is a possible algebraic representation of $X$, then
449    \begin{equation}
450    x\in\mathscr{X}=[x_1,x_2]\equiv X~~.\nonumber
451    \end{equation}
452   
453    \noindent  This statement may be abbreviated as $x\in X$ while $x\equiv X$ specifies the case of $x_1=x_2$.
454\end{defin}
455 
456 
457 
458 
459 
460\begin{defin}\label{def:fet7}
461    A point $C$ is called a midpoint of a line segment $AB$ if and only if
462    \begin{equation}
463    \cfrac{AC}{ AB} =\cfrac{CB}{ AB} =\frac{1}{2}~~.\nonumber
464    \end{equation}
465   
466    \noindent Alternatively, $C$ is a midpoint of $AB$ if and only if
467    \begin{equation}
468    AC=CB~~,\quad\text{and}\qquad AC+CB=AB~~.\nonumber
469    \end{equation}
470\end{defin}
471 
472 
473\begin{defin}
474    Hilbert's discarded axiom \cite{HILB} states the following: Any four points $\{A,B,C,D\}$ of a line can always be labeled so that $B$ shall lie between $A$ and $C$ and also between $A$ and $D$, and, furthermore, that $C$ shall lie between $A$ and $D$ and also between $B$ and $D$.
475\end{defin}
476 
477\begin{rem}
478    Hilbert's discarded axiom is discarded not because it wrong but rather because it is implicit in Hilbert's other axioms \cite{HILB}.  It is discarded by redundancy rather than invalidity.  
479\end{rem}
480 
481 
482\begin{thm}\label{def:fet17}
483    All line segments have at least one midpoint.
484\end{thm}
485 
486\begin{proof}
487    Let there be a line segment $AB$ and two circles of equal radii centered on the points $A$ and $B$.  Let the two radii be less than $AB$ but great enough such that the circles intersect at exactly two points $S$ and $T$.  The geometric configuration shown in Figure \ref{fig:twocirc} is guaranteed to exist by Hilbert's discarded axiom pertaining to $\{A,X_1,X_2,B\}$.  It follows by construction that
488    \begin{equation}
489    AS=AT=BS=BT~~.\nonumber
490    \end{equation}
491   
492    \noindent   Let the line segment $ST$ intersect $AB$ at $C$.  By the Pythagorean theorem, $C$ is a midpoint of $AB$ because
493    \begin{equation}
494    AC^2+CS^2=AS^2~~,\quad\text{and}\qquad BC^2+CS^2=BS^2~~,\nonumber
495    \end{equation}
496   
497    \noindent together yield
498    \begin{equation}
499    AC=BC~~.\nonumber
500    \end{equation}
501   
502    \noindent  $C$ separates $AB$ into two line segments so
503    \begin{equation}
504    AC+CB=AB~~.\nonumber
505    \end{equation}
506   
507    \noindent These two conditions, $AC=BC$ and $AC+CB=AB$, jointly conform to Definition \ref{def:fet7} so $C$ is a midpoint of an arbitrary line segment $AB$.
508\end{proof}
509 
510\begin{figure}[t]
511    \makebox[\textwidth][c]{
512        \includegraphics[scale=.25]{twocirc2.png}}
513    \caption{This figure proves that every line segment $AB$ has one and only one midpoint.}
514    \label{fig:twocirc}
515\end{figure}
516 
517\begin{exa}\label{ex:confrmal}
518    Theorem \ref{def:fet17} regards an arbitrary line segment $AB$.  Therefore, the theorem holds in the case of an arbitrary line segment $AB$.  One might be afflicted, however, with the assumption that it is not possible to define two such intersecting circles centered on the endpoints of an arbitrary line segment such as $\mathbf{AB}\equiv[0,\infty]$.  To demonstrate how the arbitrary case of any line segment $AB$ generalizes to the specific case of $\mathbf{AB}$, let $AB\equiv[0,\tfrac{\pi}{2}]$ and let $x'\in\mathscr{X}$ be a number in the algebraic representation of $X\in AB$.  We say that $[0,\tfrac{\pi}{2}]$ is the algebraic representation of  $AB$ charted in $x'$.  Let $x$ be such that
519    \begin{equation}
520    x=\tan(x')~~,\nonumber
521    \end{equation}
522   
523    \noindent so that $x$ and $x'$ are two charts related by a conformal transformation.  Using
524    \begin{equation}
525    \tan(0)=0~~,\quad\text{and}\qquad\tan\left(\cfrac{\pi}{2}\right)=\infty~~,\nonumber
526    \end{equation}
527   
528    \noindent where the latter follows from Definition \ref{def:RRRinf}, it follows that $[0,\infty]$ is the algebraic representation of $AB$ charted in $x$.  Therefore, $AB=\mathbf{AB}$ with respect to the $x$ chart.  
529   
530    Hilbert's discarded axiom guarantees the existence of two points $X_1\in AB$ and $X_2\in AB$ with algebraic representations $\mathscr{X}_1'$ and $\mathscr{X}_2'$ such that
531    \begin{equation}
532    x'=\cfrac{\pi}{ 6}\in\mathscr{X}_1'~~,\quad\text{and}\qquad x'= \cfrac{\pi}{ 3}\in\mathscr{X}_2'~~.\nonumber
533    \end{equation}
534   
535    \noindent If the radius of the circle centered on $A$ is $AX_2$ and the radius of the circle centered on $B$ is $X_1B$, then it is guaranteed that these circles will intersect at two points $S$ and $T$, as in Figure \ref{fig:twocirc}.  Since $AB=\mathbf{AB}$ in the $x$ chart, it is required that $X_1\in\mathbf{AB}$ and $X_2\in\mathbf{AB}$.  Therefore, circles centered on the endpoints of $\mathbf{AB}$ with radii $AX_2$ and $X_1B$ will intersect at exactly two points.  \textbf{\textit{The chart on the line segment cannot affect the line segment's basic geometric properties!}}  It is unquestionable that the points $X_1$ and $X_2$ exist and are well-defined in the $x'$ chart, and it is not possible to disrupt the geometric configuration by introducing a second chart onto $AB$.  A chart can no more disrupt the geometric configuration than erasing an island from a map might make the physical island disappear from the sea.  $X_1$ and $X_2$ do not cease to exist simply because we define a conformal chart $x=\tan(x')$.  If they ceased to exist, then that would violate Hilbert's discarded axiom.  This example demonstrates that Theorem \ref{def:fet17} is even valid for the specific case of the infinite line segment $AB=\mathbf{AB}$.
536\end{exa}
537 
538 
539 
540 
541\begin{thm}\label{thm:onemid}
542    All line segments have one and only one midpoint.
543\end{thm}
544 
545\begin{proof}
546    For proof by contradiction, suppose $C$ and $D$ are two different midpoints of a line segment $AB$.  $C$ and $D$ are midpoints of $AB$ so we may derive from Definition \ref{def:fet7}
547    \begin{equation}
548    AC=CB=\cfrac{AB}{2}~~,\quad\text{and}\qquad AD=DB=\cfrac{AB}{2}~~.\nonumber
549    \end{equation}
550   
551    \noindent It follows that $AC=AD$.  By Axiom \ref{def:2points}, therefore, $C=D$ and we invoke a contradiction having assumed that $C$ and $D$ are different.  
552\end{proof}
553 
554 
555\section{Fractional Distance}
556 
557\subsection{Fractional Distance Functions}\label{sec:FD}
558 
559 
560If there are two circles with equal radii whose centers are separated by an infinite distance, then what numerical radii less than infinity will allow the circles to intersect at exactly two points?  To answer this question, we will introduce fractional distance functions.  We will use these functions to demonstrate the existence of real numbers in the neighborhood of infinity.
561 
562 
563 
564\begin{defin}\label{def:gfdf}
565    For any point $X$ on a real line segment $AB$, the geometric fractional distance function $\mathcal{D}_{\!AB}$ is a continuous bijective map
566    \begin{equation}
567    \mathcal{D}_{\!AB}(AX):AB\to[0,1]  ~~.\nonumber
568    \end{equation}
569   
570    \noindent which takes $AX\subseteq AB$ and returns real numbers.  This function returns $AX$ as a fraction of $AB$.  Emphasizing the geometric construction,  the geometric fractional distance function  $\mathcal{D}_{\!AB}$ is defined as
571    \begin{equation}
572    \mathcal{D}_{\!AB}(AX)=\begin{cases}
573    ~~1\qquad\quad\text{for}\quad X=B\\[8pt]
574    \cfrac{AX}
575    {AB}~\qquad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
576    ~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~.\nonumber
577    \end{equation}
578   
579    \noindent The quotient of two real line segments is defined as a real number.
580\end{defin}
581 
582\begin{rem}
583    The domain of $\mathcal{D}_{\!AB}(AX)$ is defined as subsets of real line segments.  This allows $AX=AA$ which would be excluded from a domain of real line segments because $AA$ does not have two different endpoints.
584\end{rem}
585 
586 
587 
588\begin{thm}\label{thm:35353g0}
589    For any point $X\in AB$, the bijective geometric fractional distance function $\mathcal{D}_{\!AB}(AX):AB\to R$ has range $R=[0,1]$.
590\end{thm}
591 
592 
593\begin{proof}
594    Assume $\mathcal{D}_{\!AB}(AX)<0$.  Then one of the lengths in the fraction must be negative and we invoke a contradiction with the length of a line segment defined as a positive number (Definition \ref{def:metspa}.)  If $\mathcal{D}_{\!AB}(AX)>1$, then $AX>AB$ and we invoke a contradiction by the implication $AX\nsubseteq AB$.  We have excluded from $R$ all numbers less than zero and greater than one.  Since $\mathcal{D}_{\!AB}(AX)$ is a continuous function taking the values zero and one at the endpoints of its domain, the intermediate value theorem requires that the range of $\mathcal{D}_{\!AB}(AX):AB\to R$ is $R=[0,1]$.  
595\end{proof}
596 
597 
598\begin{cor}
599    All line segments have at least one midpoint.  
600\end{cor}
601 
602\begin{proof}
603    (Reproof of Theorem \ref{def:fet17}.)  $\mathcal{D}_{\!AB}(AX)$ is a continuous function on the domain $AB$ taking finite values zero and one at the endpoints of its domain.  By the intermediate value theorem, there exists a point $C$ in the domain $AB$ for which $\mathcal{D}_{\!AB}(AC)=0.5$.  By Definition \ref{def:fet7}, $C$ is a midpoint of $AB$.
604\end{proof}
605 
606 
607\begin{thm}
608    Every midpoint of a line segment $AB$ is an interior point of $AB$.
609\end{thm}
610 
611\begin{proof}
612    If $X\in AB$ is not an interior point of $AB$, then $X=A$ or $X=B$.  In each case respectively, the geometric fractional distance function returns 
613    \begin{equation}
614    \mathcal{D}_{\!AB}(AA)=0~~,\quad\text{or}\qquad \mathcal{D}_{\!AB}(AB)=1~~.\nonumber
615    \end{equation}
616   
617    \noindent A point $C$ is a midpoint of $AB$ if and only if  
618    \begin{equation}
619    \mathcal{D}_{\!AB}(AC)=0.5~~.\nonumber
620    \end{equation}
621   
622    \noindent No midpoint can be an endpoint.  
623\end{proof}
624 
625\begin{rem}
626    Given the geometric fractional distance function, it is not clear how to compute $\mathcal{D}_{\!AB}(AX)$ when $X$ is an arbitrary interior point.  By Definition \ref{def:gfdf}, we know that the fraction $\frac{AX}{AB}$ is a real number but we have not developed any tools for finding the numerical value.   The quotient notation required for computing fractional distance calls for an algebraic notion of distance.  
627\end{rem}
628 
629\begin{defin}\label{def:iuuuttgrr488}
630    $\mathcal{D}^\dagger_{\!AB} $ is the algebraic fractional distance function.  It is an algebraic expression which totally replicates the behavior of the geometric fractional distance function $\mathcal{D}_{\!AB} $ on an arbitrary line segment $AB\equiv[a,b]$, \textit{and} it has the added property that its numerical output is easily simplified.  The algebraic fractional distance function $\mathcal{D}^\dagger_{\!AB} $ is constrained to be such that
631    \begin{equation}
632    \mathcal{D}^\dagger_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
633    \end{equation}
634   
635    \noindent for every point $X\in AB$.
636\end{defin}
637 
638 
639\begin{rem}
640    In Definitions \ref{def:algfracdis} and \ref{def:algfracdisq1}, we will define two kinds of algebraic fractional distance functions (FDFs.)  The purpose in defining two kinds of FDFs will be so that we may compare their properties and then choose the one that exactly replicates the behavior of the geometric FDF $\mathcal{D}_{\!AB} $.  
641\end{rem}
642 
643\begin{defin}\label{def:algfracdis}
644    The algebraic FDF of the first kind  
645    \begin{equation}
646    \mathcal{D}'_{\!AB}(AX):AB\to[0,1] ~~,\nonumber
647    \end{equation}
648   
649    \noindent   is a map on subsets of real line segments
650    \begin{equation}
651    \mathcal{D}'_{\!AB}(AX)=\begin{cases}
652    ~~~1\qquad\quad\text{for}\quad X=B\\[8pt]
653    \cfrac{\|AX\|}{ \|AB\|}\,~~\quad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
654    ~~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~,\nonumber
655    \end{equation}
656   
657    \noindent where
658    \begin{equation}
659    \cfrac{\|AX\|}{ \|AB\|}  =   \cfrac{\text{len}[a,x]}{\text{len}[a,b]}    ~~,\nonumber
660    \end{equation}
661   
662    \noindent and $[a,x]$ and $[a,b]$ are the line segments $AX$ and $AB$ expressed in interval notation.  
663\end{defin}
664 
665 
666\begin{defin} \label{def:agree}
667    The norm $\|AX\|=\text{len}[a,x]$ which appears in  $\mathcal{D}'_{\!AB}(AX)$ is defined so that
668    \begin{equation}
669    \mathcal{D}'_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
670    \end{equation}
671   
672    \noindent Specifically, the length function is defined as the Euclidean distance between the endpoints of the algebraic representation.  Per Definition \ref{def:metspa}, we have
673    \begin{equation*}
674    \text{len}[a,b]=d(a,b)=\big|b-a\big|~~.
675    \end{equation*}
676\end{defin}
677 
678 
679 
680\begin{defin}\label{def:algfracdisq1}
681    An algebraic fractional distance function of the second kind  
682    \begin{equation}
683    \mathcal{D}''_{\!AB}(AX):[a,b]\to[0,1] ~~,\nonumber
684    \end{equation}
685   
686    \noindent  is a map on intervals of the form
687    \begin{equation}
688    \mathcal{D}''_{\!AB}(AX)=\begin{cases}
689    ~~~~~1\qquad&\text{for}\quad X=B\\[8pt]
690    \cfrac{\text{len}[a,x]}{\text{len}[a,b]}\,~~&\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
691    ~~~~~0 \qquad&\text{for}\quad X=A \end{cases}~~.\nonumber
692    \end{equation}  
693\end{defin}
694 
695\begin{rem}
696     Take note of the main difference between the two algebraic FDFs.  The first kind has a geometric domain
697    \begin{equation*}
698    \mathcal{D}'_{\!AB}(AX):AB\to\mathbb{R}~~,
699    \end{equation*}
700   
701    \noindent but the second kind has an algebraic domain
702    \begin{equation*}
703    \mathcal{D}''_{\!AB}(AX):[a,b]\to\mathbb{R}~~.
704    \end{equation*}
705   
706    \noindent As a matter of consistency of notation, we have written $\mathcal{D}''_{\!AB}(AX)$ even when the notation $\mathcal{D}''_{AB}([a,x])$ might better illustrate that the domain of $\mathcal{D}_{\!AB}''$ is intervals rather than line segments.  The reader is so advised.  
707\end{rem}
708 
709 
710 
711\begin{axio}\label{def:order}
712    The ordering of $\mathbb{R}$ is such that for any $x,y\in\mathbb{R}$, if   
713    \begin{equation}
714    x\in[x_1,x_2]=\mathscr{X}\equiv X~~,\quad\text{and}\qquad y\in[y_1,y_2]=\mathscr{Y}\equiv Y~~,\nonumber
715    \end{equation}
716   
717    \noindent then 
718    \begin{equation}
719    \mathcal{D}_{\!AB}(AX)>\mathcal{D}_{\!AB}(AY)\quad\implies\quad x>y ~~.\nonumber
720    \end{equation}
721\end{axio}
722 
723\begin{thm}\label{thm:injjj}
724    The geometric fractional distance function $\mathcal{D}_{\!AB}$ is injective (one-to-one) on all real line segments.
725\end{thm}
726 
727\begin{proof}
728    By Definition \ref{def:gfdf}, the geometric FDF is  
729    \begin{equation}
730    \mathcal{D}_{\!AB}(AX)=\begin{cases}
731    ~~1\qquad\quad\text{for}\quad X=B\\[8pt]
732    \cfrac{AX}
733    {AB}~\qquad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
734    ~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~.\nonumber
735    \end{equation}
736   
737   
738    \noindent For proof by contradiction, assume $\mathcal{D}_{\!AB}$ is not always injective.  Then there exists some $X_1\neq X_2$ such that  
739    \begin{equation}
740    \cfrac{AX_1}{AB}=\cfrac{AX_2}{AB}~~.\nonumber
741    \end{equation}
742   
743    \noindent The range of $\mathcal{D}_{\!AB}$ is $[0,1]$ and it is known that all such $0\leq x\leq1$ have an additive inverse element.  This allows us to write 
744    \begin{equation}
745    0=\cfrac{AX_2}{AB}-\cfrac{AX_1}{AB}=\cfrac{AX_2-AX_1}{AB} \quad\iff\quad AX_2=AX_1~~.\nonumber
746    \end{equation}
747   
748    \noindent Axiom \ref{def:2points} gives $AX=AY$ if and only if $X=Y$ so the implication $X_1=X_2$ contradicts the assumed condition $X_1\neq X_2$.  The geometric fractional distance function $\mathcal{D}_{\!AB}(AX)$ is injective on all real line segments.
749\end{proof}
750 
751\begin{rem}
752    In Theorem \ref{thm:injjj}, we have not considered specifically the case in which $AB$ is a line segment of infinite length.  There are many numbers $x_1$ and $x_2$ such that zero being equal to their difference divided by infinity does not imply that $x_1=x_2$, \textit{e.g.}:  
753    \begin{equation}\label{eq:r344nnmm}
754    0=\cfrac{5-3}{\infty}\quad\centernot\iff\quad 5=3~~.
755    \end{equation}
756   
757   
758    \noindent  However, $\mathcal{D}_{\!AB}(AX)$ does not have numbers in its domain.  The fraction in Equation (\ref{eq:r344nnmm}) can never appear when computing $\frac{AX}{AB}$ because $\mathcal{D}_{\!AB}(AX)$ takes line segments or simply the point $A$ (written as $AA$ in abused line segment notation.)  
759   
760    To be clear, simplifying the expression $\mathcal{D}_{\!AB}(AX)$ in the general case requires some supplemental constraint like $AB=cAX$ for some scalar $c$.  With a such a constraint, and by way of Axiom \ref{def:li5t3f5y5y5y2}, we may evaluate the quotient as
761    \begin{equation*}
762    \cfrac{AX}{AB}=\cfrac{cAB}{AB}=c~~.
763    \end{equation*}
764   
765    \noindent Without such auxiliary constraints, we have no general method for the evaluation of the quotient.  Theorem \ref{thm:injjj} holds, however, because numbers such as the $\infty$ in the denominator of Equation (\ref{eq:r344nnmm}) will be used only to compute $\mathcal{D}^\dagger_{\!AB}(AX)$ when we introduce the norm $\|AX\|$.  The main feature distinguishing the algebraic FDF $\mathcal{D}^\dagger_{\!AB}$ from the geometric FDF $\mathcal{D}_{\!AB}$ is that the former allows us to compute the quotient in the general case with no requisite auxiliary constraints.  Therefore, we might write $\mathcal{D}^\dagger_{\!AB}(AX;x)$ to show that is is a function of $AX$ \textit{and} a chart $x$ on $AB$ but we will not write that explicitly.  In the absence of words to the contrary and if $AB$ is a real line segment, then it should be assumed  that the chart is the standard Euclidean coordinate.
766\end{rem}
767 
768\begin{thm}\label{thm:surjjj}
769    The geometric fractional distance function $\mathcal{D}_{\!AB}$ is surjective (onto) on all real line segments.
770\end{thm}
771 
772\begin{proof}
773    Given the range $R=[0,1]$ proven in Theorem \ref{thm:35353g0}, proof follows from the notion of geometric fractional distance.
774\end{proof}
775 
776\begin{rem}
777    Now that we have shown a few of the elementary properties of the geometric FDF, we will continue to do so and also examine the similar behaviors of the algebraic FDFs of the first and second kinds.
778\end{rem}
779 
780\begin{conj}\label{conj:dv2324}
781    The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ is injective (one-to-one) on all real line segments.  (This is proven in Theorem \ref{thm:injjj2}.)
782\end{conj}
783 
784 
785\begin{thm}\label{thm:injjj2222}
786    The algebraic fractional distance function of the second kind $\mathcal{D}''_{\!AB}$ is not injective (one-to-one) on all real line segments.
787\end{thm}
788 
789 
790\begin{proof}
791    Recall that Definition \ref{def:algfracdisq1} gives $\mathcal{D}''_{\!AB}:[a,b]\to [0,1]$ as   
792    \begin{equation}
793    \mathcal{D}''_{\!AB}(AX)=\begin{cases}
794    ~~~~~1\qquad&\text{for}\quad X=B\\[8pt]
795    \cfrac{\text{len}[a,x]}{\text{len}[a,b]}\,~~&\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
796    ~~~~~0 \qquad&\text{for}\quad X=A \end{cases}~~.\nonumber
797    \end{equation}
798   
799    \noindent Injectivity requires that
800    \begin{equation}
801    \mathcal{D}''_{\!AB}(AX)=\mathcal{D}''_{\!AB}(AY)\quad\iff\quad [a,x]=[a,y]\quad\iff\quad x=y~~.\nonumber
802    \end{equation}
803   
804    \noindent Let $n,m\in\mathbb{N}$ be such that $n\neq m$, and also such that $n\in\mathscr{N}\equiv N$ and $m\in\mathscr{M}\equiv M$.  We have  
805    \begin{equation}
806    \mathcal{D}''_{\!\mathbf{AB}}(AN)=\cfrac{\text{len}[0,n]}{\text{len}[0,\infty]}=0~~,\quad\text{and}\qquad \mathcal{D}''_{\!\mathbf{AB}}(AM)=\cfrac{\text{len}[0,m]}{\text{len}[0,\infty]}=0~~.\nonumber
807    \end{equation}
808   
809    \noindent   Therefore, the algebraic FDF of the second kind is not injective on all real line segments because  
810    \begin{equation}
811    \mathcal{D}''_{\!AB}(AN)=\mathcal{D}''_{\!AB}(AM)\quad\centernot\iff\quad n=m~~.\nonumber
812    \end{equation}
813\end{proof}
814 
815 
816\begin{rem}
817    At this point, we can rule out $\mathcal{D}''_{\!AB}$ as the definition of $\mathcal{D}^\dagger_{\!AB}$ because the geometric FDF $\mathcal{D}_{\!AB}$ which constrains $\mathcal{D}^\dagger_{\!AB}$ is one-to-one.  If $\mathcal{D}_{\!AB}$ is one-to-one on all real line segments, then so is $\mathcal{D}^\dagger_{\!AB}$.  Carefully note that the domain of the algebraic FDF of the first kind is line segments rather than algebraic intervals.  We have   
818    \begin{equation}
819    \mathcal{D}'_{\!AB}(AX):AB\to[0,1] ~~,\quad\text{and}\qquad\mathcal{D}''_{\!AB}(AX):[a,b]\to[0,1]  ~~. \nonumber
820    \end{equation}
821   
822    \noindent Taking for granted that we will prove the injectivity of $\mathcal{D}'_{\!AB}$ in Theorem \ref{thm:injjj2}, this distinction of domain---$AB$ versus $[a,b]$---will prohibit the breakdown in the one-to-one property when a point $X\in AB$ can have many different numbers in its algebraic representation.  An assumption that the domain of the algebraic FDF is an algebraic interval $[a,b]$ is likely a root cause of \textbf{\textit{much pathology in modern analysis}}.
823\end{rem}
824 
825 
826 
827 
828 
829\begin{thm}
830    The geometric fractional distance function $\mathcal{D}_{\!AB}$ is continuous everywhere on the domain $\mathbf{AB}$.
831\end{thm}
832 
833\begin{proof}
834    To prove that $\mathcal{D}_{\!AB}$ is continuous on $\mathbf{AB}\equiv[0,\infty]$, it will suffice to show that $\mathcal{D}_{\!\mathbf{AB}}$ is continuous at the endpoints and an interior point.
835   
836   
837    $~$
838   
839    \noindent $\bullet$ (\textit{Interior point})  A function $f(x)$ is continuous at an interior point $x_0$ of its domain $[a,b]$ if and only if
840    \begin{equation}
841    \lim\limits_{x\to x_0}f(x)=f(x_0)~~.\nonumber
842    \end{equation}
843   
844    \noindent In terms of the geometric FDF, the statement that $\mathcal{D}_{\!\mathbf{AB}}$ is continuous at an interior point $X'\in\mathbf{AB}$ becomes
845    \begin{equation}
846    \lim\limits_{X\to X_0}\mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{D}_{\!\mathbf{AB}}(AX_0)~~.\nonumber
847    \end{equation}
848   
849    \noindent Obviously, $\mathcal{D}_{\!\mathbf{AB}}$ satisfies the definition of continuity on the interior of $\mathbf{AB}$.
850   
851    $~$
852   
853    \noindent  $\bullet$ (\textit{Endpoint A}) A function $f(x)$ is continuous at the endpoint $a$ of its domain $[a,b]$ if and only if
854    \begin{equation}
855    \lim\limits_{x\to a^+}f(x)=f(a)~~.\nonumber
856    \end{equation}
857   
858    \noindent We conform to this definition of continuity with 
859    \begin{equation}
860    \lim\limits_{X\to A^+}\mathcal{D}_{\!\mathbf{AB}}(AX)=\lim\limits_{X\to A^+}\cfrac{AX}{\mathbf{AB}}=\cfrac{AA}{\mathbf{AB}}=\mathcal{D}_{\!\mathbf{AB}}(AA)~~.\nonumber
861    \end{equation}
862   
863   
864    $~$
865   
866    \noindent   $\bullet$ (\textit{Endpoint B})  A function $f(x)$ is continuous at the endpoint $b$ of its domain $[a,b]$ if and only if  
867    \begin{equation}
868    \lim\limits_{x\to b^-}f(x)=f(b)~~.\nonumber
869    \end{equation}
870   
871    \noindent We conform to this definition with   
872    \begin{equation}
873    \lim\limits_{X\to B^-}\mathcal{D}_{\!\mathbf{AB}}(AX)=\lim\limits_{X\to B^-}\cfrac{AX}{\mathbf{AB}}=\cfrac{\mathbf{AB}}{\mathbf{AB}}=\mathcal{D}_{\!\mathbf{AB}}(\mathbf{AB})~~.\nonumber
874    \end{equation}
875   
876    \noindent The geometric FDF is continuous everywhere on its domain.
877\end{proof}
878 
879 
880\begin{thm}\label{thm:algfracdisnotcont}
881    The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ is not continuous everywhere on the domain $\mathbf{AB}$.
882\end{thm}
883 
884 
885\begin{proof}
886    A function $f(x)$ with domain $x\in[a,b]$ is continuous at $b$ if
887    \begin{equation}
888    \lim\limits_{x\to b^-}f(x) =f(b)~~,  \nonumber
889    \end{equation}
890   
891   
892    \noindent In terms of $\mathcal{D}'_{\!AB}$, the statement that $\mathcal{D}'_{\!\mathbf{AB}}$ is continuous at $B$ becomes
893    \begin{equation}
894    \lim\limits_{X\to B}\mathcal{D}'_{\!\mathbf{AB}}(AX)=\mathcal{D}'_{\!\mathbf{AB}}(\mathbf{AB})=1~~.\nonumber
895    \end{equation}
896   
897    \noindent Evaluation yields
898    \begin{equation}
899    \lim\limits_{X\to B}\mathcal{D}'_{\!\mathbf{AB}}(AX) =  \lim\limits_{x\to \infty}\cfrac{\text{len}[0,x]}{\text{len}[0,\infty]}     =\lim\limits_{x\to \infty}x\cfrac{1}{\infty} =\lim\limits_{x\to \infty} 0\neq1=\mathcal{D}'_{\!\mathbf{AB}}(AB)~~.\nonumber
900    \end{equation}
901   
902    \noindent The algebraic FDF of the first kind is not continuous everywhere on all real line segments.  
903\end{proof}
904 
905\begin{rem}\label{rem:kdvjuur84}
906    In Theorem \ref{thm:algfracdisnotcont}, we have shown that the limit approaches zero rather than the unit value required for $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AB)$ to agree with $\mathcal{D}_{\!\mathbf{AB}}(AB)$.  However, we may also write this limit as  
907    \begin{equation}
908    \lim\limits_{x\to \infty}\cfrac{1}{\infty} x= \lim\limits_{\substack{x\to \infty\\y\to \infty}}\cfrac{x}{y}= \lim\limits_{y\to \infty}\infty\cfrac{1}{y}= \cfrac{\infty}{\infty}=\text{undefined}~~.\nonumber
909    \end{equation}
910   
911    \noindent  Perhaps, then, it would be better to write simply   
912    \begin{equation}
913    \lim\limits_{x\to \infty}\frac{x}{\infty} = \frac{\infty}{\infty} = \text{undefined}\neq 1~~.\nonumber
914    \end{equation}
915   
916    \noindent  In any case, we have shown that an elementary evaluation does not produce the correct limit at infinity.  Therefore, we should also examine the Cauchy definition of the limit relying on the $\varepsilon$--$\delta$ formalism.  
917\end{rem}
918 
919 
920\begin{thm}\label{thm:algfracdisnotcont3}
921    The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ does not converge to a Cauchy limit at infinity.
922\end{thm}
923 
924 
925\begin{proof}
926    According to the Cauchy definition of the limit of $f:D\to R$ at infinity, we say that  
927    \begin{equation}
928    \lim\limits_{x\to \infty} f(x) = l~~,\nonumber
929    \end{equation}
930   
931    \noindent if and only if   
932    \begin{equation}
933    \forall\varepsilon>0\quad\exists\delta>0\quad\text{s.t}\quad\forall x\in D~~,\nonumber
934    \end{equation}
935   
936    \noindent we have  
937    \begin{equation}
938    0<|x-\infty|<\delta\quad\implies\quad|f(x)-l|<\varepsilon~~.\nonumber
939    \end{equation}
940   
941    \noindent There is no $\delta>\infty$ so $\mathcal{D}'_{\!AB}(AX)$ fails the Cauchy criterion for convergence to a limit at infinity.  
942\end{proof}
943 
944\begin{rem}
945    In general, the above Cauchy definition of a limit fails for any limit at infinity because there is never a $\delta$ greater than infinity.  Usually this issue is worked around with the metric space definition of a limit at infinity but it is a \textbf{\textit{main result of this paper}} that we will develop a technique for taking a limit at infinity with the normal Cauchy prescription.  This result appears in Section \ref{sec:cont2}.
946\end{rem}
947 
948 
949\begin{rem}
950    The algebraic FDF  $\mathcal{D}^\dagger_{\!AB}$ exists by definition.  It is a function which has every behavior of the geometric FDF $\mathcal{D}_{\!AB}$ and also adds the ability to compute numerical ratios between the lengths of any two real line segments.  Numbers being generally within the domain of algebra, the geometric FDF returns a fraction that we have no general way to simplify.  Since it is hard to conceive of an irreducible analytic form for the algebraic FDF other than $\mathcal{D}'_{\!AB}$ and $\mathcal{D}''_{\!AB}$, it is somewhat paradoxical that neither of them replicate the global behavior of the algebraic FDF $\mathcal{D}^\dagger_{\!AB}$.  After developing some more material, we will show in Section \ref{sec:cont2} that $\mathcal{D}^\dagger_{\!AB}$ is $\mathcal{D}'_{\!AB}$ after all.  We will prevent an unwarranted assumption about infinity from sneakily propagating into the present analysis.  Then we will fix the discontinuity of $\mathcal{D}'_{\!AB}$ which we have demonstrated in Theorems \ref{thm:algfracdisnotcont} and \ref{thm:algfracdisnotcont3}.
951\end{rem}
952 
953 
954\begin{thm}\label{thm:4f134134t34t}
955    If $x$ is a real number in the algebraic representations of both $X\in AB$ and $Y\in AB$, then $X=Y$.
956\end{thm}
957 
958\begin{proof}
959    If $X\neq Y$, then     
960    \begin{equation}
961    \mathcal{D}^\dagger_{\!AB}(AX)\neq\mathcal{D}^\dagger_{\!AB}(AY)~~.\nonumber
962    \end{equation}
963   
964    \noindent If $x\in X$ and $x\in Y$, then it is possible to make cuts at $X$ and $Y$ such that  
965    \begin{equation}
966    \mathcal{D}^\dagger_{\!AB}(AX)=\cfrac{\text{len}[a,x]}{\text{len}[a,b]}=\mathcal{D}^\dagger_{\!AB}(AY)  ~~.\nonumber
967    \end{equation}
968   
969    \noindent This contradiction requires $X=Y$.   
970\end{proof}
971 
972 
973 
974\subsection{Comparison of Real and Natural Numbers}\label{sec:RN}
975 
976The main result of this section is to prove via analysis of FDFs that there exist real numbers greater than any natural number.  Consequently, $\mathbb{R}_\infty=\mathbb{R}\setminus\mathbb{R}_0$ cannot be the empty set.
977 
978 
979\begin{defin}\label{def:3533553}
980    Every interval has a number at its center.  The number at the center of an interval $[a,b]$ is defined as the average of $a$ and $b$.  This holds for all intervals $[a,b)$, $(a,b]$, and $(a,b)$.
981\end{defin}
982 
983 
984\begin{thm}\label{thm:halfway}
985    There exists a unique real number halfway between zero and infinity.
986\end{thm}
987 
988\begin{proof}
989    By Theorem \ref{thm:onemid} and by Definition \ref{def:fet7}, there exists one midpoint $C$ of every line segment $AB$ such that    
990    \begin{equation}
991    \mathcal{D}_{\!AB}(AC)=0.5~~.\nonumber
992    \end{equation}
993   
994    \noindent Recalling that we have defined $\mathcal{D}_{\!AB}(AX)=\mathcal{D}^\dagger_{\!AB}(AX)$ for all $X\in AB$, and recalling that $\mathbf{AB}\equiv[0,\infty]$, it follows that
995    \begin{equation}
996    \mathcal{D}^\dagger_{\!\mathbf{AB}}(AC)=0.5~~.\nonumber
997    \end{equation}
998   
999    \noindent Using $C\equiv\mathscr{C}=[c_1,c_2]$, Axiom \ref{def:5y5dddy2} and Definition \ref{def:XrepR} require
1000    \begin{equation}
1001    \mathbf{AB}=AC+CB\quad\iff\quad[0,\infty]=[0,c_1)\cup\mathscr{C}\cup(c_2,\infty]~~.\nonumber
1002    \end{equation}
1003   
1004    \noindent It follows that  
1005    \begin{equation}
1006    \mathscr{C}\subset\mathbb{R}~~.\nonumber
1007    \end{equation}
1008   
1009    \noindent Every possible number that can be in the algebraic representation of the point $C$ is a real number.  If $c_1=c_2=c$, then $c\in\mathbb{R}$ is the unique real number halfway between zero and infinity.  If $c_1\neq c_2$, then, by Definition \ref{def:3533553}, the average of $c_1$ and $c_2$ is the unique real number halfway between zero and infinity.
1010\end{proof}
1011 
1012 
1013\begin{rem}
1014    How can $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AC)=0.5$ when Definition \ref{def:algfracdis} gives
1015    \begin{equation}
1016    \mathcal{D}'_{\!\mathbf{AB}}(AC)=\frac{\text{len}[0,c]}{\infty}~~?\nonumber
1017    \end{equation}
1018   
1019    \noindent The prevailing assumption about infinity is
1020    \begin{equation}\label{imp:f24f4f}
1021    x\in\mathbb{R}\quad \implies \quad\frac{x}{\infty}=0~~.
1022    \end{equation}
1023   
1024    \noindent If Equation (\ref{imp:f24f4f}) is true, then either (\textit{a}) there exists a line segment without a midpoint, or (\textit{b}) the geometric and algebraic fractional distance functions do not agree for every $X$ in an arbitrary $AB$.  
1025   
1026    Every line segment does have a midpoint (Theorem \ref{thm:onemid}) and our fractional distance functions are defined to always agree (Definition \ref{def:iuuuttgrr488}).  Therefore, Equation (\ref{imp:f24f4f}), which is a statement dependent on the assumed properties of $\infty$, must be reformulated.  In Section \ref{sec:Rneighb}, we will define notation for subsets of $\mathbb{R}$ consisting of all numbers having fractional distance $\mathcal{X}$ with respect to $\mathbf{AB}$.  The sets will be labeled $\mathbb{R}_\aleph^\mathcal{X}$ most generally with $0<\mathcal{X}<1$ but it will follow that $\mathbb{R}^0_\aleph$ is the set of all real numbers having zero fractional distance with respect to $\mathbf{AB}$.  We know that $\mathbb{R}_0\subset\mathbb{R}_\aleph^0$ but it shall remain to be determined whether or not there are real numbers greater than any natural number yet still having zero fractional distance with respect to $\mathbf{AB}$.  In Section \ref{sec:bbbhh}, we will closely examine whether or not such numbers ought to exist.  
1027   
1028    While we will postpone the definition of $\mathbb{R}^\mathcal{X}_\aleph$ to Section \ref{sec:Rneighb}, and while the formal construction of $\mathbb{R}_\aleph^\mathcal{X}$ by equivalence classes of Cauchy sequences will not appear until Section \ref{sec:fconst}, here we will go ahead and answer the question, ``How can $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AC)=0.5$ when Definition \ref{def:algfracdis} gives
1029    \begin{equation}
1030    \mathcal{D}'_{\!\mathbf{AB}}(AC)=\frac{\text{len}[0,c]}{\infty}~~?\text{''}\nonumber
1031    \end{equation}
1032   
1033    \noindent The answer is that Equation (\ref{imp:f24f4f}) must be reformulated as    
1034    \begin{equation*}
1035    x\in\mathbb{R}^0_\aleph  \quad \implies \quad     \cfrac{x}{\infty}=0 ~~.
1036    \end{equation*}
1037   
1038    \noindent Regarding Theorem \ref{thm:halfway} and the present question which follows, the real numbers in the algebraic representation of the geometric midpoint of $\mathbf{AB}$ shall be
1039    \begin{equation}
1040    x\in\mathbb{R}^{0.5}_\aleph\quad \implies \quad\cfrac{x}{\infty}=0.5~~.\nonumber
1041    \end{equation}
1042   
1043    \noindent  In addition to motivating the soon-to-be-defined $\mathbb{R}^\mathcal{X}_\aleph$ notation, the present remark illustrates the reasoning behind allowing geometric points to be represented as entire intervals $X\equiv\mathscr{X}$.  The reason is that many real numbers divided by infinity give zero but only the geometric left endpoint of $\mathbf{AB}$ will have vanishing fractional distance.  For instance, if $n$ is a natural number having zero fractional magnitude with respect to infinity, then
1044    \begin{equation*}
1045    \cfrac{x+n}{\infty}=\cfrac{x}{\infty}+\cfrac{n}{\infty}=0.5+0~~.
1046    \end{equation*}
1047   
1048    \noindent Obviously, $x\in\mathbb{R}^{0.5}_\aleph$ is not a unique number though the midpoint $C$ is a unique point.
1049\end{rem}
1050 
1051\begin{defin}
1052    If $\mathbb{R}^\mathcal{X}_\aleph$ is the set of all numbers whose fractional distance with respect to $\mathbf{AB}$ is $\mathcal{X}$, and if $0<\mathcal{X}<1$, then $\aleph_\mathcal{X}$ is the number in the center of the interval $\mathbb{R}^\mathcal{X}_\aleph=(a,b)$ in the sense that for every $\aleph_\mathcal{X}+n\in\mathbb{R}_\aleph^\mathcal{X}$ there exists a $\aleph_\mathcal{X}-n\in\mathbb{R}_\aleph^\mathcal{X}$.
1053\end{defin}
1054 
1055\begin{rem}
1056    The reader is invited to recall that Euler often employed the letter $i$ to refer to an infinitely large integer.  Euler made use of the number $\tfrac{i}{2}$ for proofs in his most seminal works \cite{EULER1748,EULER1988,OLDINF}.  Therefore, we are certainly introducing nothing new with the $\aleph_{\mathcal{X}}$ notation because $\tfrac{i}{2}\sim\aleph_{0.5}$.  
1057\end{rem}
1058 
1059\begin{mainthm} \label{thm:ef2424t24cc}
1060    Some elements of $\mathbb{R}$ are greater than every element of $\mathbb{N}$.
1061\end{mainthm}
1062 
1063\begin{proof}
1064    Let $\mathbf{AB}$ have a midpoint $C$ so that $\mathcal{D}_{\!\mathbf{AB}}(AC)=0.5$.  Then every real number $c\in[c_1,c_2]\equiv C$ is greater than any $n\in\mathbb{N}$ because $\frac{n}{\infty}=0$ implies $n\in\mathscr{A}\equiv A$ through the definition $\mathcal{D}_{\!\mathbf{AB}}(AA)=0$.  $\mathcal{D}_{\!\mathbf{AB}}$ is one-to-one so by Axiom \ref{def:order} giving for $x\in X$ and $y\in Y$
1065    \begin{equation}
1066    \mathcal{D}_{\!AB}(AX)>\mathcal{D}_{\!AB}(AY)\quad\implies\quad x>y ~~,\nonumber
1067    \end{equation}
1068   
1069    \noindent we find that every $c\in\mathscr{C}\subset\mathbb{R}$ is greater than every $n\in\mathbb{N}$.  Generally, every $x\in\mathbb{R}^\mathcal{X}_\aleph$ is greater than any natural number whenever $\mathcal{X}>0$.
1070\end{proof}
1071 
1072\begin{cor}
1073    $\mathbb{R}_\infty=\mathbb{R}\setminus\mathbb{R}_0$ is not the empty set.
1074\end{cor}
1075 
1076\begin{proof}
1077    Definition \ref{def:orig} defines $\mathbb{R}_0$ as the subset of $\mathbb{R}$ whose elements are less than some element of $\mathbb{N}$.  We have proven in Main Theorem \ref{thm:ef2424t24cc} that some elements of $\mathbb{R}$ are not in $\mathbb{R}_0$.  It follows that  
1078    \begin{equation}
1079    \mathbb{R}_\infty\neq\varnothing~~,\quad\text{because}\qquad \mathbb{R}_\infty= \mathbb{R}\setminus\mathbb{R}_0~~.\nonumber
1080    \end{equation}
1081\end{proof}  
1082 
1083 
1084\subsection{Comparison of Cuts in Lines and Points in Line Segments}\label{sec:3322442}
1085 
1086 
1087In this section, we will make clarifications regarding the cases in which an interior point of a line segment can or cannot be identified with a unique real number. Namely, we distinguish cases in which $X\equiv x$ and $X\equiv\mathscr{X}=[x_1,x_2]$ with $x_1\neq x_2$.
1088 
1089\begin{thm}\label{thm:finlinseggg}
1090    If $AB$ is a real line segment of finite length $L\in\mathbb{R}_0$, then every point $X\in AB$ has a unique algebraic representation as one and only one real number.
1091\end{thm}
1092 
1093\begin{proof}
1094    Let $a,b\in\mathbb{R}_0$ and $AB\equiv[a,b]$.  The algebraic FDF $\mathcal{D}^\dagger_{\!AB}$ is defined to behave exactly as the geometric FDF $\mathcal{D}_{\!AB}$.  Therefore, $\mathcal{D}^\dagger_{\!AB}$ must be one-to-one (injective.)  By Definition \ref{def:XrepR}, every point in a real line segment has an algebraic representation
1095    \begin{equation}
1096    X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
1097    \end{equation}
1098   
1099    \noindent Therefore, the present theorem will be proven if we show that $x_1=x_2$ for all $X\in AB$ with $L\in\mathbb{R}_0$.  To initiate proof by contradiction, assume $x_1,x_2\in\mathbb{R}_0$ and that $x_1\neq x_2$.  (The validity of this condition follows from $L\in\mathbb{R}_0$.)  Then
1100   
1101    \begin{equation}
1102    \min[\mathcal{D}^\dagger_{\!AB}(AX)]=\cfrac{\text{len}[a,x_1]}{\text{len}[a,b]} =\frac{x_1-a}{b-a}~~,\nonumber
1103    \end{equation}
1104   
1105    \noindent and
1106    \begin{equation} \max[\mathcal{D}^\dagger_{\!AB}(AX)]=\cfrac{\text{len}[a,x_2]}{\text{len}[a,b]} =\frac{x_2-a}{b-a}~~.\nonumber
1107    \end{equation}
1108   
1109    \noindent The one-to-one property of $\mathcal{D}^\dagger_{\!AB}$ requires that
1110    \begin{equation} \frac{x_1-a}{b-a}=\frac{x_2-a}{b-a}\quad\iff\quad x_1=x_2~~.\nonumber
1111    \end{equation}
1112   
1113    \noindent This contradicts the assumption $x_1\neq x_2$.  The theorem is proven.
1114\end{proof}
1115 
1116\begin{thm}\label{thm:finlinseggg2}
1117    If $AB$ is a real line segment of infinite length $L=\infty$, then no point $X\in AB$ has a unique algebraic representation as one and only one real number.
1118\end{thm}
1119 
1120\begin{proof}
1121    By Definition \ref{def:XrepR}, every point in a line segment has an algebraic representation
1122    \begin{equation}
1123    X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
1124    \end{equation}
1125   
1126    \noindent It follows that
1127    \begin{equation}
1128    \min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\cfrac{\text{len}[0,x_1]}{\text{len}[0,\infty]} =\frac{x_1}{\infty}~~,\nonumber
1129    \end{equation}
1130     
1131   
1132    \noindent Now suppose $x_0\in\mathbb{R}_0^+$ (throughout this paper, the superscript ``+'' indicates the positive-definite subset.), and $z=x_1+x_0$ so that $z> x_1$.  Then
1133    \begin{equation}
1134    \cfrac{\text{len}[0,z]}{\text{len}[0,\infty]} =\frac{z}{\infty}=\frac{x_1+x_0}{\infty} =\frac{x_1}{\infty}=\min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]~~.\nonumber
1135    \end{equation}
1136   
1137    \noindent Invoking the single-valuedness of bijective functions, we find that
1138    \begin{equation*}
1139    \min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\max[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\frac{x_2}{\infty}\quad\implies\quad x_1<z\leq x_2~~.
1140    \end{equation*}
1141   
1142    \noindent Therefore $x_1\neq x_2$ and the theorem is proven.
1143\end{proof}
1144 
1145\begin{exa}\label{ex:333}
1146    This example illustrates some of the underlying machinations associated with the many-to-one relationship between numbers and points in an infinitely long line segment.  If we separate an endpoint from a closed algebraic interval, then we may write
1147    \begin{equation}
1148    [a,b]=\{a\}\cup(a,b]~~.\nonumber
1149    \end{equation}
1150   
1151    \noindent To separate an endpoint from a line segment we write
1152    \begin{equation}
1153    AB=A+AB~~.\nonumber
1154    \end{equation}
1155   
1156    \noindent If $A$ has an algebraic representation $\mathscr{A}$ such that $\text{len}(\mathscr{A})>0$, then the only way that we can leave the length of $AB$ unchanged after removing $A$ is for $AB$ to have infinite length.  Given $\text{len}(\mathscr{A})>0$, observe that
1157    \begin{equation}
1158    \|AB\|-\text{len}\,\mathscr{A}=\|AB\|\quad\iff\quad \|AB\|=\infty~~.\nonumber
1159    \end{equation}
1160\end{exa}
1161 
1162\begin{rem}\label{rem:f32232323}
1163    Theorems \ref{thm:finlinseggg} and \ref{thm:finlinseggg2} do not cover all cases of $\text{len}(AB)=L$.  For instance, four course bins of $L$ are
1164    \begin{itemize}
1165        \item $L\in\mathbb{R}_0$
1166        \item $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$ ($L$ larger than any $n\in\mathbb{N}$ yet not so large that $\frac{L}{\infty}\neq0$.)
1167        \item  $L\in\mathbb{R}_\infty\setminus\mathbb{R}^0_\aleph$ (which is also written $L\in\mathbb{R}_\aleph^\mathcal{X}\cup\mathbb{R}^1_\aleph$ when $\mathbb{R}^\mathcal{X}_\aleph$ is understood to be $0<\mathcal{X}<1$, as in Section \ref{sec:Rneighb})
1168        \item $L=\infty$~~.
1169    \end{itemize}
1170 
1171    \noindent  We have not considered the two intermediate cases of finite $L$.  The lesser case is finite $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$.  Since we have not yet introduced numbers through which to describe the lesser case, and we will not decide $\mathbb{R}^0_\aleph\setminus\mathbb{R}_0=\varnothing$ until Section \ref{sec:bbbhh}, we cannot at this time prove the result regarding the multi-valuedness of points in line segments having $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$.  The limit of the third case as $L\in\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}^1_\aleph$ is proven to be many-to-one in Theorem \ref{thm:finlinseggg3}.
1172\end{rem}
1173 
1174 
1175 
1176 
1177 
1178\section{The Neighborhood of Infinity}\label{sec:rg8hr8y8hh}
1179 
1180 
1181\subsection{Intermediate Neighborhoods of Infinity}\label{sec:Rneighb}
1182 
1183In this section, we will develop notation useful for describing real numbers whose fractional magnitude with respect to infinity is greater than zero.
1184 
1185\begin{defin}\label{def:hh4hh4h4h4}
1186    The number $\aleph_\mathcal{X}$ is defined to have the property
1187    \begin{equation*}
1188    \frac{\aleph_{\mathcal{X}}}{\infty}=\mathcal{X}~~.  
1189    \end{equation*}
1190   
1191    \noindent  Equivalently, if $\aleph_\mathcal{X}\in \mathscr{X}\equiv X$, then
1192    \begin{equation*}
1193    \mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}~~.  
1194    \end{equation*}
1195\end{defin}
1196 
1197\begin{rem}\label{rem:kkkekk33}
1198    We have shown in Theorem \ref{thm:finlinseggg2} that there are many real numbers in the algebraic representation of $X\in\mathbf{AB}$.  When $X$ is not an endpoint of $\mathbf{AB}$, $\aleph_\mathcal{X}$ can be thought of the as the number in the center of the interval $(x_1,x_2)=\mathscr{X}\equiv X$.  Definition  \ref{def:3533553} defines the number in the center of $\mathscr{X}$ as the average of $x_1$ and $x_2$, but here we have no way to determine least and greatest numbers in the algebraic representation of $X$.  It is useful, however, to think of $\aleph_\mathcal{X}$ as the number in the center of $\mathscr{X}$ in the sense that for every $\aleph_{\mathcal{X}}+|b|\in\mathscr{X}$ there exists a $\aleph_{\mathcal{X}}-|b|\in\mathscr{X}$.  For the special cases of $\aleph_0$ and $\aleph_1$, we should not think of them as being in the centers of the intervals $\mathscr{A}\equiv A$ and $\mathscr{B}\equiv B$.  Instead, $\aleph_0$ is the least number in $\mathscr{A}\equiv A\in\mathbf{AB}$ and $\aleph_1$ is the greatest number in $\mathscr{B}\equiv B\in\mathbf{AB}$.
1199\end{rem}
1200 
1201 
1202\begin{defin}\label{def:gtg34535335}
1203    For $0<\mathcal{X}<1 $, $\mathbb{R}^\mathcal{X}_\aleph$ is a subset of positive real numbers $\mathbb{R}^+$  such that
1204    \begin{equation}
1205    \mathbb{R}^\mathcal{X}_\aleph=\big\{ \aleph_\mathcal{X}+ b~\big|~ |b|\in  A\in\mathbf{AB},~\mathcal{D}_{\!\mathbf{AB}}(AA)=0 \big\} ~~.\nonumber
1206    \end{equation}
1207 
1208    \noindent The set $\mathbb{R}^{\mathcal{X}}_\aleph$ is called the whole neighborhood of $\aleph_{\mathcal{X}}$.  The set $\{\mathbb{R}_\aleph^\mathcal{X}\}$ of all $\mathbb{R}_\aleph^\mathcal{X}$, meaning the union of $\mathbb{R}^\mathcal{X}_\aleph$ for every $0<\mathcal{X}<1$, is called the set of all intermediate neighborhoods of $\mathbb{R}$.  We will also call $\mathbb{R}^\mathcal{X}_\aleph$ the neighborhood of numbers that are $100\times\mathcal{X}$\% of the way down the real number line.  (These conventions ignore the negative branch of $\mathbb{R}$.)
1209\end{defin}
1210 
1211\begin{defin}\label{def:ffmdmmdd3md}
1212    It will also be useful to define a set $\mathbb{R}^\mathcal{X}_0\subset \mathbb{R}^\mathcal{X}_\aleph$ such that $0<\mathcal{X}<1$ and
1213    \begin{equation}
1214    \mathbb{R}^\mathcal{X}_0=\big\{ \aleph_\mathcal{X}+ b~\big| ~ b\in  \mathbb{R}_0    \big\} ~~.\nonumber
1215    \end{equation}
1216   
1217    \noindent The set $\mathbb{R}^{\mathcal{X}}_0$ is called the natural neighborhood of $\aleph_{\mathcal{X}}$ because here we have constrained $b$ to be less than some $n\in\mathbb{N}$.   $\{\mathbb{R}^\mathcal{X}_0\}$ is the union of $\mathbb{R}_0^\mathcal{X}$ for every $0<\mathcal{X}<1$.
1218\end{defin}
1219 
1220 
1221 
1222\begin{defin}\label{def:15525252525}
1223    Every number of the form $x=\aleph_\mathcal{X}+b$ has a big part $\aleph_\mathcal{X}$ and a little part $b$.  It is understood that $b<\aleph_{\mathcal{X}}$ for any $\mathcal{X}>0$.  If $b\in\mathbb{R}_0$, then $b$ is also called the natural part of $x$.  We define notations
1224    \begin{equation*}
1225    \text{Big}(\aleph_\mathcal{X}+b)=\aleph_\mathcal{X}~~,\quad\text{and}\qquad \text{Lit}(\aleph_\mathcal{X}+b)=b~~.
1226    \end{equation*}
1227\end{defin}
1228 
1229\begin{rem}
1230    We have omitted from Definitions \ref{def:gtg34535335} and \ref{def:ffmdmmdd3md} the cases of $\mathcal{X}=0$ and $\mathcal{X}=1$ though they do follow more or less directly.  The main issue is that we must restrict the sign of $b$ to keep the elements of the set within the totally real interval $[0,\infty)\subset\mathbb{R}$.  For $\mathcal{X}=0$, the little part $b$ is non-negative and for $\mathcal{X}=1$ it must be negative.
1231   
1232    The difference between the natural neighborhoods $\mathbb{R}^\mathcal{X}_0$ and the whole neighborhoods $\mathbb{R}^\mathcal{X}_\aleph$ is that $b$ is not restricted to $\mathbb{R}_0$ in the latter.  In Definition \ref{def:ffmdmmdd3md}, we do not give the condition on $b$ in terms of the absolute value, as in Definition \ref{def:gtg34535335}, because $\mathbb{R}_0$ contains negative numbers while $b\in\mathbf{AB}\equiv[0,\infty]$ is strictly non-negative.  The main purpose in defining distinct sets $\{\mathbb{R}_0^\mathcal{X}\}$ and $\{\mathbb{R}_\aleph^\mathcal{X}\}$ is this: we know there exist numbers larger than any $b\in\mathbb{R}_0$ (Main Theorem \ref{thm:ef2424t24cc}) but we do not know that all such numbers have greater than zero fractional magnitude with respect to $\mathbf{AB}$.  We will revisit this issue in Section \ref{sec:bbbhh}.  In the meantime, we will be careful to treat $\mathbb{R}_0^\mathcal{X}$ and $\mathbb{R}_\aleph^\mathcal{X}$ as distinct sets which may or may not be equal.
1233\end{rem}
1234 
1235 
1236\begin{defin}
1237    The whole neighborhood of the origin is
1238    \begin{equation*}
1239    \mathbb{R}_\aleph^0=\big\{ x~\big|~x\in \mathscr{A}\equiv A\in\mathbf{AB} \big\}~~,
1240    \end{equation*}
1241   
1242    \noindent and the natural neighborhood of the origin is
1243    \begin{equation*}
1244    \mathbb{R}_0^0=\big\{ x~\big|~x\in \mathbb{R}_0,~x\geq0 \big\}~~,
1245    \end{equation*}
1246\end{defin}
1247 
1248\begin{rem}
1249    Note that $\mathbb{R}_0\not\subset\mathbb{R}_0^0\subset\mathbb{R}_\aleph^0$ because $\mathbb{R}_0$ contains positive and negative numbers, as per Definition \ref{def:orig}.
1250\end{rem}
1251 
1252 
1253 
1254 
1255\begin{defin}\label{def:defoig777}
1256    A real number $x$ is said to be in the neighborhood of the origin if and only if
1257    \begin{equation}
1258    x\in X~~,\quad\text{and}\qquad \mathcal{D}_{\!\mathbf{AB}}(AX)=0 ~~.\nonumber
1259    \end{equation}
1260   
1261    \noindent  All such numbers are said to be $x\in\mathbb{R}_\aleph^0$.   Every real number not in the neighborhood of the origin is said to in the neighborhood of infinity.  A positive real number $x$ is said to be in the neighborhood of infinity if and only if
1262    \begin{equation}
1263    x\in  X~~,\quad\text{and}\qquad \mathcal{D}_{\!\mathbf{AB}}(AX)\neq0 ~~.\nonumber
1264    \end{equation}
1265\end{defin}
1266 
1267 
1268 
1269\begin{rem}\label{rem:v3444}
1270    Definition \ref{def:Rinf} states that $\mathbb{R}_\infty=\mathbb{R}\setminus\mathbb{R}_0$.  Therefore, if $\mathbb{R}_\aleph^0\setminus\mathbb{R}_0^0\neq\varnothing$, meaning that there do exist real numbers greater than any natural number yet not great enough to have non-zero fractional distance with respect to $\mathbf{AB}$, then the set $\mathbb{R}_\infty$ will contain numbers in the neighborhood of the origin \textit{and} numbers in the neighborhood of infinity.  To avoid ambiguity, we will not use the symbol $\mathbb{R}_\infty$ and instead we will mostly use the detailed set enumeration scheme given in the present section.  With this scheme of distinct whole and natural neighborhoods, we have left room judiciously for numbers in the neighborhood of the origin which are larger than any natural number.  In other work \cite{RINF,ZEROSZZ}, we used the semantic convention that every number in the neighborhood of the origin is less than some natural number.  That meant $\mathbb{R}_0$ was the set of all real numbers in the neighborhood of the origin.  The present convention, however, is better suited to the fuller analysis presently given.  The reader should carefully note that the present neighborhood of the origin $\mathbb{R}^0_\aleph$ includes all numbers which have zero fractional distance along the real number line, even if some of those numbers are larger than any $n\in\mathbb{N}$.
1271\end{rem}
1272 
1273 
1274\begin{defin}\label{def:delta0x00}
1275    The $\delta$-neighborhood of a number $x\in\mathbb{R}$ is an interval $(x-\delta,x+\delta)$ or some closed or half-open permutation thereof.  While there is no inherent constraint on the magnitude of $\delta$, here we will take ``$\delta$-neighborhood'' to imply $\delta\in\mathbb{R}_0$.  We will use the convention that the Ball function defines an open $\delta$-neighborhood as
1276    \begin{equation*}
1277    \text{Ball}(x,\delta)=(x-\delta,x+\delta)~~.
1278    \end{equation*}
1279\end{defin}
1280 
1281\begin{defin}\label{def:delta000}
1282    The $\delta$-neighborhood of an interior point $X\in AB$ is a line segment $YZ$ where  
1283    \begin{equation}
1284    \big|\mathcal{D}_{\!AB}(AX)-\mathcal{D}_{\!AB}(AY)\big|=\big|\mathcal{D}_{\!AB}(AX)-\mathcal{D}_{\!AB}(AZ)\big|=\delta~~.\nonumber
1285    \end{equation}
1286\end{defin}
1287 
1288\begin{rem}
1289    Without regard to the $\delta$-neighborhood of any point or number, we have defined neighborhoods with the geometric FDF, as in Definition \ref{def:defoig777}.  If $\mathcal{D}_{\!\mathbf{AB}}(AX)=0$, then the numbers in the algebraic representation of $X$ are said to be in the neighborhood of the origin.  They are said to be in the neighborhood of infinity otherwise.  Neither of these neighborhoods, neither that of the origin nor that of infinity, are defined formally as $\delta$-neighborhoods though such a definition may be inferred.
1290\end{rem}
1291 
1292 
1293 
1294\begin{defin}\label{def:35e44e3}
1295    By Definition \ref{def:3533553}, every interval has a number at its center.  If $\mathbb{R}^\mathcal{X}_\aleph= (a,b)$, then the number at the center of $(a,b)$ is $\aleph_\mathcal{X}$, as in Remark \ref{rem:kkkekk33}.  (Carefully note that $\mathbb{R}^0_\aleph\not\subset\{\mathbb{R}^\mathcal{X}_\aleph\}$.)
1296\end{defin}
1297 
1298 
1299 
1300\begin{defin} \label{def:fh93ry983y9y222}
1301    An alternative definition for $\mathbb{R}^\mathcal{X}_\aleph$ valid in the neighborhood of infinity, meaning for $0<\mathcal{X}<1$, is 
1302    \begin{equation}
1303    \mathbb{R}^\mathcal{X}_\aleph=\big\{ \aleph_\mathcal{X}\pm b~\big|~  b\in\mathbb{R}^0_\aleph  \big\} ~~.\nonumber
1304    \end{equation}
1305     
1306    \noindent  This definition is totally equivalent to Definition \ref{def:gtg34535335}.
1307\end{defin}
1308 
1309 
1310\subsection{Equivalence Classes for Intermediate Natural Neighborhoods of Infinity}\label{sec:fconst}
1311 
1312 
1313Euclid's definition of $\mathbb{R}$ is inherently a geometric one based on measurement.  The purpose of Cantor's definition by Cauchy equivalence classes \cite{CANTOR,PUGH,BRUDIN,CAUCHYR} is to give an algebraic definition based on rationals.  In this section, we will append the algebraic Cauchy definition to the Euclidean definition given in Section \ref{sec:RN1}.  This totally algebraic hybrid construction will not unduly exclude the neighborhood of infinity from $\mathbb{R}$.  In its ordinary incarnation, the Cauchy definition of $\mathbb{R}$ contradicts the axiom that $\mathbb{R}=(-\infty,\infty)$ because it precludes the existence of numbers larger than any natural number.  We have shown that if every number in the interval $(-\infty,\infty)$ is to be a real number, then there must exist numbers such as $\aleph_{0.5}$ which are greater than any natural number.  In this section, we will modify the Cauchy definition so that it will support the underlying geometric construction and facilitate the algebraic construction of numbers in the neighborhood of infinity.  Here we will only construct the natural neighborhoods because the equality or inequality of $\mathbb{R}^\mathcal{X}_0$ and $\mathbb{R}^\mathcal{X}_\aleph$ is not treated until Section \ref{sec:bbbhh}.
1314 
1315 
1316\begin{defin}
1317    The rational numbers $\mathbb{Q}$ are an Archimedean number field satisfying all of the well-known field axioms given in Section \ref{sec:fieldAX}.
1318\end{defin}
1319 
1320\begin{defin}\label{def:CSq}
1321    A sequence $\{x_n\}$ is a Cauchy sequence if and only if
1322    \begin{equation*}
1323    \forall \delta\in\mathbb{Q}\quad\exists m,n,N\in\mathbb{N}\quad\text{s.t.}\quad m,n>N\quad\implies\quad\big|x_n-x_m\big|<\delta~~.
1324    \end{equation*}
1325\end{defin}
1326 
1327\begin{defin}
1328    We say a relation is an equivalence relation if and only if $S$ is a set, if every $x\in S$ is related to $x$ (reflexive), if for every $x,y\in S$ the relation of $x$ to $y$ implies the relation of $y$ to $x$ (symmetric), and if for every $x,y,z\in S$ the relation of $x$ to $y$ and the relation of $y$ to $z$ together imply the relation of $x$ to $z$ (transitive).  The equivalence class of $x\in S$, namely the set of all objects which are related to $x$ by an equivalence relation, is denoted $[x]$.  At times we will write $[x]=[\{x_n\}]$ or $[x]=[(x_n)]$ to emphasize that the equivalence relation is among Cauchy sequences where $\{x_n\}$ and $(x_n)$ have the same meaning.
1329\end{defin}
1330 
1331\begin{defin}
1332    $C_\mathbb{Q}$ is the set of all Cauchy sequences of rational numbers.
1333\end{defin}
1334 
1335\begin{rem}
1336    Usually the Cauchy construction of $\mathbb{R}$ is formulated as, ``Every $x\in\mathbb{R}$ is some Cauchy equivalence class $[x]\subset C_\mathbb{Q}$,'' but here we will take a slightly different approach.
1337\end{rem}
1338 
1339\begin{axio}\label{ax:constaxcjco}
1340    Every $x\in\mathbb{R}$ may be constructed algebraically as a Cartesian product of Cauchy equivalence classes of rational numbers, or as a partition of all such products.
1341\end{axio}
1342 
1343\begin{axio}\label{def:lkkd33kkdzz}
1344    Every $x\in\mathbb{R}_0\subset\mathbb{R}$ is a Cauchy equivalence class of rationals $x=[x]\subset C_\mathbb{Q}$ and also a Dedekind partition of $\mathbb{Q}$ in canonical form $x=(L,R)$.  This axiom grants that the reals are constructed by Cauchy equivalence classes or Dedekind partitions in the most canonical sense \textbf{\textit{if one takes the complementary axiom that every real number is less than some natural number}}.  We do not take that axiom so we specify $x\in\mathbb{R}_0$ as the object of relevance.
1345\end{axio}
1346 
1347 
1348\begin{rem}\label{rem:hgy88877}
1349    Cantor's Cauchy construction of $\mathbb{R}$, like the Dedekind construction, is said to be ``rigorous'' because it begins with the rationals $\mathbb{Q}$.  However, before one may assume the existence of $\mathbb{Q}$, one must define zero because $0\in\mathbb{Q}$ but $0\not\in\mathbb{N}$.  Therefore, to be rigorous, one simply may not assume $\mathbb{Q}$ as a consequence of $\mathbb{N}$.  To introduce zero, we will introduce the line segment $\mathbf{AB}$ and define zero as the least number in the algebraic representation of the geometric point $A$.  In other words, $0=\aleph_0$.  It is true that this present approach can be criticized as being ``not rigorous'' because we have assumed $\mathbf{AB}$ in the same way that others assumed $\mathbb{Q}$ but the present construction is ``more rigorous'' because it bumps that which is assumed down to a more primitive level, \textit{i.e.}: Euclid's principles of geometry \cite{EE}.
1350\end{rem}
1351 
1352 
1353\begin{defin}\label{def:i9696969a}
1354    The symbol $\hat0$ is an instance of the number zero with the instruction not to do any of zero's absorptive operations.  The absorptive operations of zero are
1355    \begin{equation*}
1356    0+x=x~~,\quad\text{and}\qquad x\cdot0=0~~.
1357    \end{equation*}
1358   
1359    \noindent Expressions containing $\hat0$ are not to be simplified by either of these operations.
1360\end{defin}
1361 
1362 
1363 
1364\begin{axio}\label{ax:y98tguh}
1365    For every Cauchy sequence $\{x_n\}$ in the equivalence class $[x]\subset C_\mathbb{Q}$, there exists another Cauchy sequence $\{ \hat0+x_n\}=\{x_n\}$.  This is to say
1366    \begin{equation*}
1367    \big\{x_n\big\}\in[x]\quad\iff\quad\big\{\, \hat0+x_n\big\}\in[x]~~,
1368    \end{equation*}
1369   
1370    \noindent or that, equivalently, there exists an additive identity element for every $x\in\mathbb{Q}$.
1371\end{axio}
1372 
1373\begin{exa}\label{ex:yy89779aaa}
1374    With Axiom \ref{ax:y98tguh}, we have associated every element of $C_\mathbb{Q}$ with the endpoint $A$ of the real line segment $\mathbf{AB}$.  This is done because every $x\in\mathbb{Q}$ has zero fractional magnitude with respect to infinity.  Therefore, we may mingle the geometric and algebraic notations to write
1375    \begin{equation*}
1376    \big\{\, \hat0+x_n\big\}\equiv\big\{A+x_n\big\}\in[A+x]~~.
1377    \end{equation*}
1378   
1379    \noindent By extending the line segment in consideration from $\mathbf{AB}\equiv[0,\infty]$ to $ZB\equiv[-\infty,\infty]$, the number zero is now in the center of $A$ which is an interior point of $ZB$.  Therefore, we may give an algebraic construction by Cauchy equivalence classes for all 
1380    \begin{equation}
1381    \mathbb{R}^\mathcal{X}_0=\big\{ \aleph_\mathcal{X}+ b~\big|~ b\in  \mathbb{R}_0   \big\} ~~,\nonumber
1382    \end{equation}
1383   
1384    \noindent by changing the interior point attached to the sequences in the equivalence classes.  For any interior point $X\in\mathbf{AB}$, there is an equivalence class $[X+x]$ such that
1385    \begin{equation*}
1386    \mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}  ~~,~~[x]=b\in\mathbb{R}_0\quad\implies\quad [X+x]\equiv[\aleph_{[\mathcal{X}]}+x]=\aleph_\mathcal{X}+b~~.
1387    \end{equation*}
1388   
1389    \noindent  In this notation, the comma is a logical ``and'' ($\land$) so the implication follows if both conditions on the left are true.  Note the number $\mathcal{X}$ indicating that $\aleph_\mathcal{X}$ has $100\times\mathcal{X}$\% fractional distance with respect to $\mathbf{AB}$ is an equivalence class $\mathcal{X}=[\mathcal{X}]\subset C_\mathbb{Q}$ with no requisite geometric part because $0<\mathcal{X}<1$ implies $\mathcal{X}\in\mathbb{R}_0$.
1390\end{exa}
1391 
1392 
1393\begin{defin}\label{def:CABQ}
1394    $C_\mathbb{Q}^{\mathbf{AB}}$ is a Cartesian product of $C_\mathbb{Q}$ with the set of all $X\in\mathbf{AB}$.  Specifically,
1395    \begin{equation*}
1396    C_\mathbb{Q}^{\mathbf{AB}}=\big\{X\big\}\times C_\mathbb{Q}=\big\{X+[x]~\big|~X\in\mathbf{AB},~X\neq A,~X\neq B,~[x]\subset C_\mathbb{Q}\big\}~~.
1397    \end{equation*}
1398   
1399    \noindent Since it is considered desirable to give a totally algebraic construction, we may give the equivalent definition
1400    \begin{equation*}
1401    C_\mathbb{Q}^{\mathbf{AB}}=\big\{\aleph_{\mathcal{X}}\big\}\times C_\mathbb{Q}=\big\{\aleph_{[\mathcal{X}]}+[x]~\big|~[x],[\mathcal{X}]\subset C_\mathbb{Q},~0<[\mathcal{X}]<1\big\}~~.
1402    \end{equation*}
1403\end{defin}
1404 
1405\begin{rem}
1406    In Definition \ref{def:CABQ}, the second definition of $C_\mathbb{Q}^{\mathbf{AB}}$ avoids any ambiguity related to the many-to-one relationship between points in $\mathbf{AB}$ and the numbers in the algebraic representations of those points.  For instance, there is no single equivalence class of rationals containing all of $\mathbb{R}^0_0$ so there is no inherently well-defined notion of the equivalence class of a geometric point.  
1407\end{rem}
1408 
1409\begin{defin}\label{def:kkqkqqk11}
1410    The equivalence class of a geometric point $X$ is the equivalence class of the number in the center of its algebraic representation $\mathscr{X}\equiv X$.  That is
1411    \begin{equation*}
1412    \mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}\quad\implies \quad [X]\equiv[\aleph_{\mathcal{X}}]=\aleph_{[\mathcal{X}]}=\aleph_{\mathcal{X}}~~.
1413    \end{equation*}
1414   
1415    \noindent This notation is redundant because $X$ is nothing like a Cauchy sequence.  In general, we will use the $\aleph_{[\mathcal{X}]}=[\aleph_\mathcal{X}]$ notation.  The main purpose of the present definition is to formalize the identical sameness of the definitions of $C_\mathbb{Q}^{\mathbf{AB}}$ given in Definition \ref{def:CABQ}.
1416\end{defin}
1417 
1418 
1419\begin{defin}\label{def:jhjjjjj}
1420    Every $\aleph_\mathcal{X}\in\mathbb{R}_\aleph^\mathcal{X}\subset\mathbb{R}$ is a Cauchy equivalence class $\aleph_\mathcal{X}=[\aleph_\mathcal{X}]=\aleph_{[\mathcal{X}]}\subset C_\mathbb{Q}^{\mathbf{AB}}$ where $\aleph_\mathcal{X}\in\mathbb{R}$ implies $0<\mathcal{X}<1$ so that $\mathcal{X}=[\mathcal{X}]\subset C_\mathbb{Q}$.  If $\mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}$, then $[X]\equiv[\aleph_\mathcal{X}]$.
1421\end{defin}
1422 
1423\begin{axio}\label{def:it759595}
1424    Every $x\in\{\mathbb{R}^\mathcal{X}_0\}$ is a Cauchy equivalence class $x=\aleph_{[\mathcal{X}]}+[b]=[x]\subset C_\mathbb{Q}^\mathbf{AB}$.  $\text{Big}(x)$ is defined by $[\mathcal{X}]\in C_{\mathbb{Q}}$ and $\text{Lit}(x)$ is defined by $[b]\in C_\mathbb{Q}$.  As in Definition \ref{def:15525252525}, $x$ is defined as the sum of its big and little parts.  In other words, without inventing the object $C_\mathbb{Q}^\mathbf{AB}$, we have the equivalent axiom that every $x\in\{\mathbb{R}^\mathcal{X}_0\}$ is an ordered pair of Cauchy equivalence classes
1425    \begin{equation*}
1426    x=\big([\mathcal{X}],[b]\big)\subset C_{\mathbb{Q}}\times C_{\mathbb{Q}}~~,
1427    \end{equation*}
1428   
1429    \noindent where the Cartesian product is
1430    \begin{equation*}
1431     C_\mathbb{Q} \times  C_\mathbb{Q} : \big([\mathcal{X}],[b]\big)\to\aleph_{[\mathcal{X}]}+[b]~~.
1432    \end{equation*}
1433\end{axio}
1434 
1435\begin{rem}
1436    Axiom \ref{def:it759595} is totally compliant with Axiom \ref{ax:constaxcjco} which requires that all real numbers can be constructed from Cartesian products of subsets of $C_\mathbb{Q}$.
1437\end{rem}
1438 
1439 
1440 
1441\begin{exa}
1442    This example gives a Cauchy equivalence class definition of $\aleph_\mathcal{X}$, as in Definition \ref{def:jhjjjjj}.  Suppose $0\leq x\leq1$ and that
1443    \begin{equation*}
1444    x=[x]=[\{x_n\}]=\big\{x_1,x_2,x_3,...\big\}~~.
1445    \end{equation*}
1446   
1447    \noindent Then, moving the iterator into the superscript position for notation purposes, $\aleph_x$ is a Cauchy equivalence class
1448    \begin{equation*}
1449    \aleph_x=\aleph_{[x]}=[\aleph_x]=[\{\aleph_x^n\}]=[\aleph_{\{x_n\}}]=\big\{\aleph_{x_1},\aleph_{x_2},\aleph_{x_3},...\big\}
1450    \end{equation*}
1451\end{exa}
1452 
1453 
1454\begin{thm}
1455    If $X$ and $Y$ are two interior points of $\mathbf{AB}$, then two Cauchy equivalence classes $[X+x]$ and $[Y+y]$ are equivalent if and only if $X=Y$ and $x=y$.
1456\end{thm}
1457 
1458\begin{proof}
1459    By Definition \ref{def:kkqkqqk11}, we have $[X+x],[Y+y]\subset C_\mathbb{Q}^\mathbf{AB}$.  Every element of $C_\mathbb{Q}^\mathbf{AB}$ can be expressed as the Cartesian product of two elements of $C_\mathbb{Q}$
1460    \begin{equation*}
1461    \big\{[\mathcal{X}]\subset C_\mathbb{Q}\big\}\times \big\{  [b]\subset C_\mathbb{Q}\big\}: \big([\mathcal{X}],[b]\big)\to\aleph_{[\mathcal{X}]}+[b]~~.
1462    \end{equation*}
1463   
1464    \noindent By the definition of the equivalence class, every element of $C_\mathbb{Q}$ is such that
1465    \begin{equation*}
1466    [x]=[y]\quad\iff\quad x=y~~,
1467    \end{equation*}
1468   
1469    \noindent so the same must be true for the ordered pairs:
1470    \begin{equation*}
1471    \big([\mathcal{X}],[x]\big)=\big([\mathcal{Y}],[y]\big)\quad\iff\quad \big(\aleph_\mathcal{X},x\big)=\big(\aleph_\mathcal{Y},y\big)~~.
1472    \end{equation*}
1473   
1474    \noindent Per Definition \ref{def:kkqkqqk11}, the equivalence class of $[X]$ is uniquely determined by the equivalence class of $\aleph_\mathcal{X}$ so it follows that $X=Y$ if and only if $[X]=[Y]$.  The theorem is proven.
1475     
1476\end{proof}
1477   
1478 
1479 
1480\subsection{The Maximal Neighborhood of Infinity}\label{sec:nbinf}
1481 
1482The main purpose of this section is to treat the properties of real numbers $x\in\mathbb{R}^\mathcal{X}_\aleph$ for the special case of $\mathcal{X}=1$.  Again, the reader must note that formally $\mathbb{R}_\aleph^1\not\subset\{\mathbb{R}^\mathcal{X}_\aleph\}$ due to the restriction $0<\mathcal{X}<1$ given by Definition \ref{def:gtg34535335}.  Whenever $\mathbb{R}^\mathcal{X}_0$ or $\mathbb{R}^\mathcal{X}_\aleph$ is taken to mean $\mathcal{X}=0$ or $\mathcal{X}=1$, referring the neighborhood of the origin and the maximal neighborhood of infinity respectively, we will always make an explicit statement indicating $0\not<\mathcal{X}\not<1$.
1483 
1484\begin{defin}\label{def:y969tfff22}
1485    The whole maximal neighborhood of infinity is
1486    \begin{equation*}
1487    \mathbb{R}_\aleph^1=\big\{ \aleph_1-b ~\big|~ b\in\mathbb{R}^0_\aleph  \big\}~~.
1488    \end{equation*}
1489\end{defin}
1490 
1491 
1492\begin{rem}
1493    We have defined $\aleph_1$ as the greatest number in the algebraic representation $\mathscr{B}$ of $B\in\mathbf{AB}\equiv[0,\infty]$.  Therefore, $\aleph_1$ is an infinite element not in the real numbers.  As the arithmetic of $\infty$ is usually defined, if we set $\aleph_1=\infty$, then it would follow that $\infty-b=\infty$ and $\mathbb{R}^1_\aleph\cap\mathbb{R}=\varnothing$.  This is not the desired behavior so we will make special notation custom tailored to deliver what is desired.
1494\end{rem}
1495 
1496 
1497\begin{defin}
1498    $\infty$ is called geometric infinity or simply infinity.
1499\end{defin}
1500 
1501 
1502\begin{defin}
1503    $\widehat\infty$ is called algebraic infinity.  It shall be called infinity hat as well.
1504\end{defin}
1505 
1506 
1507 
1508\begin{defin}\label{def:adabs}
1509    Additive absorption is a property of $\infty$ such that all $x\in\mathbb{R}$ are additive identities of $\infty$.  The additive absorptive property is
1510    \begin{equation}
1511    \infty\pm x=\infty~~. \nonumber
1512    \end{equation}
1513   
1514    \noindent Multiplicative absorption is a property of $\pm\infty$ such that all non-zero $x\in\mathbb{R}$ are multiplicative identities of $\pm\infty$.  The multiplicative absorptive property is
1515    \begin{equation}
1516    \infty\cdot x=\begin{cases}
1517    ~~\infty&\text{for}\quad x>0\\
1518    -\infty&\text{for}\quad x<0
1519    \end{cases}. \nonumber
1520    \end{equation}
1521\end{defin}
1522 
1523 
1524\begin{rem}
1525    Note that infinity and zero are both multiplicative absorbers while zero's additive absorptive property is that it gets absorbed.  Indeed, the contradiction inherent to mutual multiplicative absorption may be identified as a reason contributing to the canonical non-definition of the $0\cdot\infty$ operation.
1526\end{rem}
1527 
1528 
1529\begin{defin}\label{def:hat33}
1530    The symbol $\widehat\infty$ refers to an infinite element
1531    \begin{equation}
1532    \lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\big|\widehat\infty\big|~~, \quad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \big|\widehat\infty \big| ~~,\nonumber
1533    \end{equation}
1534   
1535    \noindent together with an instruction not to perform the additive or multiplicative operations usually imbued to infinite elements.  
1536\end{defin}
1537 
1538\begin{rem}
1539    What we have done in Definition \ref{def:hat33} is exactly what we have done with $\hat0$ in Definition \ref{def:i9696969a}.  In the case of $\hat0$, it was not in any way strange to entertain the notion that one might simply choose not to do the absorptive operations of zero and neither should the present convention for $\widehat\infty$ be considered in any way strange or ill-defined.  In Section \ref{sec:fconstM}, we will construct an infinite element---what might be called an instance of infinity---stripped of its absorptive operations by considering the invariance of $\mathbf{AB}$ under the permutations of the labels of its endpoints.  As in Sections \ref{sec:Rneighb} and \ref{sec:fconst}, we will define some objects in the present section to facilitate a formal construction in Section \ref{sec:fconstM}.
1540\end{rem}
1541 
1542\begin{thm}\label{def:dvvj8yv8r}
1543    The two open intervals $(-\infty,\infty)$ and $(-\widehat\infty,\widehat\infty)$ are identically equal.  In other words, the real number line may be expressed identically as $\mathbb{R}=(-\widehat\infty,\widehat\infty)$.
1544\end{thm}
1545 
1546\begin{proof}
1547    For $a,b\in\mathbb{R}^+$, it may be taken for granted that
1548    \begin{equation*}
1549    (-a,b)=(-|a|,|b|)~~,
1550    \end{equation*}
1551   
1552    \noindent and it follows, therefore, that this is true for $a,b\in\overline{\mathbb{R}}^+$.  Then
1553    \begin{equation}
1554    \pm\big|\widehat\infty\big|=\pm\big|\infty\big|\quad\implies\quad \mathbb{R}=(-\widehat\infty,\widehat\infty)~~,\nonumber
1555    \end{equation}
1556   
1557    \noindent and the theorem is proven.
1558\end{proof}
1559 
1560 
1561 
1562 
1563\begin{exa}\label{exa:588585557aaa}
1564    This example demonstrates the arithmetic constraints that would have to be placed on the limit definition of infinity if it was said to define $\widehat\infty$ rather than $|\widehat\infty|$.  This example also demonstrates the general motivation for such notation by demonstrating the large burden that would imposed not using the absolute value in Definition \ref{def:hat33}.  In its limit incarnation, the additive absorptive property of $\infty$ is demonstrated as
1565    \begin{equation*}
1566    a+\infty=a+\lim\limits_{x\to0 }\dfrac{1}{x}=\lim\limits_{x\to0 }\dfrac{1+ax}{x}=\text{diverges}=\infty~~.
1567    \end{equation*}
1568   
1569    \noindent Therefore, if the limit were said to define $\widehat\infty$, then the arithmetic constraint ``don't simplify this expression'' would mean to keep $a$ out of the limited expression.  Similarly, multiplicative absorption is demonstrated as
1570    \begin{equation*}
1571    a\cdot\infty=a\cdot\lim\limits_{x\to0 }\dfrac{1}{x}=\lim\limits_{x\to0 }\dfrac{a}{x}=\text{diverges}=\infty~~.
1572    \end{equation*}
1573   
1574    \noindent In either case, the limit expression diverges in $\mathbb{R}$ and no contradiction is obtained by keeping $a$ out of the expression to avoid it being ``absorbed.''  
1575   
1576    The utility in adding the hat to infinity is that it supports the notion that a number lying $x$ units of Euclidean distance away from the least number $0=\aleph_0$ in the algebraic representation of $A\in\mathbf{AB}$ should, under permutation of the labels of the endpoints of $\mathbf{AB}$, be mapped to another number $x'$ lying $x$ units of distance away from the greatest number $\aleph_1$ in the algebraic representation of $B\in\mathbf{AB}$.  By suppressing the additive absorption, we let $x'=\aleph_1-x=\widehat\infty-x\neq\infty$.  Per Definition \ref{def:y969tfff22}, this number is $x'\in\mathbb{R}_\aleph^1$.  By suppressing the multiplicative absorption of $\widehat\infty$, we introduce notation by which it is possible to complement Definition \ref{def:hh4hh4h4h4} with the statement
1577    \begin{equation*}
1578    \frac{\aleph_{\mathcal{X}}}{\infty}=\mathcal{X}\quad\iff\quad \aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty~~.  
1579    \end{equation*}
1580   
1581    \noindent In the former part this paper, we have demonstrated a requirement for numbers such as $x'$ and $\aleph_\mathcal{X}$, and $\widehat\infty$ is a notation for an infinite element tailored to the requirement.  Indeed, where ``algebra'' is called the study of mathematical symbols and the rules for manipulating them, algebraic infinity $\widehat\infty$ is a perfectly ordinary algebraic object and well-defined.  
1582\end{exa}
1583 
1584 
1585 
1586\begin{defin}\label{def:ugy8re8r7777}
1587    For any $\mathcal{X}\in\mathbb{R}$, the symbol $\aleph_\mathcal{X}$ is defined as
1588    \begin{equation*}
1589    \aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty~~.
1590    \end{equation*}
1591\end{defin}
1592 
1593 
1594\begin{defin}\label{def:zzzzz86et}
1595    In terms of $\widehat\infty$, the whole maximal neighborhood of infinity is defined as
1596    \begin{equation}
1597    \mathbb{R}^1_\aleph=\big\{  \widehat{\infty}-b~\big|~b\in\mathbb{R}^0_\aleph,~b\neq0 \big\}     ~~.\nonumber
1598    \end{equation}
1599\end{defin}
1600 
1601 
1602 
1603\begin{defin}\label{def:dhgw86et86et}
1604    The maximal natural neighborhood of infinity is defined as
1605    \begin{equation}
1606    \mathbb{R}^1_0=\big\{  \widehat\infty-b ~\big|~b\in\mathbb{R}_0^+ \big\}     ~~.\nonumber
1607    \end{equation}
1608\end{defin}
1609 
1610 
1611\subsection{Equivalence Classes for the Maximal Natural Neighborhood of Infinity}\label{sec:fconstM}
1612 
1613 
1614We could easily construct $\mathbb{R}^1_0$ following the prescription in Section \ref{sec:fconst}.  There, we introduced zero as the least number in the algebraic representation of $A\in\mathbf{AB}\equiv[0,\infty]$ and then we made the extension to an arbitrary interior point by considering $A$ as the midpoint of $ZB\equiv[-\infty,\infty]$.  However, we could have easily left $A$ as an endpoint and then extended the construction to the other endpoint $B$ via a symmetry argument.  For breadth, here we will use a similar symmetry argument to take a slightly different approach to the Cauchy construction of the maximal neighborhood infinity.  The material in the present section will constitute an independent motivation for the intermediate neighborhoods, separate from the main fractional distance approach.  We will generate a non-absorbing infinite element $\widehat\infty$ and then we will define the $\aleph_\mathcal{X}$ as its algebraic fractional parts.
1615 
1616In Section \ref{sec:fconst}, we defined a real number as an ordered pair of Cauchy equivalence classes of rationals: one for the big part and one for the small part.  This approach requires that we assume the $\aleph$ notation before we can define an equivalence class $[\aleph_\mathcal{X}]=\aleph_{[\mathcal{X}]}=\aleph_\mathcal{X}$.  We were very well-motivated to assume numbers in this form, particularly by Main Theorem \ref{thm:ef2424t24cc} proving that some real numbers are larger than any real number, and by Theorem \ref{thm:halfway} proving that there exists at least one real number having $50\%$ fractional magnitude with respect to $\mathbf{AB}$.  However, it remains that $\aleph_\mathcal{X}$ is inherently foreign to what is called real analysis.  Therefore, in the present section, we will give an alternative construction for $\mathbb{R}_0^1$ based on the geometric invariance of line segments under the permutations of their endpoints.  These numbers are defined with $\infty$: a number not at all foreign to real analysis.  Then, with $|\widehat\infty|=\infty$ defined as in the previous section, and with a formal construction given here for the maximal neighborhood of infinity, we will use $\widehat\infty$ as an independent constructor for $\aleph_\mathcal{X}$.
1617 
1618 
1619 
1620\begin{axio}
1621    A Euclidean line segment $AB$ \cite{EE} is invariant under permutations of the labels of its endpoints, \textit{e.g.}: $AB=BA$.  
1622\end{axio}
1623 
1624 
1625\begin{defin}\label{def:NCP}
1626    Define a geometric permutation operator $\hat P$ such that
1627    \begin{equation*}
1628    \hat P (AB)=BA~~.
1629    \end{equation*}
1630\end{defin}
1631 
1632 
1633\begin{rem}
1634    In this section, we will construct $\mathbb{R}^1_0$ from the operation of $\hat P$ on Cauchy equivalence classes of rational numbers, \textit{e.g.}: $\hat P([x])$.  To do so, we must develop the induced operation of $\hat P$ on the algebraic interval representation $[a,b]\equiv AB$, where $[a],[b]\subset C_\mathbb{Q}$.  (It is a pleasant coincidence that the equivalence class bracket notation is exactly consistent with the abused notion of a closed one-point interval $[x,x]=[x]$.)  As in Section \ref{sec:fconst}, our departure from the usual Cauchy construction of $\mathbb{R}$ begins with an acknowledgment that $0\in\mathbb{Q}$ does not follow from $\mathbb{N}$.  Again,  we introduce $0=\aleph_0$ as the least number in the algebraic representation of $A\in\mathbf{AB}$.  Then we assume zero is an additive identity element of every $n\in\mathbb{N}$ to obtain
1635    \begin{equation*}
1636     \frac{m}{n}\in\mathbb{Q} \quad\implies\quad \frac{m}{n}=\frac{0+m}{n}=\frac{0}{n}+\frac{m}{n}=0+\frac{m}{n}~~.
1637    \end{equation*}
1638   
1639    \noindent Finally, we put the hat on $\hat 0$ to remind us not to simplify the expression.  The elements of $C_\mathbb{Q}$ now have an interpretation as Euclidean magnitudes measured relative to the origin of $\mathbb{R}$.  Specifically, $\frac{m}{n}$ is an abstract element of $\mathbb{Q}$ but $\hat0+\frac{m}{n}$ is the rational length of a real line segment whose left endpoint has zero as the least number in it algebraic representation.  This follows from Definition \ref{def:kkqkqqk11} giving $[A]=[\aleph_0]=\aleph_{[0]}=0=\hat0$.
1640\end{rem}
1641 
1642\begin{defin}\label{def:kkk3kk3111}
1643    The Euclidean chart $x$ on $\mathbf{AB}$ is such that $\min(x\in A)=0$ and $\max(x\in B)=\aleph_1$ regardless of the permutation of the labels of the endpoints.
1644\end{defin}
1645 
1646\begin{defin}
1647    Define an operator $\hat{\mathcal{P}}_0([x];\hat 0)$ which formalizes the notion of $\hat P([x])$.  Per Definition \ref{def:NCP}, the domain of $\hat P$ is not in $C_\mathbb{Q}$ so we introduce a special algebraic permutation operator $\hat{\mathcal{P}}_0([x];\hat 0)$ dual to $\hat P$ which formally operates on equivalence classes.  The definition is
1648    \begin{equation*}
1649    \hat{\mathcal{P}}_0:\hat0\times C_\mathbb{Q}\to\widehat\infty\times C_\mathbb{Q}~~,
1650    \end{equation*}
1651   
1652    \noindent where
1653    \begin{equation*}
1654    \hat0\times C_\mathbb{Q}=\big\{\hat0+[x]~\big|~[x]\subset C_\mathbb{Q}\big\}~~,\quad\text{and}\qquad\widehat\infty\times C_\mathbb{Q}=\big\{\widehat\infty-[x]~\big|~[x]\subset C_\mathbb{Q}\big\}~~.
1655    \end{equation*}
1656\end{defin}
1657 
1658\begin{exa}
1659    This example demonstrates the working of $\hat P$ and $\hat{\mathcal{P}}_0$ to give a formal construction of $\mathbb{R}^1_0$ by Cauchy sequences of rational numbers.  Suppose $b\in\mathbb{R}_0$ is a well-defined equivalence class of rationals lying within the algebraic representation $\mathscr{A}$ of $ A\in\mathbf{AB}$.  Now operate on $\mathbf{AB}$ with $\hat P$ so that
1660    \begin{equation*}
1661    \hat P(\mathbf{AB})=\mathbf{BA}~~.
1662    \end{equation*}
1663   
1664    \noindent The permutation of the labels of the endpoints has not changed the geometric position of $b$ along the line segment.  Definition \ref{def:kkk3kk3111} requires that the orientation of the Euclidean coordinate along the line segment has been reversed, so, therefore, we no longer have the property $b=[x]\subset C_\mathbb{Q}$ for the following reason.  Every rational number is less than some natural number and all such numbers have zero fractional distance with respect to $\mathbf{AB}$.  Before operating with $\hat P$, $b$ was in the algebraic representation of the the point $A$ but by operating with the geometric permutation operator $\hat P$ it becomes a number in the algebraic representation of $B$.  The FDFs are defined so that
1665    \begin{equation*}
1666    \mathcal{D}_{\!\mathbf{AB}}(\mathbf{AB})=\mathcal{D}^\dagger_{\!\mathbf{AB}}(\mathbf{AB})=1~~,
1667    \end{equation*}
1668   
1669    \noindent meaning that $b$ must now have unit fractional magnitude with respect to $\mathbf{AB}$.  Every $[x]\subset C_\mathbb{Q}$ has zero fractional magnitude so if $b\neq[x]$, what number is it?  The number is given by
1670    \begin{equation*}
1671    b=\hat{\mathcal{P}}_0(\hat 0,[x])=\widehat\infty-[x]~~.
1672    \end{equation*}
1673   
1674    \noindent Under permutation of the labels of the endpoints of a line segment, a number having distance $[x]\subset C_\mathbb{Q}$ from one endpoint becomes another number having the same distance relative to the other endpoint.
1675\end{exa}
1676 
1677\begin{rem}
1678    We take it for granted that if there exists a real number $x$ separated by distance $L$ from the least number in the algebraic representation of the endpoint $A$ of an arbitrary real line segment $AB\equiv[a,b]$---with $x$ interior in the sense that $x\in(a,b)$---then it is guaranteed by the geometric mirror symmetry of all line segments that there must exist another real number separated from the endpoint $B$ by the same distance $L$.  If we bestowed $\widehat\infty$ with the property of additive absorption, then there would be no such number.  Similarly, if there exists a real number lying one third of the way from $A$ to $B$, then the there must exist another real number lying one third of the way from $B$ to $A$.  This follows from the cut-in-a-line definition of $\mathbb{R}$ given by Definition \ref{def:real}.  For the case of $\mathbf{AB}$, it will be impossible to express these third fraction numbers if $\widehat\infty$ has the property of multiplicative absorption.  Since the third numbers \textit{must} exist, $\aleph_\mathcal{X}$ \textit{does} exist.  Therefore, the existence of an instance of infinity devoid of any absorptive properties is absolutely granted if the mirror symmetry of a geometric line segment is to be preserved in its interpretation as an algebraic interval of numbers.  
1679   
1680    The thesis of the present treatise is that  we should preserve the underlying geometric construction of $\mathbb{R}$ without invoking a contradictory algebraic construction.  Under this thesis, $\widehat\infty$ is forced into existence.  Often times, the position is taken that infinity is absolutely absorptive due to the limit definition of infinity and the attendant absorptive properties of limits (Example \ref{exa:588585557aaa}.)  As an indirect consequence of such reasoning, the mirror symmetry of line segments must be rejected in the algebraic realm of mathematics.  But why should it be preferred that the algebraic construction overrides the geometric construction?  Is it not equally valid to override the algebraic construction with the geometric one?  Considering the history of mathematics, it is, in the opinion of this writer, far more appropriate to preserve the geometric construction at all costs.  It is very easy to do so when the symbol $\widehat\infty$ is given by the limit definition of infinity as
1681    \begin{equation}
1682    \lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\big|\widehat\infty\big| ~~,\nonumber
1683    \end{equation}
1684   
1685    \noindent without $\widehat\infty$ itself being interchangeably equal with the limit expression.
1686 
1687    In Definition \ref{def:it759595}, we gave the definition of $x\in\{\mathbb{R}^\mathcal{X}_0\}$ in terms of ordered pairs of elements of $C_\mathbb{Q}$.  The purpose of the present alternative treatment for the maximal neighborhood $\mathbb{R}^1_0$ is not to replace that definition but to complement it with a different equivalence class construction for the maximal neighborhood from which the constructions of the intermediate neighborhoods may be extracted.  In this present section, we have used the permutation operator $\hat P$ which is quite similar to the implicit translation operator by which we were able to attach elements of $C_\mathbb{Q}$ to different interior points of $\mathbf{AB}$ in Section \ref{sec:fconst}.  The main utility in developing the idea of a number in the neighborhood of infinity as the operation of $\hat{\mathcal{P}}_0$ on an equivalence class of rationals is that it independently generates the requirement for an infinite element lacking the usual absorptive properties of infinity.  With $\widehat\infty$ granted, it gives a separate means by which we may construct the $x\in\{\mathbb{R}_0^\mathcal{X}\}$ without invoking the direct ordered pair definition: the $\aleph_\mathcal{X}$ in such numbers are the fractions of the non-absorbing infinite element $\widehat\infty$.
1688\end{rem}
1689 
1690\begin{defin}
1691    Every $x\in\mathbb{R}_0^1$ is defined as the output of $\hat{\mathcal{P}}_0$ operating on an element of $C_\mathbb{Q}$.  This is the Cauchy equivalence class construction of real numbers in the maximal natural neighborhood of infinity.
1692\end{defin}
1693 
1694 
1695\begin{exa}\label{exa:2r233535533535}
1696    In this example, we complement the separate definitions for $\infty$ and $\widehat\infty$ heretofore given.  We will show, for example, how they might be more fully conceptually distinguished as two mutually distinct kinds of infinite elements with markedly different qualia beyond their separate technical definitions.  While we will offer these qualia as an example, we will not alter the technical definitions with the supplemental considerations proposed here.  To that end, it is sometimes claimed, without proof, that one cannot place endpoints at the ends of $\mathbb{R}=(-\infty,\infty)$ because the notion of an endpoint contradicts the notion of the infinite geometric extent of a line extending infinitely far in both directions.   Infinite geometric extent is the main principle that we will look at in this example.  
1697   
1698    Suppose geometric infinity $\infty$ is a number which cannot be included as an endpoint without contradicting the notion of the infinite geometric extent of a number line.  Definition \ref{def:metspa} defines a number line as a 1D metric space in the Euclidean metric
1699    \begin{equation}
1700    d(x,y)=
1701    \big|y-x\big|~~.\nonumber
1702    \end{equation}
1703   
1704    \noindent If we did include geometric infinity as an endpoint, then we could invoke the invariance of line segments under permutations of their endpoints to demonstrate a contradiction.  Given
1705    \begin{equation*}
1706    (x,y)=(x_0,y_0)~~,\quad\text{and}\qquad(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=(\infty-x_0,\infty-y_0)~~,
1707    \end{equation*}
1708   
1709    \noindent then not only do the points lose their unique identity when attached to $B$ instead of $A$, but if we put $(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))$ into the Euclidean metric, then we get
1710    \begin{equation}
1711    d(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=
1712    \big|\infty-x_0-\big(\infty-y_0\big)\big|=
1713    \big|\infty-\infty\big|=\text{undefined}~~.\nonumber
1714    \end{equation}
1715   
1716    \noindent Clearly, this does not gel well with our intention to define a number line as a line equipped with a metric.  The line is supposed to have some metrical distance between any two points but now, under the permutation of the labels $A$ and $B$, we find two points who don't even have vanishing distance between them.  The distance has become undefined even though this does not follow from the invariance of Euclidean line segments under such permutations.
1717   
1718    Algebraic infinity is a number which avoids all of the problems here listed.  Under permutation, we have
1719    \begin{equation*}
1720    (x,y)=(x_0,y_0)~~,\quad\text{and}\qquad(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=(\widehat\infty-x_0,\widehat\infty-y_0)~~.
1721    \end{equation*}
1722   
1723    \noindent Jumping ahead to the arithmetic of such numbers axiomatized in Section \ref{sec:aritAX}, we find
1724    \begin{equation}
1725    d(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=
1726    \big|\widehat\infty-x_0-\big(\widehat\infty-y_0\big)\big|=
1727    \big|y_0-x_0\big|=d(x_0,y_0)~~,\nonumber
1728    \end{equation} 
1729   
1730    \noindent exactly as expected.  The only issue which remains is to revisit is the construction for $\mathbf{AB}\equiv[0,\infty]$ that we have given by a conformal chart $x=\tan(x')$ on the line segment $AB\equiv[0,\frac{\pi}{2}]$ whose endpoints unquestioningly exist.  For this, we propose a semantic convention to distinguish the geometric infinite element $\infty$ from the algebraic one $\widehat\infty$.  Let algebraic infinity be such that it can be embedded in a larger space but let geometric infinity be such that it is totally maximal and cannot be embedded in something larger than itself.  For example, the interval $[0,\frac{\pi}{2}]\subset[-\pi,\pi]$ is such that the conformal chart which sends $\frac{\pi}{2}$ to an infinite element implicitly places that element within the parent interval $[-\pi,\pi]$.  The convention proposed here would require that the infinite element to which $\frac{\pi}{2}$ is conformally mapped must be algebraic infinity $\widehat\infty$.  If we take the convention that geometric infinity $\infty$ is always totally geometrically maximal, then that would forbid its existence on the interior of the interval $[-\pi,\pi]$ which contains points to the right of $\frac{\pi}{2}$.  In a formal adoption of the distinctions made here, one would examine the merits of a supplemental ordering relation $\widehat\infty<\infty$.
1731\end{exa}
1732 
1733 
1734 
1735 
1736\begin{rem}
1737    If we wish to construct $\mathbf{AB}\equiv[0,\widehat\infty]$ directly from $AB\equiv[0,\frac{\pi}{2}]$ as in Example \ref{ex:confrmal}, wherein we cite the limit definition of infinity (Definition \ref{def:RRRinf}) as motivating the identity
1738    \begin{equation}
1739    \tan\left(\cfrac{\pi}{2}\right)=\infty~~,\nonumber
1740    \end{equation}
1741   
1742    \noindent then we need to make rigorous the relationship between $\infty$ and $\widehat\infty$.  This was the purpose of Theorem \ref{def:dvvj8yv8r} proving $\mathbb{R}=(-\widehat\infty,\widehat\infty)$.  Since the absolute value, or the magnitude, of $\widehat\infty$ is the same as that of $\infty$, the algebraic intervals $[a,\infty]$ and $[a,\widehat\infty]$ must be the same interval.  Though we cannot directly construct $[0,\widehat\infty]$ from $[0,\frac{\pi}{2}]$, we may indirectly construct it by using the limit definition of infinity to write
1743    \begin{equation*}
1744    \lim\limits_{\theta\to\frac{\pi}{2}}\tan(
1745    \theta)=\lim\limits_{\theta\to\frac{\pi}{2}}\frac{\sin(
1746        \theta)}{\cos(\theta)}=\lim\limits_{\substack{x\to0\\y\to1}}\frac{y}{x}=\infty~~.
1747    \end{equation*}
1748   
1749    \noindent Now we may directly infer the existence of conformal $\mathbf{AB}\equiv[0,\widehat\infty]$ from the assumed interval $[0,\frac{\pi}{2}]$.  Due to the transitivity of the equivalence relation, however, we must be very careful about the definition of $\widehat\infty$.   Note that Definition \ref{def:hat33} gives
1750    \begin{equation*}
1751    \big|\widehat\infty\big|=\infty=\lim\limits_{x\to0}\dfrac{1}{x}\quad\centernot\implies\quad \widehat\infty=\lim\limits_{x\to0}\dfrac{1}{x}~~.
1752    \end{equation*}
1753   
1754    \noindent Therefore, we must be careful about whether $\aleph_1$ is equal to geometric infinity, or algebraic.  If we take the convention that geometric infinity $\infty$ is imbued with the notion of infinite geometric extent such that an infinite line cannot have an endpoint there, as in Example \ref{exa:2r233535533535}, then we should not let $\aleph_1$ be defined by $\infty$ when it is said to be the greatest number in the algebraic representation of the endpoint $B\in\mathbf{AB}$.  Due to the possibility of constructing $\mathbf{AB}$ from any other line segment by one conformal chart transformation or another, $\mathbf{AB}$ ought to be taken as $[0,\widehat\infty]=[0,\aleph_1]$ in the absence of explicit words to the contrary.
1755\end{rem}
1756 
1757\begin{defin}
1758    The symbol $\aleph_1$ is an alternative notation for algebraic infinity.  We have
1759    \begin{equation*}
1760    \aleph_1=\widehat\infty~~,\quad\text{and}\qquad\aleph_1\neq\infty~~.
1761    \end{equation*}
1762\end{defin}
1763 
1764\begin{rem}
1765    All of the contradictions which forbid additive and multiplicative inverses for $\infty$ stem from its limit definition.  Should we then bestow these inverse on $\widehat\infty=\aleph_1$?  To the extent that the notion of fractional distance requires that $100\%-100\%=0\%$, or that $100\%/100\%=1$, the answer is yes.  Similarly, all of the contradictions which disallow a definition for the operation $0\cdot\infty$ are rooted in the limit definition of infinity.  Note that $0\cdot\widehat\infty=\aleph_0=0$ follows as a special of $\aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty$.  We should not expect any contradictions because $\widehat\infty\neq\infty$ and the limit definition is out of scope.
1766\end{rem}
1767 
1768 
1769\begin{axio} \label{ax:undefinby}
1770    $\aleph_1$ is such that
1771    \begin{align}
1772    \aleph_1-\aleph_1=0~~,\quad\text{and}\qquad\cfrac{\aleph_1}{\aleph_1}=1~~.\nonumber
1773    \end{align}
1774\end{axio}
1775 
1776 
1777 
1778 
1779\begin{thm}
1780    The maximal whole neighborhood of infinity is a subset of the real numbers.
1781\end{thm}
1782 
1783\begin{proof}
1784    Taking for granted that $x\in\mathbb{R}^1_\aleph$ does not have any infinitesimal part, which is obvious, it suffices to show the compliance with Definition \ref{def:real}: A real number $x\in\mathbb{R}$ is a cut in the real number line.  This follows directly from the Definition \ref{def:zzzzz86et} giving
1785    \begin{equation}
1786    \mathbb{R}^1_\aleph=\big\{  \widehat{\infty}-b~\big|~b\in\mathbb{R}^0_\aleph,~b\neq0 \big\}     ~~.\nonumber
1787    \end{equation}
1788   
1789    \noindent We clearly have
1790    \begin{equation*}
1791    (-\infty,\infty)=(-\infty,\aleph_\mathcal{X}+b]\cup(\aleph_\mathcal{X}+b,\infty)~~.
1792    \end{equation*}
1793   
1794   
1795    \noindent Even though we do not yet have an equivalence class construction of $b\in\mathbb{R}_\aleph^0\setminus\mathbb{R}_0$, it is obvious that $\widehat\infty-b$ is a cut in the real number line because $b$, whatever its algebraic construction, is such that it has less than unit zero fractional magnitude with respect to $\mathbf{AB}$ and is also such that $b>0$.  (The intuitive ordering assumed in this theorem is formalized in Axiom \ref{ax:order}.)
1796\end{proof}
1797 
1798 
1799\begin{cor}
1800    All numbers $x\in\{\mathbb{R}^\mathcal{X}_\aleph\}$ are real numbers.
1801\end{cor}
1802 
1803\begin{proof}
1804    The ordering of $\mathbb{R}$ given by Axiom \ref{def:order} is such that $0<\mathcal{X}<1$ guarantees
1805    \begin{equation*}
1806    (0,\infty)=(0,\aleph_\mathcal{X}\pm b]\cup(\aleph_\mathcal{X}\pm b,\infty)~~.
1807    \end{equation*}
1808   
1809    \noindent Definition \ref{def:real} is satisfied trivially and the theorem is proven.
1810\end{proof}
1811 
1812\begin{rem}
1813    As a final aside in this section, note the curious condition under which algebraic infinity $\aleph_1$ has its foundation in the geometric properties of a line segment while geometric infinity $\infty$ has its foundation in the limit of an algebraic expression.  The reciprocity among these two constructions of an infinite element might indicate some deeply fundamental issues extending beyond the semantic convention of our having chosen to call one infinite element geometric and the other algebraic.  We will not proceed along that analytical direction in this paper but the reciprocity of the cross-sampling of the concepts is interesting and tantalizing.
1814\end{rem}
1815 
1816\section{Arithmetic} \label{sec:5}
1817 
1818 
1819 
1820 
1821\subsection{Operations for Infinity}
1822 
1823Here we give arithmetic operations for $\widehat\infty\not\in\mathbb{R}$ to support the axioms for real numbers $x\in\mathbb{R}$ with non-zero big parts to appear in Section \ref{sec:aritAX}.
1824 
1825\begin{axio}\label{def:infOPSa}
1826    The operations for $\widehat\infty=\aleph_1$ with $b\in\mathbb{R}_0^+$ are
1827    \begin{align}
1828    \widehat\infty\pm b&= \pm b+\widehat\infty\nonumber\\
1829    \widehat\infty\pm\big( -b \big)&=\widehat\infty\mp b\nonumber\\
1830    -\big(\pm\widehat\infty\big)&=\mp\widehat\infty\nonumber\\
1831    \widehat\infty\cdot b=b\cdot\widehat\infty&=\aleph_{b}\nonumber\\
1832    \cfrac{\widehat\infty}{ b}&=\aleph_{(b^{-1})}\nonumber\\
1833    \cfrac{b}{\vphantom{ \widehat{A} }\widehat\infty}&=0\nonumber~~.
1834    \end{align}
1835\end{axio}
1836 
1837 
1838 
1839\begin{axio}\label{ax:opo2p2}
1840    We give the following supplemental axioms for zero and $ \widehat\infty$.
1841    \begin{align}
1842    \widehat\infty+0=0+\widehat\infty&=\widehat\infty\nonumber\\
1843    \widehat\infty\cdot0=0\cdot\widehat\infty&=0\nonumber\\
1844    \cfrac{0}{\vphantom{ \widehat{A} }\widehat\infty}&=0\nonumber\\
1845    \cfrac{\vphantom{ \widehat{A} } \widehat\infty}{0}&=\text{undefined}\nonumber~~.
1846    \end{align}
1847\end{axio}
1848 
1849\begin{rem}
1850    The most important facet of Axiom \ref{ax:opo2p2} is the $0\times\widehat\infty$ operation (with $\{\times\}=\{\cdot\}$) contrary to the undefined $0\times\infty$ operation.  This is required to preserve the notion of fractional distance: zero times $100\%$ is $0\%$.  To facilitate this definition, it will be required that we define division as separate operation distinct from multiplication by an inverse.  This will be one of the major distinctions of the axioms of Section \ref{sec:aritAX} from the well-known field axioms.  We demonstrate the principle in Example \ref{ex:788788m}.
1851\end{rem}
1852 
1853\begin{exa}\label{ex:788788m}
1854    This example gives a common argument in favor of the non-definition of a product between an infinite element and zero.  Then we will show how the contradiction is avoided by taking away the assumed associativity among multiplication and division.   Suppose $c\in\mathbb{R}^0_\aleph$ so that
1855    \begin{equation*}
1856    \cfrac{c}{\widehat\infty}=0~~.
1857    \end{equation*}
1858   
1859    \noindent Now suppose $0\cdot\widehat\infty$ is a defined operation so that
1860    \begin{equation*}
1861    z=0\cdot\widehat\infty~~.
1862    \end{equation*}
1863   
1864    \noindent Substitute $\frac{c}{\widehat\infty}=0$ and use the $\frac{\widehat\infty}{\widehat\infty}=1$ property of Axiom \ref{ax:undefinby} to obtain by association of multiplication and division the expression
1865    \begin{equation*}
1866    z=0\cdot\widehat\infty=\cfrac{c}{\widehat\infty}\cdot\widehat\infty=c\cdot \cfrac{\widehat\infty}{\widehat\infty}=c~~.
1867    \end{equation*}
1868   
1869    \noindent This shows that $0\cdot\widehat\infty$ is not a well-defined operation because $z=c$ is not a unique output.  When we define division as a third operation beyond multiplication and addition, we should not assume associativity among the distinct divisive and multiplicative operations, and neither will we axiomatize it in Section \ref{sec:aritAX}.  Without assumed associativity among the terms, we cannot show that $z$ fails to be a well-defined output of the product $0\cdot\widehat\infty$.  In that case, we will assume there is no problem with the definition $0\cdot\widehat\infty=0$.
1870\end{exa}
1871 
1872 
1873 
1874 
1875\subsection{Arithmetic Axioms for Real Numbers in Natural Neighborhoods}\label{sec:aritAX}
1876 
1877When one defines $\mathbb{R}$ such that the set $\mathcal{R}=\{\mathbb{R},+,\times,\leq\}$ conforms the field axioms, it is a natural progression to prove that Cauchy equivalence classes satisfy the field axioms.  We do \textit{not} presently presume that $\mathbb{R}$ is such that $\mathcal{R}$ obeys the field axioms so we will not make any such proofs.  Instead, we will list in this section the axiomatized arithmetic operations obeyed by numbers whose little parts are less than some natural number.  For disambiguation with the well-known ``field axioms,'' the axioms given in this section are called the ``arithmetic axioms.''  In Section \ref{sec:cons}, we will make proofs of certain operations given in these arithmetic axioms, and give examples.  In Section \ref{sec:consXXX}, we will define the operations in terms of the numbers' underlying equivalence classes.  All of the axioms given here pertain only to the natural neighborhoods $\mathbb{R}_0^\mathcal{X}$.  When we give the treatment leading to  $\mathbb{R}_\aleph^\mathcal{X}\setminus\mathbb{R}_0^\mathcal{X}=\varnothing$ (Section \ref{sec:bbbhh}), these axioms will be fairly comprehensive.  However, when impose the usual topology on $\mathbb{R}$ in Section \ref{sec:topo}, we will find that these axioms are not yet totally comprehensive.  
1878 
1879The equivalence class constructions given in the preceding sections were only for natural neighborhoods  and here we will follow with the axiomatized arithmetic for the elements of those neighborhoods.  Almost everything about the field axioms shall be preserved in the natural neighborhoods.  The major exception is that we will not enforce the global closure of $\mathbb{R}$ under its operations.  Among the other departures from the field axioms will be the identification of division as a separate operation from multiplication by an inverse.  Closure is nice for group theoretical applications but it is not needed for most applications in arithmetic.  For example, the set $\{3,4,5;+\}$ is not closed under addition and yet it remains a perfectly sound algebraic structure with which we can do summation mathematics in the usual way.  If one were to claim, ``Non-closure doesn't break arithmetic because $\{3,4,5;+\}$ is a subset of $\{\mathbb{R};+\}$ which is an algebraic group as defined by the field axioms,'' then we could make an easy rebuttal by defining a set $\mathbb{T}$ to be
1880\begin{equation*}
1881\mathbb{R}\subset\mathbb{T}=\big\{  x~\big|~-\aleph_\infty<x<\aleph_\infty \big\}~~.
1882\end{equation*}
1883 
1884\noindent Then the present convention for non-closed $\{\mathbb{R};+\}$ defined with the Euclidean magnitude (Definition \ref{def:real}) and supplemental arithmetic axioms is such that $\{\mathbb{R};+\}$ is a subset of the closed additive group of 1D transfinitely continued real numbers $\{\mathbb{T};+\}$.  
1885 
1886 
1887 
1888\begin{axio}\label{ax:fieldssss33}
1889    All $\mathbb{R}_0$ numbers obey the well-known axioms of a complete ordered field: Axioms \ref{ax:fieldplus}, \ref{ax:fieldtimes}, and \ref{ax:fieldord}.
1890\end{axio}
1891 
1892\begin{rem}
1893    To make a distinction between the intermediate neighborhoods of infinity and the maximal neighborhood, in this section we will use the symbol $\widehat\infty$ rather than the symbol $\aleph_1$.  However, the reader should note that the arithmetic of the maximal neighborhood follows from the arithmetic of the intermediate neighborhoods as a special case of $\aleph_\mathcal{X}$ with $\mathcal{X}=1$.
1894\end{rem}
1895 
1896\begin{axio}\label{ax:plus}
1897    Addition is commutative and associative.  There exists an additive identity element $0$ and an additive inverse $x^{-1}$ for every $x\in\mathbb{R}$.  The operations for  $+$ are given as follows when $a,b,x,y\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.
1898        \begin{center}{\normalsize
1899            \begin{longtable}{| c || c | c | c | c |  }
1900                \hline
1901                  +& 0 & $~y\in\mathbb{R}_0~$ & $~\big(\aleph_\mathcal{Y}+ a\big)\in\mathbb{R}_0^\mathcal{X}~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $~\big(\widehat\infty- |a|\big)\in\mathbb{R}_0^1\cup\widehat\infty~   $  \\ [4pt]
1902                \hline\hline
1903                 $0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $&0& $y$  &$\aleph_\mathcal{Y}+ a$ & $\widehat\infty- |a|$ \\ [4pt]
1904                \hline
1905                 $~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $x$ & $x+y$ &$\aleph_\mathcal{Y}+ \big(a+ x\big)$ &$\widehat\infty- \big(|a|- x\big)$ \\[4pt]
1906                \hline
1907                 $~\big(\aleph_\mathcal{X}+ b\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $  & $\aleph_\mathcal{X}+ b$ & $\aleph_\mathcal{X}+ \big(b+y\big)$ &$\aleph_{(\mathcal{X+Y})}+ \big(b+a\big)$ &$\aleph_{(\mathcal{X}+1)}+ \big(b-|a|\big)$\\[4pt]
1908                \hline
1909                  $~\big(\widehat\infty- |b|\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}}   $   & $\widehat\infty- |b|$ &   $\widehat\infty- \big(|b|-y\big)$ &$\aleph_{(1+\mathcal{ Y})}- \big(|b|-a\big)$  &$\aleph_{2}- \big(|b|+|a|\big)$\\[4pt]
1910                \hline
1911            \end{longtable}     }
1912    \end{center}
1913\end{axio}  
1914 
1915\begin{rem}
1916    The most important property given by Axiom \ref{ax:plus} is
1917    \begin{equation*}
1918    \aleph_\mathcal{X}+\aleph_\mathcal{Y}=\aleph_{(\mathcal{X}+\mathcal{Y})}~~.
1919    \end{equation*}
1920   
1921    \noindent This equality follows from the geometric notion of addition.  If, for instance, $\aleph_\mathcal{X}$ is a number with 10\% fractional distance along $\mathbf{AB}$ and $\aleph_\mathcal{Y}$ is a number with 20\% fractional distance, then it follows that their sum is a number with 30\% fractional distance along $\mathbf{AB}$.  Axiom \ref{ax:plus} makes clear that $\mathbb{R}$ does not satisfy the usual understanding that the reals are closed under their operations.  Any number $\aleph_\mathcal{X}+b$ with $\mathcal{X}>1$ is not a real number.  For example, the sum of two positive numbers with $99\%$ fractional magnitude is not a real number; no $x$ with big part $\aleph_{1.98}$ can be $x\in\mathbb{R}$.  
1922\end{rem}
1923 
1924 
1925\begin{axio}\label{ax:1g1g1g1}
1926    Multiplication is commutative and associative, and it is distributive over addition.  It is not associative with division (which shall not be defined as multiplication by an inverse.)  There exists a multiplicative identity $1\neq0$ for every $x\in\mathbb{R}$ but there does not exist a multiplicative inverse for all $x\in\mathbb{R}$.  The operations for  $\{\cdot\}=\{\times\}$ are given as follows when $a,b\in\mathbb{R}_0$, $x,y\in\mathbb{R}_0^+$, and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.
1927        \begin{center}{\normalsize
1928            \begin{longtable}{| c || c | c | c | c | c | }
1929                \hline
1930                $\times$& 0 &$ \mp 1$&$~y\in\mathbb{R}_0^+~$ & $~\big(\aleph_\mathcal{Y}+ a\big)\in\mathbb{R}_0^\mathcal{X}~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $\big(\widehat\infty- |a|\big)\!\in\!\mathbb{R}_0^1\cup\widehat\infty  $  \\ [4pt]
1931                \hline\hline
1932                $0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $&0& $0$  &  $0$  &$0$ & $0$\\ [4pt]
1933                \hline
1934                $\pm 1 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $&0& $-1$  &  $\pm y$  & $\aleph_{(\pm\mathcal{Y})}\pm a$ & $\pm\widehat\infty\mp |a|$\\ [4pt]
1935                \hline
1936                $~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $0$ & $\mp x$&$xy$ &$\aleph_{(x\mathcal{Y})}+ ax$&$\aleph_x-|a|x$ \\[4pt]
1937                \hline
1938                $~\big(\aleph_\mathcal{X}+ b\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $  & $0$ & $\aleph_{(\mp\mathcal{X})}\mp b$ & $\aleph_{(\mathcal{X}y)}+ by$ &$\aleph_{(\aleph_\mathcal{XY}+a\mathcal{X}+b\mathcal{Y})}+ ba$ &$\aleph_{(\aleph_\mathcal{X}-|a|\mathcal{X}+b)}- b|a|$ \\[4pt]
1939                \hline
1940                $~\big(\widehat\infty- |b|\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}}   $   &$0$ & $ \mp\widehat\infty\pm |b|$&   $\aleph_{y}- |b|y$ &$\aleph_{(\aleph_\mathcal{Y}+a -|b|\mathcal{Y})}- |b|a$&$\aleph_{(\widehat\infty-|a|-|b|)}+|ba|$ \\[4pt]
1941                \hline
1942            \end{longtable} }
1943        \end{center}
1944\end{axio}
1945 
1946 
1947\begin{rem}
1948    The most important property given in Axiom \ref{ax:1g1g1g1} is
1949    \begin{equation*}
1950    \pm\aleph_{\mathcal{X}}=\aleph_{(\pm\mathcal{X})}~~.
1951    \end{equation*}
1952   
1953    \noindent This operation follows from
1954    \begin{equation*}
1955    \aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty\quad\implies\quad \pm\aleph_\mathcal{X}=\pm\big(\mathcal{X}\cdot\widehat\infty\big)=\big(\pm\!\mathcal{X}\big)\cdot\widehat\infty=\aleph_{(\pm\mathcal{X})}~~.
1956    \end{equation*}
1957   
1958    \noindent This shows that multiplication is axiomatically associative.  Certain of the products in Axiom \ref{ax:1g1g1g1} rely on Axiom \ref{ax:plus}.  For instance, the value in the lower right corner of the table is computed as
1959    \begin{align*}
1960    \big(\widehat\infty- |b|\big)\big(\widehat\infty- |a|\big)&=\widehat\infty\cdot\widehat\infty-|b|\widehat\infty-|a|\widehat\infty+|ba|\\
1961    &=\aleph_1\cdot\aleph_1-|a|\aleph_1-|b|\aleph_1+|ba|\\
1962    &=\aleph_{(\aleph_1)}-\aleph_{|a|}-\aleph_{|b|}+|ba|\\
1963    &=\aleph_{(\aleph_1)}-\big(\aleph_{|a|}+\aleph_{|b|}\big)+|ba|\\
1964    &=\aleph_{\widehat\infty}+ \aleph_{(-|a|-|b|)}+|ba|\\
1965    &=\aleph_{(\widehat\infty-|a|-|b|)} +|ba|~~.
1966    \end{align*}
1967\end{rem}
1968 
1969 
1970 
1971\begin{rem}
1972    It follows from Axioms \ref{ax:plus} and \ref{ax:1g1g1g1} that
1973    \begin{align*}
1974    \big(\widehat\infty-b\big)-\big(\widehat\infty-a\big)&=a-b ~~.
1975    \end{align*}
1976   
1977    \noindent This is the primary operation behind the original ideation for a non-absorptive infinite element.  If $a$ and $b$ are two numbers at distances $a$ and $b$ respectively from the endpoint $0$ of the interval $[0,\infty]$, then their difference $a-b$ must be equal to the difference of two numbers lying at distances $a$ and $b$ from the endpoint $\infty$ of the same interval.
1978\end{rem}
1979 
1980 
1981 
1982\begin{exa}
1983    The purpose of this example is to demonstrate that even when numbers greater than $\widehat\infty$ do not exist in real analysis, expressions implying the existence of such are numbers are generally not considered contradictory.  Consider the quadratic equation
1984    \begin{equation*}
1985    ax^2+bx+c=0~~,
1986    \end{equation*}
1987   
1988    \noindent having roots
1989    \begin{equation*}
1990    x=\frac{-b\pm\sqrt{\vphantom{\hat{B}}b^2-4ac}}{2a}~~.
1991    \end{equation*}
1992   
1993    \noindent For every case in which $4ac>b^2$, the number $x$ does not exist in real analysis and yet it is never claimed that the quadratic formula is contradictory.  Instead, we claim that there must exist an imaginary number $i\not\in\mathbb{R}$ with the property $i=\sqrt{-1}$.  Therefore, the principle of fractional distance should support a conclusion that there exist transfinite numbers $x\not\in\mathbb{R}$ with the property that $x>\widehat\infty$.  We have seen the existence of such numbers implied previously when examining algebraic infinity as the endpoint of a line segment embedded in a line extending infinitely far in both directions.  If we use $x=\tan(x')$ to define $\mathbf{AB}\equiv[0,\widehat\infty]$ on $AB\equiv[0,\frac{\pi}{2}]$, and if a number is a cut in a line as per Definition \ref{def:real}, then there should exist non-real transfinite numbers which are cuts in an infinite line lying to the right of $x=\widehat\infty$ in the algebraic representation of the point $B$.  
1994\end{exa}
1995 
1996\begin{rem}
1997    When the field axioms give the arithmetic operations of $\mathbb{R}$, the difference operations follow from the sum operations as the addition of a product with $-1$.  The quotient operations usually follow from the $\times$ operations as multiplication by an inverse.  Presently we may define the difference operations accordingly but we may not do so for the $\div$ operations.  As demonstrated in Example \ref{ex:788788m}, the preservation of the respective geometric notions of the algebraic operations---namely $\aleph_0=0\times\widehat\infty$---requires that $\{+,\cdot\,,\div\}$ is a set of three distinct arithmetic operations among which there is not mutual associativity.  Obviously, this is a major distinction of the present axioms from the field axioms.  However, Axiom \ref{ax:fieldssss33} grants that $x\in\mathbb{R}_0$ obey the usual field axioms so there is an implicit axiom regarding the associativity of $\{\times,\div\}$ which we will make explicit in Axiom \ref{ax:dssfgss33}
1998\end{rem}
1999 
2000\begin{axio}\label{ax:dssfgss33}
2001    Division and multiplication are mutually associative for any $x\in\mathbb{R}_0$.  That is, all factors which are elements of $\mathbb{R}_0$ may be moved into or out of quotients in the usual way, even if those quotients contain $x\not\in\mathbb{R}_0$.
2002\end{axio}
2003 
2004\begin{axio}\label{ax:div1g1g1g1}
2005    The operations for  $\div$ are given as follows when $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.  There exists a divisive identity $1\neq0$ for every $x\in\mathbb{R}$.  It is the same as the multiplicative identity.  There exist at least one divisive inverse for every non-zero $x\in\mathbb{R}$.  In this table the row value is the numerator and the column value is the denominator.
2006        \begin{center}{\normalsize
2007            \begin{longtable}{| c || c | c | c | c | }
2008                \hline
2009                $\div$& 0 &$~y\in\mathbb{R}_0~$ & $~\big(\aleph_\mathcal{Y}+ a\big)\in\mathbb{R}_0^\mathcal{X}~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $~\big(\widehat\infty- |a|\big)\in\mathbb{R}_0^1\cup\widehat\infty~   $  \\ [4pt]
2010                \hline\hline
2011                $0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $& nan  & $0$  & $0$  &   $0$\\ [4pt]
2012                \hline
2013                $~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ &  nan &  $\frac{x}{y}$ & 0  &  0 \\[4pt]
2014                \hline
2015                $~\big(\aleph_\mathcal{X}+ b\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $  & nan  & $\aleph_{(\mathcal{X}y^{-1})}+\frac{b}{y}$  &  $\frac{\mathcal{X}}{\mathcal{Y}}$  &  $\mathcal{X}$  \\[4pt]
2016                \hline
2017                $~\big(\widehat\infty- |b|\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}}   $   & nan  & $\aleph_{(y^{-1})}-\frac{|b|}{y}$    & $\frac{1}{\mathcal{Y}}$  & $1$  \\[4pt]
2018                \hline
2019            \end{longtable}  }
2020        \end{center}
2021\end{axio}
2022 
2023 
2024 
2025\begin{exa}
2026    This example demonstrates that the quotient operations given by Axiom \ref{ax:div1g1g1g1} are well-defined.  (This is proven rigorously in Main Theorem \ref{thm:3111r22424}.)  An operation is well-defined if it generates a unique output.  It is obvious in Axiom \ref{ax:div1g1g1g1} that each operation has one and only one output.  It is foreign to the usual understanding of the arithmetic of real numbers, however, that the operands giving the unique resultants are not themselves unique.  Consider
2027    \begin{equation*}
2028    \cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=\cfrac{\mathcal{X}}{\mathcal{Y}}~~.
2029    \end{equation*}
2030   
2031    \noindent If multiplication was associative with division, and vice versa, then we could multiply both sides by $\aleph_\mathcal{Y}+a$ to obtain a contradiction of the form
2032   
2033    \begin{align*}
2034    \cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}\cdot\big(\aleph_\mathcal{Y}+a\big)&=\cfrac{\mathcal{X}}{\mathcal{Y}}\cdot\big(\aleph_\mathcal{Y}+a\big)\\
2035    \aleph_\mathcal{X}+b&= \aleph_\mathcal{X}+\cfrac{\mathcal{X}a}{\mathcal{Y}}~~.
2036    \end{align*}
2037   
2038    \noindent This is false whenever $b\neq\frac{\mathcal{X}a}{\mathcal{Y}}$ but it is not possible to show this contradiction without assuming associativity among $\{\times,\div\}$.
2039\end{exa}
2040 
2041\begin{exa}
2042    This example demonstrates another immediate contradiction should we assume associativity among multiplication and division.  Axiom \ref{ax:div1g1g1g1} gives
2043    \begin{equation*}
2044    \cfrac{\aleph_\mathcal{Y}}{\aleph_\mathcal{X}}=\cfrac{\mathcal{Y}}{\mathcal{X}}~~,\quad\text{and}\qquad\cfrac{1}{\aleph_\mathcal{X}}=0~~.
2045    \end{equation*}
2046   
2047    \noindent If we bestow the associativity, then
2048    \begin{equation*}
2049    \cfrac{\aleph_\mathcal{Y}}{\aleph_\mathcal{X}}=\aleph_\mathcal{Y}\cdot\cfrac{1}{\aleph_\mathcal{X}}=\aleph_\mathcal{Y}\cdot0=0\neq\cfrac{\mathcal{Y}}{\mathcal{X}}~~.
2050    \end{equation*}
2051\end{exa}
2052 
2053 
2054 
2055\begin{axio}\label{ax:order}
2056    The ordering of $\mathbb{R}$ is given as follows when $a,b,c,dx,y\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.  For the table, it is granted that
2057    \begin{align*}
2058    a&>b\\
2059    c&>d>0\\
2060    x&>y\\
2061    \mathcal{X}&>\mathcal{Y}.
2062    \end{align*}
2063   
2064     
2065    \noindent This table is such that the row identity is on the left of the given relation and the column identity is on the right.
2066     
2067    \begin{center}{\normalsize
2068            \begin{longtable}{| c || c | c | c | c | c | }
2069                \hline
2070                $\leq$ & $~y\in\mathbb{R}_0~$ & $ \big(\aleph_\mathcal{Y}+ b\big)\in\mathbb{R}_0^\mathcal{Y} $ & $~\big(\aleph_\mathcal{X}+ b\big)\in\mathbb{R}_0^\mathcal{X}  \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $  & $ \big(\widehat\infty- |d|\big)\in\mathbb{R}_0^1  $&$\widehat\infty$ \\ [4pt]
2071                \hline\hline  
2072                $~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$<$ & $<$ &$<$  &$<$   \\[4pt]
2073                \hline
2074                $~\big(\aleph_\mathcal{X}+a\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$>$ & $>$ & $<$ &$<$    \\[4pt]
2075                \hline
2076                $~\big(\widehat\infty-|c|\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$>$ & $>$ & $<$ &$<$   \\[4pt]
2077                \hline
2078        \end{longtable}     }
2079    \end{center}
2080\end{axio}  
2081 
2082 
2083\begin{thm}\label{thm:noinv}
2084    Real numbers in the intermediate natural neighborhoods of infinity $x\in\{\mathbb{R}^\mathcal{X}_0\}$ do not have a multiplicative inverse.
2085\end{thm}
2086 
2087\begin{proof}  
2088    The number $x^{-1}$ is the multiplicative inverse of $x\in\mathbb{R}$ if and only if
2089    \begin{equation*}
2090    x\cdot x^{-1}=x^{-1}\cdot x=1~~.
2091    \end{equation*}
2092   
2093    \noindent The statement of the theorem requires that $x=\aleph_\mathcal{X}+b$, that $0<\mathcal{X}<1$, and that $b\in\mathbb{R}_0$.  Axiom \ref{ax:1g1g1g1} grants that multiplication is distributive over addition so the definition of the multiplicative inverse requires
2094    \begin{equation*}
2095    \big(\aleph_\mathcal{X}+b\big)x^{-1}=\aleph_{(\mathcal{X}x^{-1})}+bx^{-1}=1~~.
2096    \end{equation*}
2097   
2098    \noindent Equating the big and little parts of this expression, we obtain two requirements
2099    \begin{equation*}
2100    \aleph_{(\mathcal{X}x^{-1})}=\aleph_0\quad\iff\quad\mathcal{X}x^{-1}=0\quad\iff\quad x^{-1}=0~~,
2101    \end{equation*}
2102   
2103    \noindent and
2104    \begin{equation*}
2105    bx^{-1}=1\quad\iff\quad x^{-1}=\frac{1}{b}~~.
2106    \end{equation*}
2107   
2108    \noindent This contradicts the requirement $b\in\mathbb{R}_0$ and so, therefore, $x\in\{\mathbb{R}^\mathcal{X}_0\}$ does not have a multiplicative inverse.
2109\end{proof}
2110 
2111 
2112\begin{thm}\label{thm:noinvplus}
2113    All real numbers $x\in\{\mathbb{R}^\mathcal{X}_0\}$ have an additive inverse.
2114\end{thm}
2115 
2116\begin{proof}
2117    The number $x^{-1}$ is the additive inverse of $x$ if and only if
2118    \begin{equation*}
2119    x+x^{-1}=x^{-1}+ x=0~~.
2120    \end{equation*}
2121   
2122    \noindent The statement of the theorem requires that $x=\aleph_\mathcal{X}+b$, that $0<\mathcal{X}<1$, and that $b\in\mathbb{R}_0$.  Assume that $x^{-1}$ has the form $\aleph_{(\mathcal{X}^{-1})}+b^{-1}$.  The definition of the additive inverse requires
2123    \begin{equation*}
2124    1=\big(\aleph_{\mathcal{X}}+b\big)+\big(\aleph_{(\mathcal{X}^{-1})}+b^{-1}\big)=\aleph_{(\mathcal{X}+\mathcal{X}^{-1})}+(b+b^{-1})~~.
2125    \end{equation*}
2126   
2127    \noindent Equating the big and little parts of this expression, we obtain two requirements
2128    \begin{equation*}
2129    \aleph_{(\mathcal{X}+\mathcal{X}^{-1})}=\aleph_0\quad\iff\quad\mathcal{X}+\mathcal{X}^{-1}=0\quad\iff\quad \mathcal{X}^{-1}=-\mathcal{X}~~,
2130    \end{equation*}
2131   
2132    \noindent and
2133    \begin{equation*}
2134    b+b^{-1}=1\quad\iff\quad b^{-1}=-b~~.
2135    \end{equation*}
2136   
2137    \noindent For every $[\mathcal{X}],[b]\subset C_\mathbb{Q}$ there exists a $[-\mathcal{X}],[-b]\subset C_\mathbb{Q}$ so, therefore, every $x\in\{\mathbb{R}^\mathcal{X}_0\}$ has an additive inverse.
2138\end{proof}
2139 
2140\begin{defin}
2141    A divisive identity is a number $e$ satisfying $x\div e=x$.  The divisive identity element of $\mathbb{R}$ is $1\in\mathbb{R}_0$.
2142\end{defin}
2143 
2144 
2145\begin{thm}\label{thm:noinvdiv}
2146    All real numbers $x\in\{\mathbb{R}^\mathcal{X}_0\}$ have a non-unique divisive inverse.
2147\end{thm}
2148 
2149\begin{proof}
2150    If $x^{-1}$ is the divisive inverse of $x$, then $x\div x^{-1}=1$.  By Axiom \ref{ax:div1g1g1g1}, any two $x\in\{\mathbb{R}^\mathcal{X}_0\}$ having equal big parts are mutual divisive inverses.
2151\end{proof}
2152 
2153 
2154 
2155\subsection{Limit Considerations Regarding the Arithmetic Axioms}\label{sec:cons}
2156 
2157We have not directly defined $\widehat\infty$ with the limit definition of infinity.  Instead, we have defined infinity hat to have the same absolute value as infinity so that they are both the unincluded endpoint of the interval $[0,\mathcal{I})$ where $\mathcal{I}\not\in\mathbb{R}$ has the property that it is larger than any real number.  Although we began this paper with the notion of $\mathbf{AB}\equiv[0,\infty]$, by the introduction of the semantic conventions regarding geometric and algebraic infinity, we would now say that $\infty$ cannot be included as an endpoint so that $[0,\widehat\infty)=[0,\infty)$ but, informally, $[0,\widehat\infty]\neq[0,\infty]$ because the latter closed interval contradicts the notion of infinite geometric extent.  In general, we have only introduced this convention as a thinking device and there is no reason to directly forbid the usual extended real interval $\overline{\mathbb{R}}=[-\infty,\infty]$.  Rather we have only shown that it is better to write $\overline{\mathbb{R}}=[-\widehat\infty,\widehat\infty]$ because it doesn't suggest the non-existence of the neighborhood of infinity.
2158 
2159So, although we have not defined $\widehat\infty$ directly with the limit definition of $\infty$, having instead deduced its existence from the geometric invariance of line segments under permutations of the labels of their endpoints, it remains that the magnitude of $\widehat\infty$ is given by the limit definition.  Since the identity of real numbers is identically their magnitude, and it is only two alternative sets of arithmetic axioms which separate $\infty$ and $\widehat\infty$, in this section we will study the compliance of the limit definition of infinity with the arithmetic axioms.
2160 
2161\begin{exa}\label{ex:ewee}
2162    Although the limit definition of $\infty$ is said to be its identical definition, we cannot always substitute the limit definition of infinity to directly compute all expressions involving geometric infinity.     Consider the use of Definition \ref{def:RRRinf} to write
2163    \begin{equation*}
2164    \infty-\infty=\left(\lim\limits_{x\to0}\dfrac{1}{x}\right)-\left(\lim\limits_{y\to0}\dfrac{1}{y}\right)=\lim\limits_{\substack{x\to0\\y\to0}} \frac{y-x}{xy}~~.
2165    \end{equation*}
2166   
2167    \noindent Generally, this limit does not exist because we obtain different results on the lines $y=x$ and $y=2x$.  Presently, however, there is only one possible line: the real number line.  By making the substitution for the limit definition, we find, therefore, that
2168    \begin{equation*}
2169    \infty-\infty=\left(\lim\limits_{x\to0}\dfrac{1}{x}\right)-\left(\lim\limits_{x\to0}\dfrac{1}{x}\right)=\lim\limits_{ x\to0 } \left(\frac{1}{x}-\frac{1}{x}\right)=\lim\limits_{ x\to0 }0=0 ~~.
2170    \end{equation*}
2171   
2172    \noindent This contradicts Axiom \ref{ax:undefin7666y} which gives
2173    \begin{align*}
2174    \infty-\infty=\text{undefined}~~.
2175    \end{align*}
2176   
2177    \noindent To the contrary, if we should examine $\widehat\infty-\widehat\infty$ under the ansatz that this expression may be computed with the limit definition, then we find
2178    \begin{equation*}
2179    \widehat\infty-\widehat\infty=\left(\lim\limits_{x\to0}\dfrac{1}{x}\right)-\left(\lim\limits_{x\to0}\dfrac{1}{x}\right)=\lim\limits_{ x\to0 } \left(\frac{1}{x}-\frac{1}{x}\right)=\lim\limits_{ x\to0 }0=0 ~~.
2180    \end{equation*}
2181   
2182    \noindent This is exactly what is given in Axiom \ref{ax:undefinby} so the ansatz is borne out.  
2183\end{exa}
2184 
2185\begin{rem}
2186    Example \ref{ex:ewee} has demonstrated that although $\infty$ is directly defined with the limit definition of infinity, we cannot always use that definition to simply $\infty$'s expressions and that, sometimes, we \textit{can} use it to simplify the expressions of $\widehat\infty$.  In the present section, as in Example \ref{exa:588585557aaa}, we will take the hat on $\widehat\infty$ as a constraint on the freedom of algebraic manipulations involving the limit expression.  Particularly, the non-absorptivity of $\widehat\infty$ allows us to combine limit expressions but forbids us moving any scalars into the limit expressions.  The main purpose of this section is to demonstrate cases of the validity of the ansatz that sometimes we can correctly compute expressions involving $\widehat\infty$ by making the direct substitution with the limit definition.
2187\end{rem}
2188 
2189 
2190 
2191\begin{thm}\label{thm:kksjdj1}
2192    The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1} giving for $a,b\in\mathbb{R}_0^+$
2193    \begin{equation*}
2194    \big(\widehat\infty-b\big)-\big(\widehat\infty-a\big)=a-b~~,
2195    \end{equation*}
2196   
2197    \noindent follows from the limit definition of infinity.
2198\end{thm}
2199 
2200\begin{proof}
2201    Proof follows from direct substitution of the limit definition of infinity (Definition \ref{def:RRRinf}.)  We have
2202    \begin{align*}
2203    \big(\widehat\infty-b\big)-\big(\widehat\infty-a\big)&=\left[\left(\lim\limits_{x\to0}\frac{1}{x}\right)-b\right]-\left[\left(\lim\limits_{x\to0}\frac{1}{x}\right)-a\right]\\
2204    &=\lim\limits_{x\to0}\left(\frac{1}{x}-b-\frac{1}{x}+a\right)\\
2205    &=\lim\limits_{x\to0}\big(-b+a\big)\\
2206    &=a-b~~.
2207    \end{align*}
2208\end{proof}
2209 
2210 
2211 
2212\begin{thm}\label{thm:kksjdj11}
2213    The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2214    \begin{equation*}
2215    \big(\aleph_\mathcal{X}+b\big)-\big(\aleph_\mathcal{Y}+a\big)=
2216    \aleph_{(\mathcal{X}-\mathcal{Y})}-a+b~~,
2217    \end{equation*}
2218   
2219    \noindent follows from the limit definition of infinity.
2220\end{thm}
2221 
2222\begin{proof}
2223    Proof follows from direct substitution of the limit definition of infinity (Definition \ref{def:RRRinf}.)  We have
2224    \begin{align*}
2225    \big(\aleph_\mathcal{X}+b\big)-\big(\aleph_\mathcal{Y}+a\big)&=\big(\mathcal{X}\,\widehat\infty-b\big)-\big(\mathcal{Y}\,\widehat\infty-a\big)\\
2226    &=\left[\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)+b\right]-\left[\mathcal{Y}\left(\lim\limits_{x\to0}\frac{1}{x}\right)+a\right]\\
2227    &=\big(\mathcal{X}-\mathcal{Y}\big)\left(\lim\limits_{x\to0}\frac{1}{x}\right)-a+b\\
2228    &=\big(\mathcal{X}-\mathcal{Y}\big)\widehat\infty-a+b\\
2229    &=\aleph_{(\mathcal{X}-\mathcal{Y})}-a+b~~.
2230    \end{align*}
2231\end{proof}
2232 
2233\begin{rem}
2234    Theorem \ref{thm:kksjdj11} requires clarification because we might have written
2235    \begin{align*}
2236    \big(\aleph_\mathcal{X}+b\big)-\big(\aleph_\mathcal{Y}+a\big)&=\big(\mathcal{X}\,\widehat\infty+b\big)-\big(\mathcal{Y}\,\widehat\infty+a\big)\\
2237    &=\left[\left(\lim\limits_{x\to0}\frac{\mathcal{X}}{x}\right)+b\right]-\left[\left(\lim\limits_{x\to0}\frac{\mathcal{Y}}{x}\right)+a\right]\\
2238    &= \left(\lim\limits_{x\to0}\frac{\mathcal{X}-\mathcal{Y}}{x}\right)-a+b\\
2239    &=\widehat\infty-a+b~~.
2240    \end{align*}
2241   
2242    \noindent Since $\widehat\infty=\aleph_1$, this would necessarily be a contradiction.  The condition $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})\leq1$ forbids $\mathcal{X}-\mathcal{Y}=1$.  In the above algebraic manipulation, we have given at the second step
2243    \begin{equation*}
2244    \aleph_\mathcal{X}=\mathcal{X}\,\widehat\infty=\lim\limits_{x\to0}\frac{\mathcal{X}}{x}~~.
2245    \end{equation*}
2246   
2247    \noindent This contradicts Definition \ref{def:hat33} requiring that $\widehat\infty$ does not have absorptive properties.  Such a property is explicitly bestowed to the limit definition of infinity when we move the scalar $\mathcal{X}$ into the limit expression.  Therefore, it is implicit in the axioms that scalar multipliers of $\widehat\infty$ must not be transferred by multiplicative association into the limit expression when substituting the limit definition of algebraic infinity $\widehat\infty$.  In practice, this has little to no relevance because arithmetic follows from the arithmetic axioms rather than the limit definition of infinity.  The purpose of the present section, rather, is to show that at least many of the axioms may be derived from the limit definition, and that \textit{\textbf{the present axiomatic framework is very strong}} because many of its axioms are directly provable when we assume the usual associativities, commutativities, and distributivities constrained by the rules of non-absorptivity.  
2248\end{rem}
2249 
2250 
2251 
2252 
2253\begin{thm}\label{thm:kksjdj22}
2254    The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0^+$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})\leq1$
2255    \begin{equation*}
2256    \big(\aleph_\mathcal{X}+b\big)\cdot a=\aleph_{(\mathcal{X}a)}+ba~~,
2257    \end{equation*}
2258   
2259    \noindent follows from the limit definition of infinity.
2260\end{thm}
2261 
2262\begin{proof}
2263    Proof follows from direct substitution of the limit definition of infinity.  We have
2264    \begin{align*}
2265    \big(\aleph_\mathcal{X}+b\big)\cdot a&=\big(\mathcal{X}\,\widehat\infty+b\big)\cdot a\\
2266    &=\left[\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)-b\right]\cdot a\\
2267    &=\mathcal{X}a \left(\lim\limits_{x\to0}\frac{1}{x}\right)+ba\\
2268    &=\mathcal{X}a\widehat\infty+ba\\
2269    &=\aleph_{(\mathcal{X}a)}+ba~~.
2270    \end{align*}
2271\end{proof}
2272 
2273 
2274\begin{thm}\label{thm:kksjdj21}
2275    The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2276    \begin{equation*}
2277    \big(\aleph_\mathcal{X}-b\big)\cdot\big(\aleph_\mathcal{Y}-a\big)=\aleph_{(\aleph_{(\!\mathcal{XY}\!)}+a\mathcal{X}+b\mathcal{Y})}+ ba~~,
2278    \end{equation*}
2279   
2280    \noindent follows from the limit definition of infinity.
2281\end{thm}
2282 
2283\begin{proof}
2284    Proof of the present theorem follows from direct substitution of the limit definition of infinity.  We have
2285    \begin{align*}
2286    \big(\aleph_\mathcal{X}-b\big)\big(\aleph_\mathcal{Y}-a\big)&=\big(\mathcal{X}\,\widehat\infty-b\big)\big(\mathcal{Y}\,\widehat\infty-a\big)\\
2287    &=\left[\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)-b\right]\left[\mathcal{Y}\left(\lim\limits_{x\to0}\frac{1}{x}\right)-a\right]\\
2288    &= \mathcal{XY} \left(\lim\limits_{x\to0}\frac{1}{x}\right)^{\!2}-a\left(\lim\limits_{x\to0}\frac{1}{x}\right)-b\left(\lim\limits_{x\to0}\frac{1}{x}\right)+ba
2289    \end{align*}   
2290   
2291    \noindent If we wrote here
2292    \begin{equation*}
2293    \widehat\infty\cdot\widehat\infty=\left(\lim\limits_{x\to0}\frac{1}{x}\right)^{\!2}=\lim\limits_{x\to0}\frac{1}{x^2}=\widehat\infty~~,
2294    \end{equation*}
2295   
2296    \noindent then that would not exactly violate Definition \ref{def:hat33} because it shows infinity absorbing itself while Definition \ref{def:adabs} gives the the multiplicative absorptive property in terms of a composition between $\widehat\infty$ and $x\in\mathbb{R}$.  However, moving the exponent into the limit violates Definition \ref{def:ugy8re8r7777} requiring that
2297    \begin{equation*}
2298    \widehat\infty\cdot\widehat\infty=\widehat\infty\cdot\aleph_1=\aleph_{\widehat\infty}\neq\aleph_1=\widehat\infty~~.
2299    \end{equation*}
2300   
2301    \noindent Therefore, we finish the proof as
2302    \begin{align*}
2303    \big(\aleph_\mathcal{X}-b\big)\big(\aleph_\mathcal{Y}-a\big)a&=\mathcal{XY}\left(\lim\limits_{x\to0}\frac{1}{x}\right)\aleph_{1 } -a\aleph_1-b\aleph_1+ba\\
2304    &=\aleph_{\mathcal{XY}\left(\lim\limits_{x\to0}\frac{1}{x}\right) } -\aleph_a-\aleph_b+ba\\
2305    &=\aleph_{\mathcal{XY}}\cdot\infty-\aleph_{a+b}+ba\\
2306    &=\aleph_{(\aleph_{(\!\mathcal{XY}\!)}+a\mathcal{X}+b\mathcal{Y})}+ ba~~.
2307    \end{align*}
2308\end{proof}
2309 
2310 
2311 
2312 
2313 
2314\begin{thm}\label{thm:kksgjdj}
2315    The property of Axiom \ref{ax:div1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2316    \begin{equation*}
2317    \cfrac{\aleph_\mathcal{X}+b}{\widehat\infty}=\mathcal{X}~~,
2318    \end{equation*}
2319   
2320    \noindent follows from the limit definition of infinity.
2321\end{thm}
2322 
2323 
2324\begin{proof}
2325    We will use the property that $\mathcal{X}\in\mathbb{R}_0$ to allow us move it out of the quotient, as per Axiom \ref{ax:dssfgss33}.  We have
2326    \begin{align*}
2327    \cfrac{\aleph_\mathcal{X}+b}{\widehat\infty}=\cfrac{\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)}{\lim\limits_{x\to0}\frac{1}{x}}+\cfrac{b}{\lim\limits_{x\to0}\frac{1}{x}}=\mathcal{X}\cfrac{\lim\limits_{x\to0}\frac{1}{x}}{\lim\limits_{x\to0}\frac{1}{x}}=\mathcal{X}\lim\limits_{x\to0}1=\mathcal{X}~~.
2328    \end{align*}
2329\end{proof}
2330 
2331 
2332\begin{rem}
2333    The property of Axiom \ref{ax:div1g1g1g1} giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2334    \begin{equation*}
2335    \cfrac{a}{\aleph_\mathcal{X}+b}=0~~,
2336    \end{equation*}
2337   
2338    \noindent does not follow from the limit definition of infinity.If we wrote
2339    \begin{equation*}
2340    \cfrac{a}{\aleph_\mathcal{X}+b}=\cfrac{a}{\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)+b}=\cfrac{a}{\mathcal{X}}\cdot\cfrac{1}{\left(\lim\limits_{x\to0}\frac{1}{x}\right)+\frac{b}{\mathcal{X}}}~~,
2341    \end{equation*}
2342   
2343    \noindent then we would have way to evaluate the quotient without bringing the denominator's $\frac{b}{\mathcal{X}}$ term into the limit expression.  If we did, then the expected zero output would follow directly but moving that term into the limit expression is not allowed because doing so would give $\widehat\infty$ an additive absorptive property.
2344\end{rem}
2345 
2346\begin{thm}\label{thm:g34k3k222}
2347    The quotient of any $\mathbb{R}_0$ number divided by any number with a non-vanishing big part is identically zero.
2348\end{thm}
2349 
2350\begin{proof}
2351    Suppose $x,b\in\mathbb{R}_0^+$ and $0<\mathcal{X}<1$ and that
2352    \begin{equation*}
2353    \cfrac{x}{\aleph_\mathcal{X}+b}=z~~.
2354    \end{equation*}
2355   
2356    \noindent Axiom \ref{ax:dssfgss33} allows us to take $x\in\mathbb{R}_0$ out of the quotient so we may write
2357    \begin{equation*}
2358    \cfrac{1}{\aleph_\mathcal{X}+b}=\cfrac{z}{x}~~.
2359    \end{equation*}
2360   
2361    \noindent The quotient is only well defined for $z=0$. 
2362\end{proof}
2363 
2364 
2365\begin{thm}
2366    Quotients of the form $\mathbb{R}^\mathcal{X}_0\div\mathbb{R}^\mathcal{Y}_0$ are always equal to $\frac{\mathcal{X}}{\mathcal{Y}}$.
2367\end{thm}
2368 
2369\begin{proof}
2370    By Theorem \ref{thm:g34k3k222}, we have
2371    \begin{equation*}
2372    \cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=\cfrac{\aleph_\mathcal{X}}{\aleph_\mathcal{Y}+a}+\cfrac{b}{\aleph_\mathcal{Y}+a}=\cfrac{\aleph_\mathcal{X}}{\aleph_\mathcal{Y}+a}~~.
2373    \end{equation*}
2374   
2375    \noindent If $a=0$, then
2376    \begin{equation*}
2377    \cfrac{\aleph_\mathcal{X}}{\aleph_\mathcal{Y}}=\cfrac{\mathcal{X}\lim\limits_{x\to0}\frac{1}{x}}{\mathcal{Y}\lim\limits_{x\to0}\frac{1}{x}}=\cfrac{\mathcal{X}}{\mathcal{Y}}\cdot\cfrac{\lim\limits_{x\to0}\frac{1}{x}}{\lim\limits_{x\to0}\frac{1}{x}}=\cfrac{\mathcal{X}}{\mathcal{Y}}\cdot \lim\limits_{x\to0}1=\cfrac{\mathcal{X}}{\mathcal{Y}}~~.
2378    \end{equation*}
2379   
2380    \noindent To prove the present theorem in the general case, we will demonstrate a contradiction.  Suppose $c\neq\frac{\mathcal{X}}{\mathcal{Y}}$ and that
2381    \begin{equation*}
2382    \cfrac{\aleph_\mathcal{X}}{\aleph_\mathcal{Y}+a}=c~~.
2383    \end{equation*}
2384   
2385    \noindent Further suppose that $\mathcal{X}<\mathcal{Y}$ so that we may assume $0<c<1$.  Then $c$ has a multiplicative inverse and
2386    \begin{equation*}
2387    \cfrac{\aleph_\mathcal{X}}{\aleph_{(c\mathcal{Y})}+ca}=1~~.
2388    \end{equation*}
2389   
2390    \noindent Then
2391    \begin{equation*}
2392    \lim\limits_{a\to0}\cfrac{\aleph_\mathcal{X}}{\aleph_{(c\mathcal{Y})}+ca}=\cfrac{\aleph_\mathcal{X}}{\aleph_{(c\mathcal{Y})}}=\cfrac{\mathcal{X}}{c\mathcal{Y}}=1\quad\iff\quad c=\cfrac{\mathcal{X}}{\mathcal{Y}}~~.
2393    \end{equation*}
2394   
2395    \noindent It follows that the small part of the denominator does not contribute to the quotient.  The case of $\mathcal{X}>\mathcal{Y}$ follows from the case of $a=0$.  The theorem is proven.
2396\end{proof}
2397 
2398 
2399 
2400 \begin{exa}
2401    This example demonstrates that the associativity of multiplication and division for $\mathbb{R}_0$ numbers such as $c$.  Consider the expression
2402    \begin{equation*}
2403    c\cdot\cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=c\cdot\cfrac{\mathcal{X}}{\mathcal{Y}}=\cfrac{c\mathcal{X}}{\mathcal{Y}}~~.
2404    \end{equation*}
2405   
2406    \noindent If we move $c$ into the quotient and perform the multiplication before the division, then
2407    \begin{equation*}
2408    c\cdot\cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=\cfrac{c\cdot\big(\aleph_\mathcal{X}+b\big)}{\aleph_\mathcal{Y}+a}=\cfrac{\aleph_{(c\mathcal{X})}+cb }{\aleph_\mathcal{Y}+a}=\cfrac{c\mathcal{X}}{\mathcal{Y}}~~,
2409    \end{equation*}
2410   
2411    \noindent demonstrates that the operation remains well-defined with the special associative operations for $\mathbb{R}_0$
2412 \end{exa}
2413 
2414 \begin{exa}
2415    This example treats the negative exponent inverse notation.  We have
2416    \begin{equation*}
2417    \cfrac{x}{\aleph_\mathcal{X}+b}=0 \quad\centernot\implies\quad \cfrac{\aleph_\mathcal{X}+b}{x}=\cfrac{1}{0}~~.
2418    \end{equation*}
2419   
2420    \noindent The usual ``invert and multiply'' rule for dividing by fractions relies on an assumed associativity between multiplication and division, and so it cannot be used in certain cases of numbers with non-vanishing big parts.  We have
2421    \begin{equation*}
2422    \cfrac{\aleph_\mathcal{X}+b}{x}=\aleph_{\left(\frac{\mathcal{X}}{x}\right)}+\frac{b}{x}~~,\quad\text{and}\qquad \left(\cfrac{x}{\aleph_\mathcal{X}+b}\right)^{\!-1}=\cfrac{1}{\left(\cfrac{x}{\aleph_\mathcal{X}+b}\right)}=\text{undefined}~~.
2423    \end{equation*}
2424 \end{exa}
2425 
2426 
2427 
2428 
2429 
2430\subsection{Field Axioms}\label{sec:fieldAX}
2431 
2432 
2433    In earlier work on the neighborhood of infinity \cite{RINF}, we studied exclusively the maximal neighborhood of infinity using the symbol $\widehat{\mathbb{R}}$ to refer to what we have labeled $\mathbb{R}^1_0$ in the present conventions.  To build numbers of the form $x=\widehat\infty-b$ in the set $\widehat{\mathbb{R}}\sim\mathbb{R}^1_0$, it was only required to suppress the additive absorption of $\widehat\infty$.  The remaining multiplicative absorption resulted in certain (undesirable?) mathematical artifacts which are presently eliminated by the total suppression of all absorptive properties in $\widehat\infty$.  Here, we will list those artifacts which are cured in the present conventions and in this section we will examine that which remains yet still disagrees with the field axioms.
2434   
2435    If $\widehat\infty$ retains multiplicative absorption, then for $n,b\in\mathbb{N}$ we have
2436    \begin{equation*}
2437    n\big(\widehat\infty-b\big)\leq\big(\widehat\infty-b\big)~~.
2438    \end{equation*}
2439   
2440    \noindent This ordering relation is not supported by the geometric notion of multiplication; the product of any positive number $x$ multiplied by a natural number should be greater than or equal to $x$.  Another cured artifact is observed in the sums of numbers in maximal neighborhood of infinity.  Even without multiplicative absorption, the geometric notion of the difference is preserved with
2441    \begin{equation*}
2442    \big(  \widehat\infty-b \big)-\big(  \widehat\infty-a \big)=a-b~~,
2443    \end{equation*}
2444   
2445    \noindent but the notion of the sum is not.  With multiplicative absorption in place, adding two $\mathbb{R}_0^1$ numbers yields
2446    \begin{align}\label{eq:it96y0hho}
2447    \big(  \widehat\infty-b \big)+\big(  \widehat\infty-a \big)&=2\widehat\infty -\big( b+a \big)=\widehat\infty -\big( b+a \big)~~.   
2448    \end{align}
2449   
2450    \noindent The geometric notion of addition would require that the sum of two numbers just less than infinity would not be another number just less than infinity.  This issue is cured in the present convention with the implicit transfinite ordering $\aleph_{0.9}+\aleph_{0.9}=\aleph_{1.8}\ggg\aleph_1$.  
2451   
2452    The most undesirable artifact (most significant problem?)$\,$with allowing $\widehat\infty$ to retain multiplicative absorption is the loss of additive associativity.  Subtracting $(\widehat\infty-c)$ from both sides of Equation (\ref{eq:it96y0hho}) yields
2453    \begin{align*}
2454    \big[\big(  \widehat\infty-b \big)+\big(  \widehat\infty-a \big)\big]-\big(  \widehat\infty-c \big)&=\big[\widehat\infty -\big( b+a \big)\big]-\big(  \widehat\infty-c\big)~~. 
2455    \end{align*}
2456   
2457    \noindent Assuming the associative property of addition, we may arrange the LHS brackets to write
2458    \begin{align*}
2459    \big(  \widehat\infty-b \big)+\big[\big(  \widehat\infty-a \big)-\big(  \widehat\infty-c \big)\big]&=\big[\widehat\infty -\big( b+a \big)\big]-\big(  \widehat\infty-c\big)\\
2460    \widehat\infty+\big[c-\big( b+a \big)\big]&=c-\big(b+a\big)~~. 
2461    \end{align*}
2462   
2463    \noindent Subtracting the $\mathbb{R}_0$ part from both sides yields the plain contradiction $\widehat\infty=0$.  This was avoided, originally, by revoking additive associativity in Reference \cite{RINF}.  In the present conventions, we avoid this undesirable result by taking away the multiplicative absorption of infinity hat.
2464   
2465    While it is permissible, in principle, to have notions of addition and multiplication which are not inherently geometric, it is highly undesirable for basic arithmetic if addition is not associative.  Indeed, it is tantamount to arbitrary to say, ``$\widehat\infty$ has one kind of absorption but not the other,'' so the present convention is better because it gives operations which are inherently geometric \textit{and} wherein addition is has the highly desirable associative property.  Now that we have reviewed the issues that were cleared up, in the present section we will give a common statement of the field axioms together with the ordering and completeness axioms, and then we will make comparisons to the given arithmetic axioms.
2466 
2467\begin{defin}
2468    A field is a set $S$ together with the addition and multiplication operators which satisfies the addition and multiplication axioms for fields: Axioms \ref{ax:fieldplus} and \ref{ax:fieldtimes}.
2469\end{defin}
2470 
2471\begin{axio}\label{ax:fieldplus}
2472    The addition axioms for fields are
2473    \begin{itemize}
2474        \item (A1) $S$ is closed under addition: If $x,y\in S$, then $x+y\in S$.
2475        \item (A2) Addition is commutative: If $x,y\in S$, then $x+y=y+x$.
2476        \item (A3) Addition is associative: If $x,y,z\in S$, then $(x+y)+z=x+(y+z)$.
2477        \item (A4) There exists an additive identity element $0$ in $S$: If $x\in S$, then $x+0=x$.
2478        \item (A5) Every $x\in S$ has an additive inverse: If $x\in S$, then there exists $-x\in S$ such that $x+(-x)=0$.
2479    \end{itemize}
2480\end{axio}
2481 
2482 
2483\begin{rem}
2484    The arithmetic axioms do not obey (A1) but they do obey (A2)-(A5).
2485\end{rem}
2486 
2487 
2488 
2489\begin{axio}\label{ax:fieldtimes}
2490    The multiplication axioms for fields are
2491    \begin{itemize}
2492        \item (M1) $S$ is closed under multiplication: If $x,y\in S$, then $x\cdot y\in S$.
2493        \item (M2) Multiplication is commutative: If $x,y\in S$, then $x\cdot y=y\cdot x$.
2494        \item (M3) Multiplication is associative: If $x,y,z\in S$, then $(x\cdot y)\cdot z=x\cdot(y\cdot z)$.
2495        \item (M4) There exists a multiplicative identity element $1\neq0$ in $S$: If $x\in S$, then $x\cdot 1=x$.
2496        \item (M5) If $x\in S$ and $x\neq0$, then $x$ has a multiplicative inverse: If $x\in S$, then there exists $x^{-1}\in S$ such that $x\cdot x^{-1}=1$.
2497    \end{itemize}
2498\end{axio}
2499 
2500 
2501\begin{rem}
2502    The arithmetic axioms preserve (M2)-(M4) but both of (M1) and (M5) are lost.  The loss of (M5) was proven in Theorem \ref{thm:noinv}.
2503\end{rem}
2504 
2505 
2506\begin{defin}
2507    An ordered field is a field $F$ together with a relation $<$ which satisfies the field ordering axioms: Axiom \ref{ax:fieldord}.
2508\end{defin}
2509 
2510\begin{axio}\label{ax:fieldord}
2511    The field ordering axioms are
2512    \begin{itemize}
2513        \item (O1) Elements of $F$ have trichotomy: If $x,y\in F$, then one and only one of the following is true: $x<y$, $x=y$, or $x>y$.
2514        \item (O2) The $<$ relation is transitive: If $x,y,z\in F$, then $x<y$ and $y<z$ together imply $x<z$.
2515        \item (O3) If $x,y,z\in F$, then $x<y$ implies $x+z<y+z$.
2516        \item (O4) If $x,y,z \in F$, and if $z>0$, then $x<y$ implies $x\cdot z< y\cdot z$.
2517    \end{itemize}
2518 
2519    \noindent It is understood that $x<y$ means $y>x$.
2520\end{axio}
2521 
2522 
2523\begin{thm}\label{thm:443eee}
2524    For any $\mathcal{X}>0$, $\aleph_\mathcal{X}$ is an upper bound of $\mathbb{R}_0$.
2525\end{thm}
2526 
2527\begin{proof}
2528    An upper bound of a set is greater than or equal to every element of that set.  Suppose
2529    \begin{equation}
2530    X,Y\in\mathbf{AB}~~,~~x\in\mathbb{R}_0~~,~~x\in X~~,\quad\text{and}\qquad  \aleph_\mathcal{X}\in Y~~.\nonumber
2531    \end{equation}
2532   
2533    \noindent It follows that
2534    \begin{equation}
2535    \mathcal{D}_{\!\mathbf{AB}}(AX)=0~~,\quad\text{and}\qquad  \mathcal{D}_{\!\mathbf{AB}}(AY)=\mathcal{X}~~.\nonumber
2536    \end{equation}
2537   
2538    \noindent By the ordering of $\mathbb{R}$ (Axioms \ref{def:order} and \ref{ax:order}), $\aleph_\mathcal{X}$ is an upper bound of $\mathbb{R}_0$ whenever $\mathcal{X}>0$.
2539\end{proof}
2540 
2541 
2542 
2543\begin{cor}\label{cor:rtt35}
2544    $\mathbb{N}$ is bounded from above.
2545\end{cor}
2546 
2547\begin{proof}
2548    If $n\in\mathbb{N}$, then $n\in\mathbb{R}_0$.  By Theorem \ref{thm:443eee}, all $x\in\mathbb{R}_0$ are bounded from above.  $\mathbb{N}$ is bounded from above.
2549\end{proof}
2550 
2551 
2552 
2553\begin{rem}
2554    Proposition \ref{mthm:u9999979y} is usually presented as a theorem and it brings us to one of the most finely nuanced issues in the present treatment of $\mathbb{R}$.  This proposition makes a convincing case that $\mathbb{R}_0$ cannot have a supremum in $\mathbb{R}$.  However, if $\mathbb{R}_0$ is a subset of the connected interval $(-\widehat\infty,\widehat\infty)$, then it most certainly must have a least upper bound.  Otherwise $(-\widehat\infty,\widehat\infty)$ is not connected.  We will continue to develop the principles related to whether or not the different open neighborhoods can have suprema in $\mathbb{R}$, and then in Section \ref{sec:r3r23r23r3r} we will return to the topic of algebraic contradictions related to the suprema required for the connectedness of the interval.  If $\mathbb{R}$ is to have the usual topology overall, then it must have the least upper bound property.
2555\end{rem}
2556 
2557 
2558\begin{pro}\label{mthm:u9999979y}
2559    $\mathbb{R}_0\subset\mathbb{R}$ does not have a least upper bound $\sup(\mathbb{R}_0)\in\mathbb{R}$.  In other words, $\mathbb{R}$ does not have the least upper bound property.  
2560\end{pro}
2561 
2562 
2563\begin{just}
2564    To invoke a contradiction, suppose $s\in\mathbb{R}$ is a least upper bound of $\mathbb{R}_0$.  If $s-1$ was an upper bound of $\mathbb{R}_0$, then $s$ could not be the least upper bound because $s-1<s$.  Therefore, $s=\sup(\mathbb{R}_0)$ implies $(s-1)\in\mathbb{R}_0$.  By Axiom \ref{ax:fieldssss33}, $\mathbb{R}_0$ is closed under addition.  It follows that $(s-1+2)\in\mathbb{R}_0$ because $2\in\mathbb{R}_0$.  Since $s+1>s$, we obtain a contradiction having shown that there exist elements of $\mathbb{R}_0$ greater than the assumed supremum $s$.
2565\end{just}
2566 
2567\begin{defin}
2568    The issue described in the justification of Proposition \ref{mthm:u9999979y} shall be referred to as the least upper bound problem.
2569\end{defin}
2570 
2571 
2572\subsection{Compliance of Cauchy Equivalence Classes with the Arithmetic Axioms}\label{sec:consXXX}
2573 
2574 
2575In this section, we give the usual definitions for arithmetic operations on Cauchy equivalence classes.  We clarify the meanings for the cases of $[x]\to[X+x]=[\aleph_\mathcal{X}+x]$ and then we prove in a few cases that the arithmetic axioms are satisfied by the extended Cauchy equivalence classes $[X+x]\subset C_\mathbb{Q}^{\mathbf{AB}}\setminus C_\mathbb{Q}\implies[X+x]\not\in\mathbb{R}_0$.  The proofs in this section mostly follow References \cite{BRUDIN,CAUCHYR}.
2576 
2577\begin{thm}\label{thm:3f444042040}
2578    Every convergent rational sequence of terms $a_n\in\mathbb{Q}$ is a Cauchy sequence.
2579\end{thm}
2580 
2581\begin{proof}
2582    Per Definition \ref{def:CSq}, a sequence $\{a_n\}$ is a Cauchy sequence if and only if
2583    \begin{equation*}
2584    \forall \delta\in\mathbb{Q}\quad\exists m,n,N\in\mathbb{N}\quad\text{s.t.}\quad m,n>N\quad\implies\quad\big|a_n-a_m\big|<\delta~~.
2585    \end{equation*}
2586   
2587    \noindent By the convergence of $\{a_n\}$, it is granted that there exists some $l\in\mathbb{R}$ such that
2588    \begin{equation*}
2589    \lim\limits_{n\to\infty}a_n=l~~.
2590    \end{equation*}
2591   
2592    \noindent  Convergence then guarantees that
2593    \begin{equation*}
2594    \exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|a_n-l\big|<\frac{\delta}{2}~~.
2595    \end{equation*}
2596   
2597    \noindent Then, whenever $n,m>N$, we have
2598    \begin{equation*}
2599    \big|a_n-a_m\big|=\big|\big(a_n-l\big)-\big(a_m-l\big)\big|\leq\big|a_n-l\big|+\big|a_m-l\big|<\frac{\delta}{2}+\frac{\delta}{2}=\delta~~.
2600    \end{equation*}
2601   
2602    \noindent Therefore, every convergent rational sequence $\{a_n\}$ is a Cauchy sequence.
2603\end{proof}
2604 
2605\begin{defin}\label{def:86969696}
2606    If $x,y\in\mathbb{R}$ such that there are two Cauchy equivalence classes $x=[(x_n)]$ and $y=[(y_n)]$, then $x+y=[(x_n+y_n)]$ and $x\cdot y=[(x_n\cdot y_n)]$.
2607\end{defin}
2608 
2609\begin{thm}\label{thm:cec200e0c0ec}
2610    The additive operation for equivalence classes given by Definition \ref{def:86969696} is well-defined.
2611\end{thm}
2612 
2613\begin{proof}
2614    Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2615    \begin{equation*}
2616    [a]=[b]~~,\quad\text{and}\qquad [c]=[d]~~,
2617    \end{equation*}
2618   
2619    \noindent so that
2620    \begin{equation*}
2621    \lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\quad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2622    \end{equation*}
2623   
2624    \noindent For addition to be proven well-defined, we need to prove that $[(a_n+c_n)]=[(b_n+d_n)]$.  This requires
2625    \begin{equation*}
2626    [(a_n+c_n)]-[(b_n+d_n)]=0~~.
2627    \end{equation*}
2628   
2629    \noindent The difference being equal to zero means that for sufficiently large $n$, and for any $\delta\in\mathbb{R}$, we must have
2630    \begin{equation*}
2631    [(a_n+c_n)]-[(b_n+d_n)]=[(a_n-b_n)]-[(c_n-d_n)]<\delta~~.
2632    \end{equation*}
2633   
2634    \noindent We will prove this by the same method of Theorem \ref{thm:3f444042040}.  The limits of $a_n-b_n$ and $c_n-d_n$ approaching zero tell us that
2635    \begin{equation*}
2636    \exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|a_n-b_n\big|<\frac{\delta}{2}~~,~~\big|c_n-d_n\big|<\frac{\delta}{2}~~.
2637    \end{equation*}
2638   
2639    \noindent Then, whenever $n,m>N$, we have
2640    \begin{equation*}
2641    \big|\big(a_n-b_n\big)-\big(c_m-d_m\big)\big|\leq\big|a_n-b_n\big|+\big|c_m-d_m\big|<\frac{\delta}{2}+\frac{\delta}{2}=\delta~~.
2642    \end{equation*}
2643   
2644    \noindent This proves that $[a+c]=[b+d]$ and that, therefore, addition is a well-defined operation on Cauchy equivalence classes.
2645\end{proof}
2646 
2647\begin{exa}
2648    This example gives a specific case of Theorem \ref{thm:cec200e0c0ec} using numbers in the neighborhood of infinity.  Suppose there are four subsets of $C_\mathbb{Q}^{\mathbf{AB}}$ with the properties
2649    \begin{equation*}
2650    [\aleph_{[\mathcal{X}_1]}+x_1]=[\aleph_{[\mathcal{Y}_1]}+y_1] ~~,\quad\text{and}\qquad[\aleph_{[\mathcal{X}_2]}+x_2]=[\aleph_{[\mathcal{Y}_2]}+y_2] ~~.
2651    \end{equation*}
2652   
2653    \noindent Since the big and little parts of equal numbers are equal, we have equality among all the matched pairs of $[x_1],[x_2],[y_1],[y_2],[\mathcal{X}_1],[\mathcal{X}_2],[\mathcal{Y}_1],[\mathcal{Y}_2]\subset C_\mathbb{Q}$.     If addition is well-defined, then
2654    \begin{equation*}
2655    [\aleph_{[\mathcal{X}_1]}+x_1]+[\aleph_{[\mathcal{X}_2]}+x_2]=[\aleph_{[\mathcal{Y}_1]}+y_1]+[\aleph_{[\mathcal{Y}_2]}+y_2]~~.
2656    \end{equation*}
2657   
2658    \noindent Evaluating the left and right sides independently yields
2659    \begin{align*}
2660    [\aleph_{[\mathcal{X}_1]}+x_1]+[\aleph_{[\mathcal{X}_2]}+x_2]&=[\aleph_{[\mathcal{X}_1]}+x_1+\aleph_{[\mathcal{X}_2]}+x_2]=[\aleph_{[\mathcal{X}_1+\mathcal{X}_2]}+x_1+x_2]~~,
2661    \end{align*}
2662   
2663    \noindent and
2664    \begin{align*}
2665    [\aleph_{[\mathcal{Y}_1]}+y_1]+[\aleph_{[\mathcal{Y}_2]}+y_2]&=[\aleph_{[\mathcal{Y}_1]}+y_1+\aleph_{[\mathcal{Y}_2]}+y_2]=[\aleph_{[\mathcal{Y}_1+\mathcal{Y}_2]}+y_1+y_2]~~.
2666    \end{align*}
2667   
2668    \noindent Considering first the small parts, Definition \ref{def:86969696} gives $[x+y]=[x]+[y]$ so
2669    \begin{equation*}
2670    [x_1+x_2]=[y_1+y_2]\quad\iff\quad [x_1]+[x_2]=[y_1]+[y_2]~~.
2671    \end{equation*}
2672   
2673    \noindent This condition follows from Theorem \ref{thm:cec200e0c0ec}.  Considering the big parts yields
2674    \begin{equation*}
2675    [\aleph_{[\mathcal{X}_1+\mathcal{X}_2]}]=[\aleph_{[\mathcal{Y}_1+\mathcal{Y}_2]}]\quad\iff\quad [\mathcal{X}_1]+[\mathcal{X}_2]=[\mathcal{Y}_1]+[\mathcal{Y}_2]~~.
2676    \end{equation*}
2677   
2678    \noindent It follows as an obvious corollary of Theorem \ref{thm:cec200e0c0ec} that the additive operation is well-defined for numbers in the neighborhood of infinity.
2679\end{exa}
2680 
2681\begin{rem}
2682    To prove that the multiplicative operation is well-defined, we will rely on the boundedness of Cauchy sequences.  First, we will give the proof of boundedness.
2683\end{rem}
2684 
2685\begin{thm}\label{thm:bbbbb4b4}
2686    If $\{a_n\}$ is a Cauchy sequence of rationals, then there exists an $M\in\mathbb{R}$ such that $|a_n|<M$ for all $n\in\mathbb{N}$.  In other words, every Cauchy sequence of rationals is bounded.
2687\end{thm}
2688 
2689\begin{proof}
2690    Since $\{a_n\}$ is Cauchy, we know there is some sufficiently large $m,n\in\mathbb{N}$ such that
2691    \begin{equation*}
2692    \big|a_n-a_m\big|<1~~.
2693    \end{equation*}
2694   
2695    \noindent If follows for such $n$ that
2696    \begin{equation*}
2697    \big|a_{N+1}-a_n\big|<1\quad\implies\quad \big(a_{N+1}-1\big)<a_n<\big(a_{N+1}+1\big)~~.
2698    \end{equation*}
2699   
2700    \noindent Define $M$ as the greatest element of a set with a natural number of elements
2701    \begin{equation*}
2702    M=\max\big\{\big|a_0\big|,\big|a_1\big|,...,\big|a_N\big|,\big|a_{N+1}-1\big|,\big|a_{N+1}+1\big|\big\}~~.
2703    \end{equation*}
2704   
2705    \noindent Every $a_n$ with $n\leq N$ is in the set, and every $a_n$ with $n> N$ is less than one of the last two elements of the set.  Therefore, there exists a bound $M\in\mathbb{R}$ for every rational Cauchy sequence $\{a_n\}$.
2706\end{proof}
2707 
2708 
2709 
2710\begin{thm}\label{thm:cec20f0e0c0ec}
2711    The multiplicative operation for equivalence classes given by Definition \ref{def:86969696} is well-defined.  
2712\end{thm}
2713 
2714\begin{proof}
2715    Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2716    \begin{equation*}
2717    [a]=[b]~~,\quad\text{and}\qquad [c]=[d]~~,
2718    \end{equation*}
2719   
2720    \noindent so that
2721    \begin{equation*}
2722    \lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\quad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2723    \end{equation*}
2724   
2725    \noindent For multiplication to be proven well-defined, we need to prove that $[(a_n\cdot c_n)]=[(b_n\cdot d_n)]$, or specifically that for sufficiently large $n$
2726    \begin{equation*}
2727    [(a_n\cdot c_n)]-[(b_n\cdot d_n)]<\delta~~.
2728    \end{equation*}
2729   
2730    \noindent  To that end, insert the additive identity as a difference of cross terms so that
2731    \begin{align*}
2732    a_n\cdot c_n-b_n\cdot d_n&=a_n\cdot c_n-b_n\cdot d_n+\big(c_n\cdot b_n-c_n\cdot b_n\big)\\
2733    &=\big(a_n\cdot c_n- c_n\cdot b_n\big)+\big(c_n\cdot b_n -b_n\cdot d_n\big)\\
2734    &=c_n\cdot\big(a_n-  b_n\big)+b_n\cdot\big(c_n - d_n\big)~~.
2735    \end{align*}
2736   
2737    \noindent It follows that
2738    \begin{equation*}
2739    \big|a_n\cdot c_n-b_n\cdot d_n\big|\leq\left(\big|c_n\big|\cdot\big|a_n-  b_n\big|+\big|b_n\big|\cdot\big|c_n - d_n\big|\right)~~.
2740    \end{equation*}
2741   
2742    \noindent By Theorem \ref{thm:bbbbb4b4}, there exists bounds $|b_n|\leq B_0$ and $|c_n|\leq C_0$ for any $n\in\mathbb{N}$.  Then let $M_0=B_0+C_0$ so that
2743    \begin{equation*}
2744    \big|a_n\cdot c_n-b_n\cdot d_n\big|<M_0\left( \big|a_n-  b_n\big|+ \big|c_n - d_n\big|\right)~~.
2745    \end{equation*}
2746   
2747    \noindent Since all four sequences are Cauchy, we have
2748    \begin{equation*}
2749    \exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|a_n-b_n\big|<\frac{\delta}{2M_0}~~,~~\big|c_n-d_n\big|<\frac{\delta}{2M_0}~~.
2750    \end{equation*}
2751   
2752    \noindent We prove the theorem by writing
2753    \begin{equation*}
2754    \big|a_n\cdot c_n-b_n\cdot d_n\big|<M_0\left( \frac{\delta}{2M_0}+ \frac{\delta}{2M_0}\right)=\delta~~.
2755    \end{equation*}
2756\end{proof}
2757 
2758 
2759\begin{rem}
2760    Theorem \ref{thm:bbbbb4b4} proves the boundedness of Cauchy sequences of rationals in $C_\mathbb{Q}$ but not the boundedness of all sequences in $C_\mathbb{Q}^{\mathbf{AB}}$.  Since numbers with non-zero big parts are represented as ordered pairs of elements of $C_\mathbb{Q}$, it is obvious that such numbers are bounded because each sequence in the pair is bounded.  As a consequence of Theorem \ref{thm:cec20f0e0c0ec}, which regards general Cauchy equivalence classes and does not restrict to the rationals, it follows that multiplication is well-defined for numbers in the neighborhood of infinity.  However, one must carefully note that the boundedness of such products will not always be such that the bound is in $\mathbb{R}$.  By the identity $\aleph_\mathcal{X}\cdot\aleph_\mathcal{Y}=\aleph_{\aleph_{(\mathcal{XY})}}$, it is never in $\mathbb{R}$ when $\mathcal{X}>0$ or $\mathcal{Y}>0$.
2761\end{rem}
2762 
2763 
2764 
2765\begin{rem}
2766    Assuming the field axioms, Definition \ref{def:86969696} giving $x\cdot y=[(x_n\cdot y_n)]$ is good enough to allow us to prove the arithmetic operations are well-defined.  However, we have presently not defined division as multiplication by an inverse, so we need to give a definition for the quotient of two Cauchy equivalence classes.
2767\end{rem}
2768 
2769\begin{defin}\label{def:869696ffefe96}
2770    If $x,y\in\mathbb{R}$ such that there are two Cauchy equivalence classes $x=[(x_n)]$ and $y=[(y_n)]$, then $x\div y=[(x_n\div y_n)]$.
2771\end{defin}
2772 
2773 
2774\begin{thm}\label{thm:3111r22424}
2775    The quotient operation for equivalence classes of rationals given by Definition \ref{def:86969696} is well-defined.  
2776\end{thm}
2777 
2778\begin{proof}
2779    Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2780    \begin{equation*}
2781    [a]=[b]~~,\quad\text{and}\qquad [c]=[d]~~,
2782    \end{equation*}
2783   
2784    \noindent so that
2785    \begin{equation*}
2786    \lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\quad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2787    \end{equation*}
2788   
2789    \noindent For division to be proven well-defined, we need to prove that $[(a_n\div c_n)]=[(b_n\div d_n)]$.  Specifically, for sufficiently large $n$, we must demonstrate
2790    \begin{equation*}
2791    [(a_n\div c_n)]-[(b_n\div d_n)]<\delta~~.
2792    \end{equation*}
2793   
2794    \noindent  To that end, insert the additive identity as a difference of the cross terms so that
2795    \begin{align*}
2796    \frac{a_n}{c_n}-\frac{b_n}{d_n}&=\frac{a_n}{c_n} -\frac{b_n}{d_n}+\left(\frac{b_n}{c_n}-\frac{b_n}{c_n}\right)\\
2797    &=\left( \frac{a_n}{c_n}-\frac{b_n}{c_n}\right)+\left(\frac{b_n}{c_n}-\frac{b_n}{d_n}\right)\\
2798    &=\frac{a_n-b_n}{c_n}+\frac{b_n\cdot\big(d_n-c_n\big)}{c_n\cdot d_n}~~.
2799    \end{align*}
2800   
2801    \noindent It follows that
2802    \begin{equation*}
2803    \left|\frac{a_n}{c_n}-\frac{b_n}{d_n}\right|\leq\left(\frac{\big|a_n-  b_n\big|}{\big|c_n\big|}+\frac{\big|b_n\big|\cdot\big|c_n - d_n\big|}{\big|c_n\big|\cdot\big|d_n\big|}\right)~~.
2804    \end{equation*}
2805   
2806   
2807    \noindent By Theorem \ref{thm:bbbbb4b4}, there exists bounds $|b_n|\leq B_0$, $|c_n|\leq C_0$ and $|d_n|\leq D_0$ for any $n\in\mathbb{N}$.  Since all four sequences are Cauchy, we have
2808    \begin{equation*}
2809    \exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|a_n-b_n\big|<\frac{C_0\delta }{2}~~,~~\big|c_n-d_n\big|<\frac{C_0D_0\delta}{2B_0}~~.
2810    \end{equation*}
2811   
2812    \noindent We prove the theorem by writing
2813    \begin{equation*}
2814    \left|\frac{a_n}{c_n}-\frac{b_n}{d_n}\right|<\left( \frac{\frac{C_0\delta}{2}}{C_0}+ \frac{B_0\frac{C_0D_0\delta}{2B_0}}{C_0D_0}\right)=\frac{\delta}{2}+\frac{\delta}{2}=\delta~~.
2815    \end{equation*}
2816   
2817    \noindent Since we have assumed $[a],[b],[c],[d]\subset C_\mathbb{Q}$, we have proven the theorem with Axiom \ref{ax:dssfgss33} granting associativity among division and multiplication.
2818\end{proof}
2819 
2820 
2821 
2822\begin{mainthm}\label{thm:3111r22424}
2823    The quotient operation given by Definition \ref{def:86969696} is well-defined for equivalence classes in $C_\mathbb{Q}^\mathbf{AB}\setminus C_\mathbb{Q}$.  
2824\end{mainthm}
2825 
2826\begin{proof}
2827    Suppose there are four subsets of $C_\mathbb{Q}^{\mathbf{AB}}$ with the properties
2828    \begin{equation*}
2829    [\aleph_{[\mathcal{A}]}+a]=[\aleph_{[\mathcal{B}]}+b] ~~,\quad\text{and}\qquad[\aleph_{[\mathcal{C}]}+C]=[\aleph_{[\mathcal{D}]}+d] ~~.
2830    \end{equation*}
2831   
2832    \noindent It follows from the equality of Cauchy sequences that
2833    \begin{align*}
2834    \lim\limits_{n\to\infty}\big(\mathcal{A}_n-\mathcal{B}_n\big)&=0\\
2835    \lim\limits_{n\to\infty}\big(\mathcal{C}_n-\mathcal{D}_n\big)&=0\\
2836    \lim\limits_{n\to\infty}\big(a_n-b_n\big)&=0\\
2837    \lim\limits_{n\to\infty}\big(c_n-d_n\big)&=0~~.
2838    \end{align*}
2839   
2840    \noindent For concision in notation, introduce the symbols
2841    \begin{align*}
2842    (A_n)&=(\aleph_{[(\mathcal{A}_n)]}+a_n)\\
2843    (B_n)&=(\aleph_{[(\mathcal{B}_n)]}+b_n)\\
2844    (C_n)&=(\aleph_{[(\mathcal{C}_n)]}+c_n)\\
2845    (D_n)&=(\aleph_{[(\mathcal{D}_n)]}+d_n)~~.
2846    \end{align*}
2847   
2848   
2849    \noindent For division to be proven well-defined, we need to prove that $[(A_n\div C_n)]=[(B_n\div D_n)]$.  Specifically, for sufficiently large $n$, we must demonstrate
2850    \begin{equation*}
2851    [(A_n\div C_n)]-[(B_n\div D_n)]<\delta~~.
2852    \end{equation*}
2853   
2854    \noindent Following the form of Theorem \ref{thm:3111r22424}, we may insert the identity to obtain the inequality  
2855    \begin{equation*}
2856    \left|\frac{A_n}{C_n}-\frac{B_n}{D_n}\right|\leq \frac{\big|A_n-  B_n\big|}{\big|C_n\big|}+\frac{\big|B_n\big|\cdot\big|C_n - D_n\big|}{\big|C_n\big|\cdot\big|D_n\big|} ~~.
2857    \end{equation*}
2858   
2859    \noindent Here we make the major distinction with Theorem \ref{thm:3111r22424}: the bounds on $(A_n),(B_n),(C_n),(D_n)$ are not in $\mathbb{R}_0$ and we must be careful not to allow associativity among multiplication and division when simplifying the expression.  Since each of $(A_n),(B_n),(C_n),(D_n)$ are ordered pairs of Cauchy sequences of rationals (Axiom \ref{def:it759595}), we know the pairs of sequences are bounded.  Let the bounds be
2860    \begin{align*}
2861    [(A_n)]&=([\mathcal{A}],[a])\leq(A_0,a_0)\\
2862    [(B_n)]&=([\mathcal{B}],[b])\leq(B_0,b_0)\\
2863    [(C_n)]&=([\mathcal{C}],[c])\leq(C_0,c_0)\\
2864    [(D_n)]&=([\mathcal{D}],[d])\leq(D_0,d_0)~~,
2865    \end{align*}
2866   
2867    \noindent where the notation implies the ordering of each paired element respectively.  It follows that
2868    \begin{align*}
2869    \left|\frac{A_n}{C_n}-\frac{B_n}{D_n}\right|&\leq\frac{\big|\aleph_{A_0}+a_0-  \aleph_{B_0}-b_0\big|}{\big|\aleph_{C_0}+c_0\big|}+\frac{\big|\aleph_{B}+b_0\big|\cdot\big|\aleph_{C_0}+c_0 - \aleph_{D_0}-d_0\big|}{\big|\aleph_{C_0}+c_0\big|\cdot\big|\aleph_{D_0}+d_0\big|}\\
2870    &\leq\frac{\big|\aleph_{(A_0-B_0)}+a_0 -b_0\big|}{\big|\aleph_{C_0}+c_0\big|}+\frac{\big|\aleph_{B}+b_0\big|\cdot\big|\aleph_{(C_0-D_0)}+c_0 -d_0\big|}{\big|\aleph_{C_0}+c_0\big|\cdot\big|\aleph_{D_0}+d_0\big|}\\
2871    &\leq\frac{\big|A_0-B_0\big|}{\big|C_0\big|}+\frac{\big|\aleph_{\left(\aleph_{(B_0C_0-B_0D_0)}+B_0c_0-B_0d_0+b_0C_0-b_0D_0\right)}+b_0c_0-b_0d_0\big|}{ \big|\aleph_{\left(\aleph_{(C_0D_0)}+D_0c_0+d_0C_0\right)} +d_0c_0)\big|}\\
2872    &\leq\frac{\big|A_0-B_0\big|}{\big|C_0\big|}+\frac{\big|\aleph_{(B_0C_0-B_0D_0)}+B_0c_0-B_0d_0+b_0C_0-b_0D_0\big|}{ \big|\aleph_{(C_0D_0)}+D_0c_0+d_0C_0\big|}\\
2873    &\leq\frac{\big|A_0-B_0\big|}{\big|C_0\big|}+\frac{\big|B_0C_0-B_0D_0\big|}{ \big|C_0D_0\big|}\\
2874    &\leq\frac{\big|A_0-B_0\big|}{\big|C_0\big|}+\frac{\big|B_0\big|\cdot\big|C_0-D_0\big|}{ \big|C_0\big|\cdot\big|D_0\big|}~~.
2875    \end{align*}
2876   
2877    \noindent Since $A_0,B_0,C_0,D_0\in\mathbb{R}_0$, this is the same form achieved in Theorem \ref{thm:3111r22424} and we will conclude the proof in the same way.  Use the Cauchy property of the respective sequences to write
2878    \begin{equation*}
2879    \exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|A_n-B_n\big|<\frac{C_0\delta }{2}~~,~~\big|C_n-D_n\big|<\frac{C_0D_0\delta}{2B_0}~~.
2880    \end{equation*}
2881   
2882    \noindent We prove the theorem by writing
2883    \begin{equation*}
2884    \left|\frac{A_n}{C_n}-\frac{B_n}{D_n}\right|<  \frac{\frac{C_0\delta}{2}}{C_0}+ \frac{B_0\frac{C_0D_0\delta}{2B_0}}{C_0D_0} =\frac{\delta}{2}+\frac{\delta}{2}=\delta~~.
2885    \end{equation*}
2886\end{proof}
2887 
2888 
2889 
2890 
2891\section{Arithmetic Applications}
2892 
2893 
2894 
2895 
2896\subsection{Properties of the Algebraic Fractional Distance Function Revisited}\label{sec:cont2}
2897 
2898 
2899We have defined the algebraic FDF $\mathcal{D}^\dagger_{\!AB}$ to totally replicate the behavior of the geometric FDF $\mathcal{D}_{\!AB}$ with the added property that it should allow us to compute numerical quotients of the form $\frac{AX}{AB}$ without requiring a supplemental constraint of the form $AX=cAB$.  In verbose notation, we have
2900\begin{equation*}
2901\mathcal{D}_{\!\mathbf{AB}}:AX\to[0,1]~~,\quad\text{and}\qquad\mathcal{D}_{\!\mathbf{AB}}^\dagger:\{AX;x\}\to[0,1]~~,
2902\end{equation*}
2903 
2904\noindent so that the algebraic FDF provides more information by taking the line segment and the chart on the line segment whereas the geometric FDF doesn't know about $x$.
2905 
2906In Section \ref{sec:FD}, we found that neither the algebraic FDF of the first kind nor the second has the analytic form of $\mathcal{D}^\dagger_{\!AB}$.  The second kind was ruled out by Theorem \ref{thm:injjj2222} when we showed that $\mathcal{D}''_{\!AB}$ is not one-to-one.  $\mathcal{D}'_{\!AB}$ was provisionally eliminated based on an unallowable discontinuity at infinity.  Since $\mathcal{D}_{\!AB}$ is continuous on its domain, $\mathcal{D}^\dagger_{\!AB}$ is too.  In Theorem \ref{thm:algfracdisnotcont3} specifically, we showed that $\mathcal{D}'_{\!AB}$ cannot conform to the Cauchy criterion for continuity at infinity because that criterion always fails at infinity.  The nature of the failure of the Cauchy criterion at infinity (Theorem \ref{thm:algfracdisnotcont3}) is that it gives a requirement
2907\begin{equation}\label{eq:dfgr3r3434}
2908|x-\infty|<\delta\quad\iff\quad \delta>\infty~~.\nonumber
2909\end{equation}
2910 
2911\noindent There is no such $\delta$.  What is the source of this discrepancy?  The source is the additive absorptive property of infinity giving $\infty-x=\infty$. By now, we have shown that the absorptive properties of all infinite elements are not supported by the invariance of line segments under permutations of their endpoints and we have otherwise given an artificial construction $\widehat\infty$ which does not have the problematic properties.  In this section, we will revisit the continuity and other properties of $\mathcal{D}'_{\!AB}$.  We will show that \textbf{\textit{the algebraic FDF of the first kind \underline{does} satisfy the Cauchy criterion for a limit at infinity}}, something which has been considered historically impossible.  In the present section, we will also prove Conjecture \ref{conj:dv2324} wherein it was postulated that $\mathcal{D}'_{\!AB}$ is injective.  Having shown by the end of the present section that there are no obvious discrepancies between $\mathcal{D}^\dagger_{\!AB}$ and $\mathcal{D}'_{\!AB}$, we will assume that the algebraic FDF of the first kind is identically $\mathcal{D}^\dagger_{\!AB}$.
2912 
2913 
2914 
2915\begin{mainthm}\label{thm:algfracdisnotcont4}
2916    The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!\mathbf{AB}}(AX)$ converges to a limit $l=1$ at $B\in\mathbf{AB}$.
2917\end{mainthm}
2918 
2919 
2920\begin{proof}
2921    According to the Cauchy definition of the limit of $f(x):D\to R$ as $x$ approaches $\widehat\infty$, we say that
2922    \begin{equation}
2923    \lim\limits_{x\to \widehat\infty} f(x) = l~~,\nonumber
2924    \end{equation}
2925   
2926    \noindent if and only if
2927    \begin{equation}
2928    \forall\varepsilon>0\quad\exists\delta>0\quad\text{s.t}\quad\forall x\in D~~,\nonumber
2929    \end{equation}
2930   
2931    \noindent we have
2932    \begin{equation}
2933    0<|x-\widehat\infty|<\delta\quad\implies\quad|f(x)-l |<\varepsilon~~.\nonumber
2934    \end{equation}
2935   
2936    \noindent In Theorem \ref{thm:algfracdisnotcont3}, we attempted to show this limit in the approach to geometric infinity $x\to\infty$.  At that point, we had to stop because there is no $\delta\in\mathbb{R}$ such that $\infty-x<\delta$.  Now we may choose $x\in\mathbb{R}$ with the given arithmetic axioms to obtain, for example,
2937    \begin{equation}
2938    |( \widehat\infty-b )-\widehat\infty|= b~~,\quad\text{or}\qquad |\aleph_\mathcal{X}-\widehat\infty|=\aleph_{(1-\mathcal{X})}\nonumber~~.
2939    \end{equation}
2940   
2941    \noindent Per the ordering axiom (Axiom \ref{ax:order}), either of these can be less than some $\delta\in\mathbb{R}$.  This remedies the blockage encountered in Theorem \ref{thm:algfracdisnotcont3} where we found $\delta\in\mathbb{R}$ implies $\infty-x\not<\delta$.  Now we may follow the usual prescription for the Cauchy definition of a limit, even at infinity!  To that end, let $\delta=\aleph_{\left(\frac{\varepsilon}{2}\right)}$.  Then the Cauchy definition requires that
2942    \begin{equation}
2943    0<|x-\widehat\infty|<\aleph_{\left(\frac{\varepsilon}{2}\right)}~~,\quad\text{and}\qquad|\mathcal{D}'_{\!\mathbf{AB}}(AX)-\mathcal{D}'_{\!\mathbf{AB}}(AB)|<\varepsilon~~.\nonumber
2944    \end{equation}
2945   
2946    \noindent First we will evaluate $\delta$ expression on the left as
2947    \begin{equation}
2948    \widehat\infty-x<\aleph_{\left(\tfrac{\varepsilon}{2}\right)}\quad\iff\quad x>\aleph_{\left(1-\tfrac{\varepsilon}{2}\right)}~~.\nonumber
2949    \end{equation}
2950   
2951    \noindent Definition \ref{def:algfracdis} gives $\mathcal{D}'_{\!AB}$ as
2952    \begin{equation}
2953    \mathcal{D}'_{\!AB}(AX)=\begin{cases}
2954    ~~~1\qquad\quad\text{for}\quad X=B\\[8pt]
2955    \cfrac{\|AX\|}{ \|AB\|}\,~~\quad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
2956    ~~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~,\nonumber
2957    \end{equation}
2958   
2959    \noindent where
2960    \begin{equation}
2961    \cfrac{\|AX\|}{ \|AB\|}  =   \cfrac{\text{len}[a,x]}{\text{len}[a,b]}    ~~.\nonumber
2962    \end{equation}
2963 
2964   
2965    \noindent Evaluation of the $\varepsilon$ expression, therefore, yields
2966    \begin{equation}
2967    \left| \cfrac{\text{len}[0,x]}{\text{len}[0,\widehat\infty]} -1 \right| =   \left| \cfrac{x}{\widehat\infty} -1 \right|<    \left| \cfrac{\aleph_{\left(1-\tfrac{\varepsilon}{2}\right)}}{\widehat\infty} -1 \right|=   \left|\left(1-\cfrac{\varepsilon}{\vphantom{\hat 2}2}\right) -1 \right|=\left| -\cfrac{\varepsilon}{\vphantom{\hat 2}2} \right|<\varepsilon~~.\nonumber
2968    \end{equation}
2969   
2970    \noindent Therefore,
2971    \begin{equation}
2972    \lim\limits_{x\to \widehat\infty} \mathcal{D}'_{\!\mathbf{AB}}(AX) = 1~~.\nonumber
2973    \end{equation} 
2974   
2975    \noindent This limit demonstrates the continuity of $\mathcal{D}'_{\!\mathbf{AB}}$ at infinity.
2976\end{proof}
2977 
2978 
2979\begin{rem}
2980    When defining $\mathcal{D}'_{\!AB}$ and $\mathcal{D}''_{\!AB}$ in Section \ref{sec:FD}, we were able to show that $\mathcal{D}''_{\!AB}$ is not one-to-one but we did not yet have to tools to prove that $\mathcal{D}'_{\!AB}$ is one-to-one on all real line segments.  We conjectured it with Conjecture \ref{conj:dv2324} and now we will use Lemma \ref{thm:uniqueneighb} to prove it in Theorem \ref{thm:injjj2}.
2981\end{rem}
2982 
2983 
2984 
2985\begin{lem}\label{thm:uniqueneighb}
2986    For any point $X\equiv\mathscr{X}=[x_1,x_2]$ in a real line segment $AB$, we have $x_1\in\mathbb{R}^{\mathcal{X}_0}_\aleph$ if and only if $x_2\in\mathbb{R}^{\mathcal{X}_0}_\aleph$.
2987\end{lem}
2988 
2989 
2990\begin{proof}
2991    For proof by contradiction, suppose $x_1\in\mathbb{R}^{\mathcal{X}_1}_\aleph$ and $x_2\in\mathbb{R}^{\mathcal{X}_2}_\aleph$, and that $\mathcal{X}_1\neq\mathcal{X}_2$.  By Definition \ref{def:fh93ry983y9y222}, there exist $b_1,b_2\in\mathbb{R}_\aleph^0$ such that
2992    \begin{equation*}
2993    x_1=\aleph_{\mathcal{X}_1}+b_1~~,\quad\text{and}\qquad x_2=\aleph_{\mathcal{X}_2}+b_2~~.
2994    \end{equation*}
2995   
2996    \noindent With the $a\leq b$ condition inherent to the $[a,b]$ interval notation, the algebraic FDF tells us that
2997    \begin{equation}
2998    \min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX) ]= \cfrac{\text{len}[0,x_1]}{\text{len}[0,\infty]} = \cfrac{x_1}{\infty}=\mathcal{X}_1  ~~,\nonumber
2999    \end{equation}
3000   
3001    \noindent and
3002    \begin{equation}
3003    \max[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX) ]= \cfrac{\text{len}[0,x_2]}{\text{len}[0,\infty]} = \cfrac{x_2}{\infty} =\mathcal{X}_2 ~~.\nonumber
3004    \end{equation}
3005   
3006    \noindent It follows from the identity $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)=\mathcal{D}_{\!\mathbf{AB}}(AX)$ that
3007    \begin{equation}
3008    \min[\mathcal{D}_{\!\mathbf{AB}}(AX)] =\mathcal{X}_1~~,\quad\text{and}\qquad \max[\mathcal{D}_{\!\mathbf{AB}}(AX)]=\mathcal{X}_2~~.\nonumber
3009    \end{equation}
3010   
3011    \noindent By Definition \ref{def:gfdf}, $\mathcal{D}_{\!\mathbf{AB}}(AX)$ is one-to-one which requires 
3012    \begin{equation}
3013    \mathcal{X}_1=\mathcal{X}_2~~.\nonumber
3014    \end{equation}
3015   
3016    \noindent This contradicts the assumed condition that $\mathcal{X}_1\neq\mathcal{X}_2$.
3017\end{proof}
3018 
3019 
3020\begin{thm}\label{thm:injjj2}  
3021    The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ is injective (one-to-one) on all real line segments.
3022\end{thm}
3023 
3024\begin{proof}
3025    (Proof of Conjecture \ref{conj:dv2324}.)  Recall that $\mathcal{D}'_{\!AB}:AB\to [0,1]$ is 
3026    \begin{equation}
3027    \mathcal{D}'_{\!AB}(AX)=\begin{cases}
3028    ~~~1\qquad\quad&\text{for}\quad X=B\\[8pt]
3029    \cfrac{\|AX\|}{\|AB\|}=\cfrac{\text{len}[a,x]}{\text{len}[a,b]}\,~~\quad&\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
3030    ~~~0 \qquad\quad&\text{for}\quad X=A \end{cases}~~.\nonumber
3031    \end{equation}
3032   
3033    \noindent Injectivity requires that
3034    \begin{equation}
3035    \mathcal{D}'_{\!AB}(AX_1)=\mathcal{D}'_{\!AB}(AX_2)\quad\iff\quad AX_1=AX_2\quad\iff\quad X_1=X_2~~.\nonumber
3036    \end{equation}
3037   
3038   
3039    \noindent Even if there is an entire interval of numbers in the algebraic representations of each of $X_1$ and $X_2$, we have by Lemma \ref{thm:uniqueneighb}: 
3040    \begin{equation}
3041    \min[\mathcal{D}'_{\!AB}(AX_k)] = \max[\mathcal{D}'_{\!AB}(AX_k)]=\mathcal{X}_k~~.\nonumber
3042    \end{equation}
3043   
3044    \noindent This tells us that choosing any $x\in\mathscr{X}\equiv X$ will yield the same $\mathcal{D}'_{\!AB}(AX)$.  Therefore, the injectivity of $\mathcal{D}'_{\!AB}(AX)$ follows from the injectivity of $\mathcal{D}_{\!AB}(AX)$ through the constraint    
3045    \begin{equation}
3046    \mathcal{D}'_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
3047    \end{equation}
3048\end{proof}
3049 
3050 
3051 
3052\begin{conj}
3053    The algebraic fractional distance function $\mathcal{D}^\dagger_{\!AB}$ is an algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$.
3054\end{conj}
3055 
3056 
3057 
3058\subsection{Some Theorems for Numbers in the Neighborhood of Infinity}\label{sec:gduidt7t}
3059 
3060In Section \ref{sec:3322442}, we listed four course bins of length as distinct modes in which a line segment might have a many-to-one or one-to-one relationship between its points and the numbers in their algebraic representations.  The bins were
3061\begin{itemize}
3062    \item $L\in\mathbb{R}_0$
3063    \item $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$
3064    \item  $L\in\mathbb{R}_\aleph^\mathcal{X}\cup\mathbb{R}^1_\aleph$ (Recall that $0<\mathcal{X}<1$ is implicit in the absence of explicit statements to the contrary.)
3065    \item $L=\widehat\infty$
3066\end{itemize}
3067 
3068\noindent  In Theorems \ref{thm:finlinseggg} and \ref{thm:finlinseggg2}, we were able to prove the cases $L\in\mathbb{R}_0$ and $L=\widehat\infty$ respectively but we did not yet have sufficient tools to easily demonstrate the cases of $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$ and  $L\in\mathbb{R}_\aleph^\mathcal{X}\cup\mathbb{R}^1_\aleph$.  We still have not decided whether or not $\mathbb{R}^0_\aleph\setminus\mathbb{R}_0=\varnothing$ but, by this point, we have given the tools needed to prove the many-to-one relationship between real numbers and points in a line segment with $L\in\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}^1_0$.  This is the third case given above modified with a restriction to the natural neighborhoods of $\aleph_\mathcal{X}$ rather than the whole neighborhoods so $\text{Lit}(L)\in\mathbb{R}_0$.  We will give this result in the present section.  The present section also contains various and sundry theorems and examples, the most exciting of which is left as a surprise.
3069 
3070 
3071 
3072\begin{thm}\label{thm:finlinseggg3}
3073    If $AB$ is a real line segment with finite length $L\in\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}^1_0$, then no point $X\in AB$ has a unique algebraic representation as one and only one real number.
3074\end{thm}
3075 
3076\begin{proof}
3077	From the statement of the theorem, we have $L=\text{len}(AB)=\aleph_\mathcal{X}+b$ with $\mathcal{X}>0$.  By Definition \ref{def:XrepR}, every point in a line segment has an algebraic representation
3078    \begin{equation}
3079    X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
3080    \end{equation}
3081   
3082    \noindent It follows that
3083    \begin{equation}
3084    \min[\mathcal{D}^\dagger_{\!AB}(AX)]=\cfrac{\text{len}[0,x_1]}{\text{len}[0,\aleph_\mathcal{X}+b]} =\frac{x_1}{\aleph_\mathcal{X}+b}~~,\nonumber
3085    \end{equation}
3086   
3087    \noindent Now suppose $x_0\in\mathbb{R}_0^+$, and $z=x_1+x_0$ so that $z> x_1$.  Then
3088    \begin{equation}
3089    \cfrac{\text{len}[0,z]}{\text{len}[0,\aleph_\mathcal{X}+b]} =\frac{z}{\aleph_\mathcal{X}+b}=\frac{x_1+x_0}{\aleph_\mathcal{X}+b} =\frac{x_1}{\aleph_\mathcal{X}+b}+\frac{x_0}{\aleph_\mathcal{X}+b}~~.\nonumber
3090    \end{equation}
3091   
3092    \noindent By Axiom \ref{ax:div1g1g1g1}, the $x_0$ term vanishes so we find
3093    \begin{equation}
3094    \cfrac{\text{len}[0,z]}{\text{len}[0,\aleph_\mathcal{X}+b]} =\frac{x_1}{\aleph_\mathcal{X}+b}=\min[\mathcal{D}^\dagger_{\!AB}(AX)]~~.\nonumber
3095    \end{equation}
3096       
3097    \noindent Invoking the single-valuedness of bijective functions, we find that
3098    \begin{equation*}
3099    \min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\max[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\frac{x_2}{\aleph_\mathcal{X}+b}\quad\implies\quad x_1<z\leq x_2~~.
3100    \end{equation*}
3101   
3102    \noindent Therefore $x_1\neq x_2$ and the theorem is proven.
3103\end{proof}
3104 
3105 
3106\begin{thm}\label{thm:ijdsvoiydt97c}
3107    The derivative of $f(x)=\aleph_x$ with respect to $x$ is infinite.
3108\end{thm}
3109 
3110\begin{proof}
3111    The definition of the derivative of $f(x)$ with respect to $x$ is
3112    \begin{equation*}
3113    \dfrac{d}{dx}\,f(x)=\lim\limits_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}~~.
3114    \end{equation*}
3115   
3116    \noindent For $f(x)=\aleph_x$, we have
3117    \begin{align*}
3118    \dfrac{d}{dx}\,\aleph_x&=\lim\limits_{\Delta x\to0}\frac{\aleph_{(x+\Delta x)}-\aleph_x}{\Delta x}\\
3119    &=\lim\limits_{\Delta x\to0}\frac{\aleph_x+\aleph_{\Delta x}-\aleph_x}{\Delta x}\\
3120    &=\lim\limits_{\Delta x\to0}\frac{ 1 }{\Delta x}\aleph_{\Delta x}\\
3121    &=\aleph_1~~.
3122    \end{align*}
3123\end{proof}
3124 
3125 
3126\begin{defin}\label{def:y98t9tuigjaaaa}
3127    For $0<\mathcal{X}<1$, $\mathbb{N}_\mathcal{X}$ is a subset of real numbers such that
3128    \begin{equation*}
3129    \mathbb{N}_\mathcal{X}=\big\{ \aleph_\mathcal{X}+w~\big|~w\in\mathbb{W}  \big\}~~.
3130    \end{equation*}
3131   
3132    \noindent where the whole numbers are $\mathbb{W}=\{...,-2,-1,0,1,2,...\}$.  The set  $\{\mathbb{N}_\mathcal{X}\}$ is called the set of all $\mathbb{N}_\mathcal{X}$ such that $0<\mathcal{X}<1$.  Complementing $\mathbb{N}$ in the neighborhood of the origin, define a set
3133    \begin{equation*}
3134    \mathbb{N}_1=\big\{ \widehat\infty-n  ~\big|~ n\in\mathbb{N}\big\}~~,
3135    \end{equation*}
3136   
3137    \noindent called natural numbers in the maximal neighborhood of infinity.  The set of all extended natural numbers is
3138    \begin{equation*}
3139    \mathbb{N}_\infty=\mathbb{N}\cup\{\mathbb{N}_\mathcal{X}\}\cup\mathbb{N}_1~~.
3140    \end{equation*}
3141\end{defin}
3142 
3143 
3144\begin{defin}
3145    The function $E^x$ is defined as
3146    \begin{equation*}
3147    E^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}~~,
3148    \end{equation*}
3149   
3150    \noindent where the sum is taken to mean all $k\in\mathbb{N}_\infty\cup\{0\}$.  This function is called the big exponential function.
3151\end{defin}
3152 
3153\begin{thm}
3154    For any $x\in\mathbb{R}_0$, the big exponential function is equal to the usual exponential function:
3155    \begin{equation*}
3156    x\in\mathbb{R}_0\quad\implies\quad E^x=e^x~~.
3157    \end{equation*}   
3158\end{thm}
3159 
3160\begin{proof}
3161    The usual exponential function is
3162    \begin{equation*}
3163    e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}~~,
3164    \end{equation*}
3165   
3166    \noindent with the upper bound on $k$ meaning ``as $k$ increases without bound'' but also giving an implicit restriction $k\in\mathbb{N}_0=\mathbb{N}\cup\{0\}$.  To prove the present theorem, it will suffice to show that all terms vanish for $k\not\in\mathbb{N}$.  We have
3167    \begin{align*}
3168    E^x&=\sum_{k\in\mathbb{N}_0}\frac{x^k}{k!}+\sum_{k\in\mathbb{N}_{\mathcal{X}_1}}\frac{x^k}{k!}+\sum_{k\in\mathbb{N}_{\mathcal{X}_2}}\frac{x^k}{k!}+...\\
3169    &=e^x+\sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}^{0}}}k\in\mathbb{N}_0}}^{\infty}\frac{x^{\left(\aleph_{\mathcal{X}_1}+k\right)}}{\big(\aleph_{\mathcal{X}_1}+k\big)!}+\sum_{\substack{k=1\\\vphantom{\hat{\mathbb{N}^{0}}}k\in\mathbb{N}}}^{\infty}\frac{x^{\left(\aleph_{\mathcal{X}_1}-k\right)}}{\big(\aleph_{\mathcal{X}_1}-k\big)!}+\sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}}}k\in\mathbb{N}_0}}^{\infty}\frac{x^{\left(\aleph_{\mathcal{X}_2}+k\right)}}{\big(\aleph_{\mathcal{X}_2}+k\big)!}+...~~.
3170    \end{align*}
3171   
3172    \noindent Now it will suffice to show that the sum over $k\in\mathbb{N}_{\mathcal{X}}$ vanishes for any $\mathcal{X}>0$.  Observe that
3173    \begin{align*}
3174    \sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}^{0}}}k\in\mathbb{N}_0}}^{\infty}\frac{x^{\left(\aleph_{\mathcal{X}}\pm k\right)}}{\big(\aleph_{\mathcal{X}}\pm k\big)!}&=\sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}}}k\in\mathbb{N}_0}}^{\infty}\frac{x}{\big(\aleph_{\mathcal{X}}\pm k\big)}\frac{x}{\big(\aleph_{\mathcal{X}}\pm k-1\big)}\frac{x^{\left(\aleph_{\mathcal{X}}\pm k-2\right)}}{\big(\aleph_{\mathcal{X}}\pm k-2\big)!}\\
3175    &=\sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}}}k\in\mathbb{N}_0}}^{\infty}0\cdot0\cdot...0\cdot0\cdot\frac{x^{\left(\aleph_{\mathcal{X}}\pm k-1\right)}}{\big(\aleph_{\mathcal{X}}\pm k-1\big)!}=0\\
3176    \end{align*}
3177   
3178    \noindent We have shown that every term of $E^x$ which is not in $e^x$ vanishes whenever $x\in\mathbb{R}_0$.  This proves the theorem.
3179\end{proof}
3180 
3181 
3182\begin{exa}\label{exa:762}
3183    This example gives a good thinking device for understanding  limits $n\to\infty$ when $n$ steps in integer multiples.  Usually this is taken to mean ``as the iterator $n$ increases without bound.''  In this example, we will argue that $n\to\infty$ is better interpreted as meaning ``the sum over every $n\in\mathbb{N}_\infty$.''
3184    Definition \ref{def:RRRinf} gives two definitions for the $\infty$ symbol, one of which is
3185    \begin{equation}
3186    \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
3187    \end{equation}
3188   
3189    \noindent The $n\to\infty$ limit of the partial sums is taken to mean ``as $n$ increases without bound'' without a self-referential presupposition of the number defined by the limit.  Axiom \ref{ax:fieldssss33} (the closure of $\mathbb{R}_0$ under its operations) tells us that the partial sums will always be another $\mathbb{R}_0$ number for any $n\in\mathbb{N}$.  For any $\mathcal{X}>0$, it follows that the sum will be less than $\aleph_\mathcal{X}$ but the notion ``as $n$ increases without bound'' induces the notion of the non-convergence of the partial sums.  In turn, this allows us to think of the sum as exceeding $\aleph_\mathcal{X}$.  However, it may more plainly demonstrate the notion of non-convergence when we take $n\to\infty$ to mean the sum over all $n\in\mathbb{N}_\infty$.  In that case, the partial sums will eventually have individual terms greater than $\aleph_\mathcal{X}$ for any $0<\mathcal{X}<1$.  It is immediately obvious that $\aleph_\mathcal{X}$ cannot be an upper bound on the partial sums over $n\in\mathbb{N}_\infty$.  The big part of the partial sums will easily exceed $|\aleph_1|=\infty$.  Taking $m\in\mathbb{N}$, observe that the $n\in\mathbb{N}_\infty$ convention gives
3190    \begin{equation*}
3191     \lim\limits_{n\to\infty} \sum_{k=1}^{n}k > \lim\limits_{n\to m} \sum_{k=1}^{n}\big( \aleph_{(m^{-1})}+k  \big)>m\aleph_{(m^{-1})}=\aleph_1 ~~.\nonumber
3192    \end{equation*}
3193   
3194    \noindent Now it is plainly obvious that the limit of the partial sums diverges in $\mathbb{R}$.  Certainly, it is obvious that the partial sums diverge in either case but it may be more obvious when $n\in\mathbb{N}_\infty$.  When $n$ is said to increase without bound and is also taken as $n\in\mathbb{N}$, then there is an intuitive hiccup seeing that the sequence of the sums should diverge when every element in the sequence of partial sums is less that $\aleph_\mathcal{X}\in\mathbb{R}$ for any $\mathcal{X}>0$.  Instead, it is better to think of the $n\to\infty$ notation as meaning the sum over all $n\in\mathbb{N}_\infty$.
3195   
3196    This example has demonstrated the utility of $\mathbb{N}_\infty$ as a thinking device, and it also makes a distinction between the two formulae
3197    \begin{equation}
3198    \lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\infty~~, \quad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
3199    \end{equation}
3200   
3201    \noindent In the partial sums definition, and under the $\mathbb{N}_\infty$ convention, the distinction between geometric infinity and algebraic infinity is suggested as
3202    \begin{equation*}
3203    \widehat\infty=\aleph_1~~,\quad\text{and}\qquad\infty=\aleph_\infty~~.
3204    \end{equation*}
3205   
3206    \noindent This convention would require a significant revision of the entire paper to accommodate $|\aleph_1|\neq|\aleph_\infty|$ but we point out the possibility of the alternative convention with a nod toward future inquiry.   Note, however, that the present convention for either definition of $\infty$ is preserved with Definition \ref{def:RRRinf9999}: both sums diverge in $\mathbb{R}$ and we cannot differentiate $\aleph_1$ from $\aleph_\infty$ without first making a transfinite analytic continuation.  This continuation is surely something to be explored because it is the longitudinal continuation of $\mathbb{R}$ beyond its endpoints perfectly dual to the famous transverse continuation of $\mathbb{R}$ onto $\mathbb{C}$.  Where the latter has yielded so much fruit in the history of mathematics, the former ought to bear some fruit as well.
3207\end{exa}
3208 
3209 
3210 
3211 
3212\begin{mainthm}\label{mthm:giutqx}
3213    If $ABC$ is a right triangle such that $\angle ABC=\frac{\pi}{2}$, $\|AB\|=\aleph_\mathcal{X}+x$, $\|BC\|=\aleph_\mathcal{Y}+y$, and such that $\|AB\|\neq c_0\|BC\|$, and if the Pythagorean theorem is phrased as
3214    \begin{equation*}
3215    \|AC\|=\sqrt{\vphantom{\hat A}   \|AB\|^2+  \|BC\|^2 }~~,\quad\text{with}\qquad\|AC\|=\text{len}(AC)~~,
3216    \end{equation*}
3217   
3218    \noindent then
3219    \begin{equation*}
3220    \text{len}(AC)\not\in\mathbb{R}^0_\aleph\cup\{\mathbb{R}^\mathcal{X}_\aleph\}\cup\mathbb{R}^1_\aleph ~~.
3221    \end{equation*}
3222\end{mainthm}
3223 
3224 
3225\begin{proof}
3226    The squared lengths of the legs are
3227    \begin{equation*}
3228    \|AB\|^2=\big( \aleph_\mathcal{X}+x \big)^2=\aleph_{\left(\aleph_{(\mathcal{X}^2)}+2x\mathcal{X}\right)}+x^2~~,
3229    \end{equation*}
3230   
3231    \noindent and
3232    \begin{equation*}
3233    \|BC\|^2=\big( \aleph_\mathcal{Y}+y \big)^2=\aleph_{\left(\aleph_{(\mathcal{Y}^2)}+2y\mathcal{Y}\right)}+y^2~~.
3234    \end{equation*}
3235   
3236    \noindent If we directly state the Pythagorean theorem in terms of the lengths then we find
3237    \begin{equation*}
3238    \|AC\|^2=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2~~.
3239    \end{equation*}
3240   
3241    \noindent Assuming $\|AC\|=\aleph_\mathcal{A}+a$, we find
3242    \begin{equation*}
3243    \aleph_{\left(\aleph_{(\mathcal{A}^2)}+2a\mathcal{A}\right)}+a^2=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2~~.
3244    \end{equation*}
3245   
3246    \noindent Setting the big parts equal yields
3247    \begin{equation*}
3248    \aleph_{(\mathcal{A}^2)}+2a\mathcal{A}=\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})~~,
3249    \end{equation*}
3250   
3251    \noindent which still has separable big and little parts.  Doing the maximum possible separation of all the big and little parts yields
3252    \begin{align*}
3253    \mathcal{A}^2&=\mathcal{X}^2+\mathcal{Y}^2\\
3254    a\mathcal{A}&=x\mathcal{X}+y\mathcal{Y}\\
3255    a^2&=x^2+y^2~~.
3256    \end{align*}
3257   
3258    \noindent Here we have three inconsistent equations in two variables $a$ and $\mathcal{A}$.  No real-valued length $\|AC\|$ squared will satisfy the Pythagorean theorem as stated.   The theorem is proven.
3259\end{proof}
3260 
3261 
3262 
3263\begin{defin}\label{def:oydc986sdv8}
3264    A number is a complex number $z\in\mathbb{C}$ if and only if
3265    \begin{equation}
3266    z=x+iy~~,\quad\text{and}\qquad x,y\in\mathbb{R}~~.\nonumber
3267    \end{equation}
3268\end{defin}
3269 
3270\begin{thm}\label{mthm:gfff222x}
3271    If we assign an algebraic representation to the hypotenuse $AC\equiv z\in\mathbb{C}$ rather than the $AC\equiv \|AC\|\in\mathbb{R}$ disallowed by Main Theorem \ref{mthm:giutqx}, then the Pythagorean identity is satisfied by $AC^2\equiv \bar zz$.
3272\end{thm}
3273 
3274\begin{proof}
3275    Given two legs, we want to find the hypotenuse through the Pythagorean theorem.  We assume that the legs are real line segments so that
3276    \begin{equation*}
3277    AB^2\equiv\|AB\|^2~~,\quad\text{and}\qquad BC^2\equiv\|BC\|^2~~.
3278    \end{equation*}
3279 
3280   
3281    \noindent The geometric identity
3282    \begin{equation*}
3283    AC^2=AB^2+BC^2~~,
3284    \end{equation*}
3285   
3286    \noindent needs an algebraic interpretation if we are to do trigonometry with ``non-algebraic'' numbers like $\pi$ or $\tau$.  The present theorem concerns the ``squared,'' exponent $2$ operation being identified as multiplication by the complex conjugate in the sense that the inner product of a 1D vector $\vec z\in\mathbb{C}^1$ with itself is $\vec z^2=\langle z|z\rangle=\bar zz$.  The vector space axioms are known to be satisfied in $\mathbb{C}=\mathbb{C}^1$ so it is only an irrelevant matter of notation whether we specify a complex number $z$ or a 1D complex vector $\vec z$.  However, the satisfaction of the Pythagorean identity relies critically on the multiplicative $AC^2$ being identified as the multiplication by the complex conjugate
3287    \begin{equation*}
3288    AC^2\equiv\langle {AC}|{AC}\rangle~~.
3289    \end{equation*}
3290   
3291    \noindent Since the legs are taken as real, the algebraic representation of each is its own complex conjugate.  Again we find
3292    \begin{equation*}
3293    \big\langle AB\big|AB\big\rangle=\big( \aleph_\mathcal{X}+x \big)^2=\aleph_{\left(\aleph_{(\mathcal{X}^2)}+2x\mathcal{X}\right)}+x^2~~,
3294    \end{equation*}
3295   
3296    \noindent and
3297    \begin{equation*}
3298    \big\langle BC\big|BC\big\rangle=\big( \aleph_\mathcal{Y}+y \big)^2=\aleph_{\left(\aleph_{(\mathcal{Y}^2)}+2y\mathcal{Y}\right)}+y^2~~.
3299    \end{equation*}
3300   
3301    \noindent The present statement of the Pythagorean theorem is
3302    \begin{equation*}
3303    \big\langle AC\big|AC\big\rangle= \aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2~~.
3304    \end{equation*}
3305   
3306    \noindent Let $z\in\mathbb{C}$ be such that it conforms to Definition \ref{def:oydc986sdv8}, such that $AC\equiv z=|AC\rangle$, and such that
3307    \begin{equation*}
3308    z=\aleph_{(\mathcal{X}\pm i\mathcal{Y})}+x\pm iy=\big(\aleph_\mathcal{X}+x\big)+i\big(\aleph_\mathcal{Y}+y\big)~~.
3309    \end{equation*}
3310   
3311    \noindent We have
3312    \begin{align*}
3313    \bar zz=\big\langle AC\big|AC\big\rangle
3314    &=\big[\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}+(x\pm iy)\big]^{\!*}\big[\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}+(x\pm iy)\big]\\
3315    &=\big[\aleph_{\left(\mathcal{X}\mp i\mathcal{Y}\right)}+(x\mp iy)\big]\big[\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}+(x\pm iy)\big]\\
3316    &=\aleph_{\left(\mathcal{X}\mp i\mathcal{Y}\right)\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}}+\aleph_{\left(\mathcal{X}\mp i\mathcal{Y}\right)}(x\pm iy)\\
3317    &\qquad\qquad+\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}(x\mp iy)+(x\mp iy)(x\pm iy)\\
3318    &=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}\right)}+\aleph_{\left( x\mathcal{X}\pm iy\mathcal{X}\mp ix\mathcal{Y}+y\mathcal{Y} \right)}\\
3319    &\qquad\qquad+\aleph_{\left( x\mathcal{X}\mp iy\mathcal{X}\pm ix\mathcal{Y}+y\mathcal{Y} \right)}+x^2+y^2\\
3320    &=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2 ~~.
3321    \end{align*}
3322   
3323    \noindent Therefore,
3324    \begin{equation*}
3325    \big\langle AC\big|AC\big\rangle=\big\langle AB\big|AB\big\rangle+\big\langle BC\big|BC\big\rangle~~.
3326    \end{equation*}  
3327\end{proof}
3328 
3329\begin{cor}\label{mthm:gffff222x}
3330    If we assign an algebraic representation to the hypotenuse $AC\equiv \vec x\in\mathbb{R}^2$ rather than the $AC\equiv \|AC\|\in\mathbb{R}$ disallowed by Main Theorem \ref{mthm:giutqx}, then the Pythagorean identity is satisfied by $AC^2\equiv \vec x \cdot \vec x$.
3331\end{cor}
3332 
3333\begin{proof}
3334    This corollary follows from Theorem \ref{mthm:gfff222x} in the way that everything in $\mathbb{C}$ has two equivalent vector space representations $\mathbb{C}^1$ and $\mathbb{R}^2$.  Let $\vec x\in\mathbb{R}^2$ be the Cartesian plane equipped as a vector space.  We have three real vectors in $\mathbb{R}^2$:
3335    \begin{equation*}
3336    \vec{AB}=(\aleph_\mathcal{X}+x,0) ~~,~~ \vec{BC}=(0,\aleph_\mathcal{Y}+y~~,\quad\text{and}\qquad  \vec{AC}=(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)~~.
3337    \end{equation*}
3338   
3339    \noindent The Pythagorean theorem yields   
3340    \begin{align*}
3341    \vec {AC}\cdot\vec{AC}&=\vec {AB}\cdot\vec{AB}+\vec {BC}\cdot\vec{BC}~~.
3342    \end{align*}
3343   
3344    \noindent Again we find
3345    \begin{equation*}
3346    \vec{AB}\cdot\vec{AB}=\big( \aleph_\mathcal{X}+x \big)^2=\aleph_{\left(\aleph_{(\mathcal{X}^2)}+2x\mathcal{X}\right)}+x^2~~,
3347    \end{equation*}
3348   
3349    \noindent and
3350    \begin{equation*}
3351    \vec{BC}\cdot\vec{BC}=\big( \aleph_\mathcal{Y}+y \big)^2=\aleph_{\left(\aleph_{(\mathcal{Y}^2)}+2y\mathcal{Y}\right)}+y^2~~.
3352    \end{equation*}
3353   
3354    \noindent Checking the given form of $\vec{AC}\in\mathbb{R}^2$, we find
3355     \begin{align*}
3356     \vec{AC}\cdot\vec{AC}&=(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)\cdot(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)\\
3357     &=\vec {AB}\cdot\vec{AB}+\vec {BC}\cdot\vec{BC}
3358     \end{align*}
3359     
3360     \noindent Therefore, the Pythagorean identity is satisfied with an algebraic representation of the hypotenuse $AC$ such that $AC\equiv\vec{AC}\in\mathbb{R}^2$.  The theorem is proven.
3361\end{proof}
3362 
3363\begin{exa}
3364    If a right triangle has two equal legs $AB=BC$, then the hypotenuse $AC$ should be such that $\text{len}(AC)=\sqrt{2}\cdot\|AB\|$.  In this example, we will make the comparisons to the $z\in\mathbb{C}$ and $\vec x\in\mathbb{R}^2$ statements of the Pythagorean theorem.
3365   
3366    TAKE $\sqrt{\bar zz}$
3367   
3368\end{exa}
3369 
3370 
3371=======================
3372 
3373=======================
3374 
3375=======================
3376 
3377 
3378Is the Pythagorean Theorem a geometric statement or an algebraic one?
3379 
3380 
3381=======================
3382 
3383=======================
3384 
3385=======================
3386 
3387 
3388\begin{exa}\label{exa:f3ede1xxxx3}
3389    This example demonstrates a ramification of Example \ref{ex:626262g2vll} for the ordinary notions of trigonometry.  Consider a right triangle $ABC$ such that $\angle ABC=\frac{\pi}{2}$.  Suppose $\|AB\|=\aleph_\mathcal{X}+x$ and $\|BC\|=\aleph_\mathcal{Y}+y$.  Let $\alpha=\angle C\!AB$ so that the ordinary notions of trigonometry suggest
3390    \begin{equation}\label{eq:jjj3j3j3}
3391    \|AC\|\sin(\alpha)=\aleph_\mathcal{Y}+y~~,\quad\text{and}\qquad \|AC\|\cos(\alpha)=\aleph_\mathcal{X}+x~~.
3392    \end{equation}
3393   
3394    \noindent It follows that
3395    \begin{equation*}
3396    \|AC\|=\aleph_{\left(\frac{\mathcal{Y}}{\sin(\alpha)}\right)}+\frac{y}{\sin(\alpha)}~~,\quad\text{and}\qquad \|AC\|=\aleph_{\left(\frac{\mathcal{X}}{\cos(\alpha)}\right)}+\frac{x}{\cos(\alpha)}
3397    \end{equation*}
3398   
3399    \noindent Equating the big and little parts yields
3400    \begin{equation*}
3401    \tan(\alpha)=\frac{\mathcal{Y}}{\mathcal{X}}~~,\quad\text{and}\qquad \tan(\alpha)=\frac{y}{x}~~.
3402    \end{equation*}
3403   
3404    \noindent This is a contradiction for every case in which $\frac{\mathcal{Y}}{\mathcal{X}}\neq \frac{y}{x}$.  This is a perfectly consistent result; if the trigonometry functions in Equation (\ref{eq:jjj3j3j3}) are real-valued, and each RHS is, then the equality cannot hold for complex-valued $\|AC\|$.  
3405\end{exa}
3406 
3407 
3408 
3409\begin{rem}
3410    In leaving the real line and going onto the plane, we have exceeded the scope of this paper.  Other than a $\mathbb{C}$ application is Section \ref{sec:g54584688} to demonstrate the negation of the Riemann hypothesis, we will not presently exceed the confines of $\mathbb{R}$.  Even given the solution to that very famous problem in Section \ref{sec:g54584688}, however, it is the opinion of this writer that the principle demonstrated in Main Theorem \ref{mthm:giutqx} is certainly the most important result given herein.  It demonstrates cleanly that the extension $L\in\mathbb{R}_0\to L\in\mathbb{R}_\infty$ is not the trivial exercise that might be intuitively assumed.  Among the two interpretations for the Pythagorean identity, the $z\in\mathbb{C}$ representation of the length of the hypotenuse is more relevant because $z$ is a 1D scalar number whose real part is a cut in the real number line.  In other words, $z$ is a cut in the real number line added to a cut in the imaginary number line.  Since cuts in the real number line are known to have both zero and non-zero imaginary parts, $z\in\mathbb{C}$ is far more germane to $\mathbb{R}$ than is $\vec x\in\mathbb{R}^2$.  Vector structure requires an entire axiomatic framework for vector arithmetic but all of the arithmetic for $z\in\mathbb{C}$ can be derived from the arithmetic axioms through the Definition \ref{def:oydc986sdv8} giving complex numbers as pairs of real numbers.
3411\end{rem}
3412 
3413 
3414 
3415 
3416 
3417\begin{thm}\label{thm:t24t24t24fffft}
3418    A real number with non-vanishing big part is not the product of any real number with itself, \textit{i.e.}:
3419    \begin{equation*}
3420    \centernot\exists x\in\mathbb{R}\quad\text{s.t.}\quad x^2\in\{\mathbb{R}^\mathcal{X}_\aleph\}\cup\mathbb{R}^1_\aleph
3421    \end{equation*}
3422\end{thm}
3423 
3424\begin{proof}
3425    Assume there exists a square root $z=\aleph_\mathcal{Z}+a$ of $x=\aleph_\mathcal{X}+b$ so that
3426    \begin{equation}\label{eq:87587f55757}
3427    \big( \aleph_\mathcal{Z}+a \big)^2=\aleph_\mathcal{X}+b~~.
3428    \end{equation}
3429   
3430    \noindent We have
3431    \begin{equation*}
3432    \big( \aleph_\mathcal{Z}+a \big)^2=\aleph_{\left(\aleph_{(\mathcal{Z}^2)}+2a\mathcal{Z}\right)}+z^2~~,
3433    \end{equation*}
3434   
3435    \noindent so we should set the big and little parts of the left and right sides of Equation (\ref{eq:87587f55757}) equal to each other.  This gives
3436    \begin{equation*}
3437    \aleph_{\left(\mathcal{Z}^2\right)}+2a\mathcal{Z}=\mathcal{X}~~,\quad\text{and}\qquad z^2=b~~.
3438    \end{equation*}
3439   
3440    \noindent The former constraint equation gives $\mathcal{Z}=0$ because the RHS has zero big part.  It follows that $\mathcal{X}=0$.  This is a contradiction because we have already selected $\aleph_\mathcal{X}$ as the non-zero big part of $x$.
3441\end{proof}
3442 
3443 
3444 
3445\begin{exa}\label{exa:f42tl2444}
3446    Consider the limit
3447    \begin{equation*}
3448    \lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=l~~.
3449    \end{equation*}
3450   
3451    \noindent It remains to be clarified precisely what is meant by the notation $b\to\infty$ because we should have options for at least two distinct behaviors.  For example, one might wish to define
3452    \begin{equation*}
3453    \lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=-\infty~~,\quad\text{and}\qquad\lim\limits_{ b\to\widehat\infty} \aleph_\mathcal{X}-b=\aleph_\mathcal{X}-\widehat\infty=-\aleph_{(1-\mathcal{X})}~~,
3454    \end{equation*}
3455   
3456    \noindent where $b\to\widehat\infty$ means that $b$ approaches $\widehat\infty$ while $b\to\infty$ would mean that $b$ increases without bound---even including transfinite numbers larger than $\widehat\infty$---such that $b$ approaches some geometric infinity whose absolute value is in some sense greater than that of algebraic infinity.  We will not make such definitions here because the requisite formal definitions for $x>\widehat\infty$ are out of scope.  However, simply based on the absorption or non-absorption of $\infty$ and $\widehat\infty$ respectively, the limits given in this example should be presumed correct.
3457    \end{exa}
3458 
3459 
3460 
3461 
3462\subsection{The Archimedes Property of Real Numbers}\label{sec:archim}
3463 
3464While there are many ways to state the Archimedes property of real numbers with symbolic logic, the modern establishment has adopted
3465\begin{equation}
3466\forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists n\in\mathbb{N}\quad\text{s.t.}\quad nx>y~~.\nonumber
3467\end{equation}
3468 
3469\noindent For this statement to accurately characterize the property as it appeared in the first edition Greek language copy of Euclid's \textit{Elements}, it must depend on an unstated axiom that every real number is less than some natural number.  Without that axiom, the statement is wrong and there is no other word than ``wrong'' by which it should be described.  In this section, we will consult the original text in Euclid's \textit{Elements} \cite{EE}.  We will use the original text to prove absolutely that the above symbolic statement is not the Archimedes property of real numbers given so famously by Euclid in Reference \cite{EE}.  For the above statement to agree with that which was given by Euclid in Greek, one must first take the axiom that every real number is less than some natural number.  Without a statement or implicit acknowledgment of such an axiom, the above Latin symbolic statement is \textit{wrongly} called the Archimedes property of real numbers.
3470 
3471 
3472 
3473\begin{defin}\label{def:eudox}
3474    The statement of the Archimedes property which appears in Euclid's \textit{Elements}, and which was attributed by Archimedes to his predecessor Eudoxus, and which must be taken as \textit{the} definitive statement of the Archimedes property of real numbers, appears as Definition 4 in Book 5 of \textit{The Elements}.  The original Greek is translated as follows \cite{EE}.
3475    \begin{quote}
3476        ``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3477    \end{quote}
3478\end{defin}
3479 
3480\begin{rem}
3481    \label{rem:32322222v}
3482    As it appears in Euclid's \textit{Elements}, the straightforward mathematical statement of the property would be
3483    \begin{equation}
3484    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z\in\mathbb{R}\quad\text{s.t.}\quad zx>y~~.\nonumber
3485    \end{equation}
3486   
3487    \noindent There is no mention of multiplication by a positive integer $n\in\mathbb{N}$.  To prove that the Archimedes property of real numbers which was recorded by Euclid in \textit{The Elements} does not implicitly restrict the multiplier to $n\in\mathbb{N}$, we will examine the context of the original text.
3488\end{rem}
3489 
3490\begin{defin}
3491    In Reference \cite{EE}, Fitzpatrick translates Book 5, Definitions 1 through 5 as follows.
3492    \begin{enumerate}
3493        \item A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.
3494        \item And the greater is a multiple of the lesser whenever it is measured by the lesser.
3495        \item A ratio is a certain type of condition with respect to size of two magnitudes of the same kind.
3496        \item (Those) magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.
3497        \item Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when equal multiples of the first and third both exceed, are both equal to, or are both less than, equal multiples of the second and fourth, respectively, being taken in corresponding order, according to any kind of multiplication whatever.
3498    \end{enumerate}
3499\end{defin}
3500 
3501\begin{rem}
3502    Though we may prove directly from Euclid's own words that the multiplier in the Archimedes property of real numbers is not defined as a natural number, Fitzpatrick gives footnotes qualifying his translations of Euclid's original Greek.  These footnotes support the wrongness of the supposition that Euclid meant to imply that the multiplier in his definition must always be a natural number.  We will list the footnotes here for thoroughness though we will not rely on them in Theorem \ref{thm:euclid}.  The footnotes are as follows.
3503    \begin{enumerate}
3504        \item In other words, $\alpha$ is said to be a part of $\beta$ if $\beta=m\alpha$.
3505        \item (\textit{No footnote given.})
3506        \item In modern notation, the ratio of two magnitudes, $\alpha$ and $\beta$, is denoted $\alpha\,:\,\beta$.
3507        \item In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.
3508        \item In other words, $\alpha\,:\,\beta\,::\,\gamma\,:\,\delta$ if and only if $m\alpha>n\beta$ whenever $m\gamma>n\delta$, $m\alpha=n\beta$ whenever $m\gamma=n\delta$, and $m\alpha<n\beta$ whenever $m\gamma<n\delta$, for all $m$ and $n$.  This definition is the kernel of Eudoxus' theory of proportion, and is valid even if $\alpha$, $\beta$, \textit{etc.}, are irrational.
3509    \end{enumerate}
3510 
3511    \noindent Footnote 5 makes it exceedingly obvious that the multipliers are ``all $m$ and $n$'' in $\mathbb{R}$.
3512\end{rem}
3513 
3514\begin{thm}\label{thm:euclid}
3515    The statement
3516    \begin{equation}
3517    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists n\in\mathbb{N}\quad\text{s.t.}\quad nx>y~~.\nonumber
3518    \end{equation}
3519   
3520    \noindent is not a proper statement of the Archimedes property of real numbers as given in antiquity.  
3521\end{thm}
3522 
3523\begin{proof}
3524    It follows from Book 5, Definition 5, of Euclid's original text that if $y\in\mathbb{R}$ is a multiple of $z\in\mathbb{R}$, then there exists some ``multiplier'' $x\in\mathbb{R}$ such that $xy=z$.  For proof by contradiction, assume that Euclid meant to restrict the multiplier in his definitions as $n\in\mathbb{N}$, and then consider Definition 2:
3525    \begin{quote}
3526        ``And the greater is a multiple of the lesser whenever it is measured by the lesser.''
3527    \end{quote}
3528 
3529    \noindent Suppose $y=2$ and $z=3$ so that among the two numbers, $z$ is the greater.  If the multiplier by which $z$ is to be measured by $y$ is restricted to $n\in\mathbb{N}$ rather than $x\in\mathbb{R}$, then $z$ cannot be measured by $y$.  This is \textbf{\textit{an affront to reason}}, firstly, and it directly contradicts Definition 1:
3530    \begin{quote}
3531        ``A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.''
3532    \end{quote}
3533 
3534    \noindent It is self-evidently true that $3>2$ so for $2$ to be a part of $3$ means it must measure the greater.  ``Measure'' is defined by Definition 2 in terms of multiples which are thence defined in terms of multiplication.  For $2$ be a part of $3$ in the sense of Definition 1, we must do multiplication with a multiplier $x=1.5\not\in\mathbb{N}$.  The theorem is proven.
3535\end{proof}
3536 
3537\begin{rem}
3538    In Book 7, Definition 2, Euclid defines ``numbers'' as natural numbers but what are today called real numbers are instead the ``magnitudes'' described in Book 5.  Euclid in no way implied that the multiplier in Definition 4 should be taken strictly as $n\in\mathbb{N}$ and so neither was Euclid of the opinion that Archimedes meant to do so in his own earlier paraphrasing of Eudoxus.
3539\end{rem}
3540 
3541\begin{exa}
3542    This example demonstrates that if one presupposes the non-existence of real numbers greater than any natural number, taking it purely as an unproven axiomatic definition, one which violates the contrary proof of the existence of such numbers given in Main Theorem \ref{thm:ef2424t24cc}, then the statement
3543    \begin{equation}
3544    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists n\in\mathbb{N}\quad\text{s.t.}\quad nx>y~~.\nonumber
3545    \end{equation}
3546   
3547    \noindent does adequately encapsulate the Archimedes property of antiquity.  Proof of this statement is given by Rudin in Reference \cite{BRUDIN} as follows.
3548   
3549        \begin{quote}
3550            ``Let $A$ be the set of all $nx$, where $n$ runs through the positive integers.  If [\textit{the symbolic statement given in the present example}] were false, then $y$ would be an upper bound of $A$.  But then $A$ has a least upper bound in $\mathbb{R}$.  Put $\alpha=\sup A$.  Since $x>0$, $\alpha-x<\alpha$, and $\alpha-x$ is not an upper bound of $A$.  Hence $\alpha-x<mx$ for some positive integer $m$.  But then $\alpha<(m+1)x\in A$, which is impossible since $\alpha$ is an upper bound of $A$.''
3551        \end{quote}
3552   
3553    \noindent Here Rudin has followed the reasoning of Proposition \ref{mthm:u9999979y} in which it was claimed that $\mathbb{R}_0$ cannot have a supremum.  We will revisit this issue most specifically in Section \ref{sec:r3r23r23r3r}.
3554    \end{exa}
3555 
3556 
3557 
3558\begin{rem}
3559    If we adopt
3560    \begin{equation}
3561    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z\in\mathbb{R}\quad\text{s.t.}\quad zx>y~~.\nonumber
3562    \end{equation}
3563   
3564    \noindent as the definitive statement of the Archimedes property, as in Remark \ref{rem:32322222v}, then we will have taken away the Archimedes property of real numbers from the maximal whole neighborhood of infinity $\mathbb{R}^1_\aleph$.  For instance, if
3565    \begin{equation*}
3566    \widehat\infty-a<\widehat\infty-b~~,
3567    \end{equation*}
3568   
3569    \noindent then we cannot multiply the LHS by a number greater than one and have a real-valued product due to the identity $\aleph_\mathcal{X}=\mathcal{X}
3570    \cdot\widehat\infty$.  If we multiply by a positive number less than one, call it $\delta$, then
3571    \begin{equation*}
3572    \aleph_\delta-\delta a<\widehat\infty-a<\widehat\infty-b~~,
3573    \end{equation*}
3574   
3575    \noindent does \textit{not} conform to the Archimedean requirement that $\delta(\widehat\infty-a)>y$.  If this caused us to eject $\mathbb{R}^1_\aleph\not\in\mathbb{R}$ because such numbers were found not to exhibit the Archimedes property, then would cause a breakdown in Axiom \ref{ax:mainR} giving $\mathbb{R}=(-\infty,\infty)$.    If we suppose, correctly, that all real numbers obey the Archimedes property, then we might write concisely
3576    \begin{equation}\label{eq:699869869880}
3577    \mathbb{R}^+=(0,\infty)\setminus\mathbb{R}^1_\aleph~~.
3578    \end{equation}
3579   
3580    \noindent Even this is highly disfavorable because we lose the perfect geometric infinite line construction that we have sought to preserve by modifying the canonical algebraic construction by equivalence classes.  In terms of the topology, Equation (\ref{eq:699869869880}) breaks the usual topology of $\mathbb{R}^+$ such that its basis is all open subsets $(a,b)\subset(0,\infty)$.
3581   
3582    In what manner shall the maximal neighborhood of infinity exhibit the Archimedes property of real numbers?  How might we solve this problem?  The answer lies in Euclid's original Greek:
3583        \begin{quote}
3584        ``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3585    \end{quote}
3586 
3587    \noindent In Remark \ref{rem:32322222v}, we have adopted the convention that the multiplier must attach to the lesser part of $x<y$ but no such requirement is given by Euclid's totally symmetric statement.  For one to exceed the other upon multiplication allows us to state the property in terms of multiplication of either the greater or the lesser among $x$ and $y$.  In Euclid's own parlance, for one to exceed the other only requires that each is a ``part'' or ``multiple'' of the other without a requirement for which is which.  Taking careful note of the non-specificity of the ordering relation in Definition 4, we will preserve the highly favorable definition $\mathbb{R}=(-\infty,\infty)$ by giving a symbolic statement of the Archimedes property obeyed by $x\in\mathbb{R}^1_\aleph$.  At the end of this section, we will give a new modern statement of the Archimedes property such that its application is greatly simplified. First, we will formally show that the fractional distance model of $\mathbb{R}$ obeys the symbolically complexified restatement of Euclid's handful of original Greek words.  Once we show that the ancient definition is satisfied, will make a simplifying axiom such that demonstrating the property is simplified.
3588\end{rem}
3589 
3590\begin{defin}\label{def:lllllllllll3}
3591    The most general statement of the ancient Archimedes property of real numbers is
3592    \begin{equation}
3593    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z_1,z_2\in\mathbb{R}^+\quad\text{s.t.}\quad z_1x>z_2y~~.\nonumber
3594    \end{equation}
3595\end{defin}
3596 
3597 
3598\begin{mainthm}\label{mthm:tc87dc5}
3599    The present construction of $\mathbb{R}$ is such that every $x,y\in\mathbb{R}^0_\aleph\cup\{\mathbb{R}^\mathcal{X}_\aleph\}\cup\mathbb{R}^1_\aleph$ exhibit the ancient Archimedes property of real numbers.
3600\end{mainthm}
3601 
3602\begin{proof}
3603    By Definition \ref{def:lllllllllll3}, it suffices to demonstrate
3604    \begin{equation}
3605    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z_1,z_2\in\mathbb{R}^+\quad\text{s.t.}\quad z_1x>z_2y~~.\nonumber
3606    \end{equation}
3607   
3608    \noindent We will consider the general forms
3609    \begin{equation*}
3610    x=\aleph_\mathcal{X}+b~~,\quad\text{and}\qquad y=\aleph_\mathcal{Y}+a~~,
3611    \end{equation*}
3612   
3613    \noindent such that $x\in\mathbb{R}_\aleph^\mathcal{X}$ and $y\in\mathbb{R}_\aleph^\mathcal{Y}$, and we will assume constraints $x<y$ and $0\leq\mathcal{X}\leq\mathcal{Y}\leq1$.  Further assume that  $a$ and $b$ constrained appropriately for $\mathcal{X}$ and/or $\mathcal{Y}$ equal to one or zero so that $x$ and $y$ are always in $\mathbb{R}^+$.  The starting point for demonstrating the Archimedes property is $x<y$ which we write as
3614    \begin{equation*}
3615    \aleph_\mathcal{X}+b<\aleph_\mathcal{Y}+a~~.
3616    \end{equation*}
3617   
3618    \noindent To prove the theorem, we will consider the distinct cases.  In each equality listed below, we put $z_1x$ on the left and $z_2y$ on the right.
3619   
3620    $~$
3621   
3622    \noindent $\bullet$ ($x\in\mathbb{R}^0_\aleph$ and $y\in\mathbb{R}^0_\aleph$) Here, both $x$ and $y$ have vanishing big parts so $x<y$ defines the ordering of the little parts.  Choose $z_1=\aleph_\mathcal{Z}+z$ such that $0<\mathcal{Z}b<1$ and choose $z_2=1$.  Then
3623    \begin{equation*}
3624    \big(\aleph_\mathcal{Z}+z\big)b=\aleph_{(\mathcal{Z}b)}+zb\quad>\quad a~~.
3625    \end{equation*}  
3626   
3627    $~$
3628   
3629    \noindent $\bullet$ ($x\in\mathbb{R}^0_\aleph$ and $y\in
3630    \{\mathbb{R}^\mathcal{Y}_\aleph\}$) Here, $x$ has a vanishing big part and $y$ has a non-vanishing big part.  Choose $z_1=\aleph_{\left(\frac{1+\mathcal{Y}}{2b}\right)}+z$ and  $z_2=1$.  Then
3631    \begin{equation*}
3632    \left(\aleph_{\left(\frac{1+\mathcal{Y}}{2b}\right)}+z\right)b=\aleph_{\left(\frac{1+\mathcal{Y}}{2}\right)}+zb\quad>\quad \aleph_{\mathcal{Y}}+a~~.
3633    \end{equation*}  
3634   
3635    \noindent Since $\frac{1+\mathcal{Y}}{2}$ is the average of $\mathcal{Y}$ and $1$, it is guaranteed to be a number in the interval $(\mathcal{Y},1)$.  
3636   
3637    $~$
3638   
3639    \noindent $\bullet$ ($x\in\mathbb{R}^0_\aleph$ and $y\in
3640    \mathbb{R}^1_\aleph$) Here, $x$ has a vanishing big part and $y$ has big part $\aleph_1$.  Choose $z_1=\aleph_{\mathcal{Z}}+z$ and $z_2$ such that $0<z_2<\mathcal{Z}b<1$.  Then
3641    \begin{equation*}
3642    \left(\aleph_\mathcal{Z}+z\right)b=\aleph_{(\mathcal{Z}b)}+zb\quad>\quad z_2\big(\aleph_1-a\big)=\aleph_{z_2}-z_2a~~.
3643    \end{equation*}  
3644   
3645    $~$
3646   
3647    \noindent $\bullet$ ($x\in\mathbb{R}^\mathcal{X}_\aleph$ and $y\in\mathbb{R}^\mathcal{Y}_\aleph$ such that $\mathcal{X}<\mathcal{Y}$) Here, neither $x$ nor $y$ has a vanishing big part and the big part of $x$ is less than big part of $y$.  Choose $z_1=\frac{1+\mathcal{Y}}{2\mathcal{X}}$ and $z_2=1$.  Then
3648    \begin{equation*}
3649    \frac{1+\mathcal{Y}}{2\mathcal{X}}\left(\aleph_{\mathcal{X}}+b\right)=\aleph_{\left(\frac{1+\mathcal{Y}}{2}\right)}+b\frac{1+\mathcal{Y}}{2\mathcal{X}} \quad>\quad \aleph_{\mathcal{Y}}+a~~.
3650    \end{equation*}  
3651   
3652    $~$
3653   
3654    \noindent $\bullet$ ($x,y\in\mathbb{R}^\mathcal{X}_\aleph$ such that $\mathcal{X}=\mathcal{Y}$) Here, $x$ and $y$ have equal big parts so it follows from $x<y$ that the little parts are ordered accordingly.  Choose $z_1=z$ such that $\mathcal{X}<z\mathcal{X}<1$ and  $z_2=1$.  Then
3655    \begin{equation*}
3656    z\left(\aleph_{\mathcal{X}}+b\right)=\aleph_{(z\mathcal{X})}+zb \quad>\quad \aleph_{\mathcal{X}}+a~~.
3657    \end{equation*}  
3658   
3659    $~$
3660   
3661    \noindent $\bullet$ ($x\in\mathbb{R}^\mathcal{X}_\aleph$ and $y\in\mathbb{R}^1_\aleph$) Here, $x$ and $y$ have unequal big parts with the big part of $y$ being the greater.  Choose $z_1=1$ and $z_2=\frac{\mathcal{X}}{2}$.  Then
3662    \begin{equation*}
3663    \aleph_{\mathcal{X}}+b \quad>\quad \frac{\mathcal{X}}{2}\big(\aleph_1-a\big)=\aleph_{\left(\frac{\mathcal{X}}{2}\right)}-\frac{a\mathcal{X}}{2 }~~.
3664    \end{equation*}  
3665   
3666    $~$
3667   
3668    \noindent $\bullet$ ($x,y\in\mathbb{R}^1_\aleph$) Here, $x$ and $y$ have equal big parts $\aleph_1$ and $x<y$ defines the ordering of the little parts.  Choose $z_1=1$ and $z_2=\frac{1}{2}$.  Then
3669    \begin{equation*}
3670    \aleph_{1}+b \quad>\quad \frac{1}{2}\big(\aleph_1-a\big)=\aleph_{\left(\frac{1}{2}\right)}-\frac{a}{2 }~~.
3671    \end{equation*}  
3672   
3673     \noindent We have considered every combination of $x<y$ among the various neighborhoods and shows that they all comply with Definition \ref{def:lllllllllll3}.  The theorem is proven.  
3674\end{proof}
3675 
3676\begin{rem}
3677     If it were not for the extremal case of $x\in\mathbb{R}^0_\aleph$ and $y\in \mathbb{R}^1_\aleph$, we might have formulated the symbolic statement of the Archimedes property as
3678     \begin{equation}
3679     \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z\in\mathbb{R}^+\quad\text{s.t.}\quad zx>y\quad\text{or}\quad x>zy~~.\nonumber
3680     \end{equation}
3681     
3682     \noindent This form is nice because it uses only a single multiplication operation and exactly reflects Fitzpatrick's footnote:
3683     \begin{quote}
3684        ``In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.''
3685     \end{quote}
3686     
3687     \noindent However, it is not possible to phrase the symbolic statement of the property with only a single multiplier because of the extremal case in which $x$ is in the neighborhood of the origin and $y$ is in the maximal neighborhood of infinity.  Even then, Euclid does not precisely require a condition of the form, ``multiplication of one can exceed the other.''  As it is written:
3688      \begin{quote}
3689        ``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3690     \end{quote}
3691 
3692    \noindent This statement can be equally clarified with a similar but slightly different footnote than what Fitzpatrick has given.  An alternative footnote explaining the meaning of the property would be the following.
3693     \begin{quote}
3694        In other words, $\alpha$ has a ratio with respect to $\beta$ if $m_1\alpha>n_1\beta$ and $m_2\beta>n_2\alpha$, for some $m_1,m_2,n_1$, and $n_2$.
3695    \end{quote}
3696 
3697    \noindent This is exactly what is given in Definition \ref{def:lllllllllll3} and it is well consistent with the ratio of ratios language seen in Book 5, Definition 5.
3698   
3699    In general we have made a rather large statement of Euclid's few original words in the form
3700    \begin{equation}\label{eq:689985986f}
3701    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists z_1,z_2\in\mathbb{R}^+\quad\text{s.t.}\quad z_1x>z_2y~~.
3702    \end{equation}
3703   
3704    \noindent The reasoning behind including the Archimedes property of real numbers as a supplemental constraint on the behavior of cuts in the real number line is that it is supposed to be an elegantly simple statement of a simple behavior.  Certainly, Equation (\ref{eq:689985986f}) is not elegant.  Therefore, having already independently demonstrated rigorous compliance with Euclid in the absence of a simplifying axiom, now we will give one.
3705\end{rem}
3706 
3707 
3708\begin{axio}\label{ax:nncncncn}
3709    The Archimedes property of 1D transfinitely continued real numbers is
3710    \begin{equation*}
3711    \forall x,y\in\mathbb{R}\quad\text{s.t.}\quad x<y\quad \exists n\in \mathbb{N}\quad\text{s.t.}\quad \aleph_nx>y~~.
3712    \end{equation*}
3713   
3714    \noindent This axiom defines the implicit transfinite ordering required for $\leq$ to be a relation among real numbers and 1D transfinitely continued real numbers whose big parts are greater than $\widehat\infty$.  As a subset of the 1D transfinitely continued  real numbers, the real numbers themselves automatically inherit compliance with the Archimedes property.
3715\end{axio}
3716 
3717\begin{rem}
3718    If the real number line ends at infinity, that indicates an endpoint there.  Endpoints are associated with $\widehat\infty$ when we take the convention that the notion of infinite geometric extent precludes the existence of endpoints at $\infty$.  Therefore, the lack of a terminating point for the line at infinity automatically implies the 1D transfinite continuation of $\mathbb{R}=(-\aleph_1,\aleph_1)$ onto $\mathbb{T}=(-\aleph_{\widehat\infty},\aleph_{\widehat\infty})$.  If it didn't continue onto $\mathbb{R}$, then it would end at $\infty$, a contradiction.  There is no requirement whatsoever that $\widehat\infty=\aleph_1$ is the largest number; it is only required that it is the supremum of the real numbers.  Axiom \ref{ax:nncncncn} generates the requite definition of transfinite ordering such that given $x,y\in\mathbb{R}$ and $x<y$, $zx$ can be greater than $y$ without $zx$ itself being $zx\in\mathbb{R}$.  Here we define ordering for $zx\in\mathbb{T}$, and then we use this ordering to satisfy the $zx>y$ condition irreducibly cited in \textit{The Elements}.  In the scheme of Axiom \ref{ax:nncncncn}, all the cases of $x<y$ statements in Main Theorem \ref{mthm:tc87dc5} are replaced with elegantly simple formulae.
3719\end{rem}
3720 
3721 
3722 
3723\section{The Topology of the Real Number Line}\label{sec:topoR}
3724 
3725 
3726 
3727\subsection{Basic Set Properties}
3728 
3729In this section, we give some elementary set properties of the natural neighborhoods and begin to approach the connection to the whole neighborhoods.  Recall that the natural neighborhoods $\mathbb{R}_0^\mathcal{X}$ are defined with little part $|b|\in\mathbb{R}_0^0$ and the whole neighborhoods are defined with $|b|\in\mathbb{R}^0_\aleph$.
3730 
3731\begin{lem}\label{thm:opee}
3732    Every natural neighborhood in $\{\mathbb{R}_0^\mathcal{X}\}$ is an open set.
3733\end{lem}
3734 
3735 
3736\begin{proof}
3737    By Definition \ref{def:ffmdmmdd3md}, the set of all intermediate natural neighborhoods of infinity is
3738    \begin{equation*}
3739    \big\{\mathbb{R}^\mathcal{X}_0\big\}=\{ \aleph_\mathcal{X}+b~| ~ b\in  \mathbb{R}_0 ,~0<\mathcal{X}<1   \} ~~.
3740    \end{equation*}
3741   
3742    \noindent A given $\mathbb{R}^\mathcal{X}_0$ defined with a particular $\mathcal{X}$ is open if and only if there is a $\delta$-neighborhood of each of its elements such that every element of that neighborhood is also an element of $\mathbb{R}^\mathcal{X}_0$.  We use the ball function $\delta$-neighborhood as in Definition \ref{def:delta0x00} rather than \ref{def:delta000} because the elements of $\mathbb{R}^\mathcal{X}_0$ are numbers, not points.  This theorem is proven with a $\delta$-neighborhood of an arbitrary $x\in\mathbb{R}^\mathcal{X}_0$.  Defining $b^\pm=b\pm\delta$, we have 
3743    \begin{equation}
3744    \text{Ball}(x\in\mathbb{R}_0^\mathcal{X},\delta)=(\aleph_\mathcal{X}+b-\delta,\aleph_\mathcal{X}+b+\delta)=(\aleph_\mathcal{X}+b^-,\aleph_\mathcal{X}+b^+)~~.\nonumber
3745    \end{equation}
3746   
3747    \noindent Axiom \ref{ax:fieldssss33} requires that $\mathbb{R}_0$ is closed under the $\pm$ operations so $b,\delta\in\mathbb{R}_0$ implies $b^\pm\in\mathbb{R}_0$.  The set $\mathbb{R}^\mathcal{X}_0$ is open because    
3748    \begin{equation}
3749    (\aleph_\mathcal{X}+b^-,\aleph_\mathcal{X}+b^+)\subset\mathbb{R}^\mathcal{X}_0=\big\{\aleph_\mathcal{X}+b~\big|~b\in\mathbb{R}_0   \big\}~~.\nonumber
3750    \end{equation}
3751   
3752    Alternatively, no set in $\{\mathbb{R}_0^\mathcal{X}\}$ contains its boundary points so each such set is open.
3753\end{proof}
3754 
3755 
3756 
3757 
3758\begin{thm}\label{thm:f3444xx}
3759    Given two natural neighborhoods $\mathbb{R}^{\mathcal{X}}_0$ and $\mathbb{R}^{\mathcal{Y}}_0$ with $0\leq\mathcal{X}<\mathcal{Y}\leq1$, there exists another natural  neighborhood $\mathbb{R}^{\mathcal{Z}}_0$ such that $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$.
3760\end{thm}
3761 
3762\begin{proof}
3763    Consider the interval
3764    \begin{equation}
3765    (\aleph_{\mathcal{X}},\aleph_{\mathcal{Y}})\subset\mathbb{R}~~.\nonumber
3766    \end{equation}
3767   
3768    \noindent By Definition \ref{def:3533553}, the number at the center of this interval is
3769    \begin{equation}
3770    \cfrac{\aleph_{\mathcal{Y}}+\aleph_{\mathcal{X}}}{2}=\aleph_{\left(\!\frac{\mathcal{Y}+\mathcal{X}}{2}\!\right)}~~.\nonumber
3771    \end{equation}
3772   
3773    \noindent We have
3774    \begin{equation}
3775    \mathcal{X}<\frac{\mathcal{Y}+\mathcal{X}}{2}<\mathcal{Y}~~,\nonumber
3776    \end{equation}
3777   
3778    \noindent so let $\mathcal{Z}=\frac{\mathcal{Y}+\mathcal{X}}{2}$.  Any number $z\in Z\in\mathbf{AB}$ of the form
3779    \begin{equation}
3780    z=\aleph_{\mathcal{Z}}+z_0~~,\quad\text{for}\qquad |z_0|\in\mathbb{R}_0~~,\nonumber
3781    \end{equation}
3782   
3783    \noindent will be such that
3784    \begin{equation}
3785    \mathcal{D}_{\!\mathbf{AB}}(AZ)=\mathcal{Z}~~.\nonumber
3786    \end{equation}
3787   
3788    \noindent Since $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$, the theorem is proven.
3789\end{proof}
3790 
3791 
3792\begin{cor}
3793    Given two whole neighborhoods $\mathbb{R}^{\mathcal{X}}_\aleph$ and $\mathbb{R}^{\mathcal{Y}}_\aleph$ with $0\leq\mathcal{X}<\mathcal{Y}\leq1$, there exists another whole neighborhood $\mathbb{R}^{\mathcal{Z}}_\aleph$ such that $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$.
3794\end{cor}
3795 
3796\begin{proof}
3797    Following the proof of Theorem \ref{thm:f3444xx}, we arrive at a number $z\in Z\in\mathbf{AB}$ of the form
3798    \begin{equation}
3799    z=\aleph_{\mathcal{Z}}+z_0~~,\quad\text{for}\qquad z_0\in\mathbb{R}^0_\aleph~~,\nonumber
3800    \end{equation}
3801   
3802    \noindent Even in the whole neighborhood, $z_0$ has no fractional magnitude with respect to $\mathbf{AB}$.  Therefore, the total fractional distance is still completely determined by the big part as
3803    \begin{equation}
3804    \text{Big}(z)=\aleph_\mathcal{Z}\quad\iff\quad \mathcal{D}_{\!\mathbf{AB}}(AZ)=\mathcal{Z}~~.\nonumber
3805    \end{equation}
3806   
3807    \noindent Proof follows from $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$, as in Theorem \ref{thm:f3444xx}.
3808\end{proof}
3809 
3810 
3811 
3812\begin{defin}
3813    An open set $S$ is disconnected if and only if there exist two open, non-empty sets $U$ and $V$ such that
3814    \begin{equation*}
3815    S=U\cup V~~,\quad\text{and}\qquad U\cap V=\varnothing~~.
3816    \end{equation*}
3817   
3818    \noindent If a set is not disconnected, then it is connected.
3819\end{defin}
3820 
3821 
3822\begin{cor}\label{cor:et3443454}
3823    $\mathbb{R}^{\mathcal{X}}_0\cup\mathbb{R}^{\mathcal{Y}}_0$ is a disconnected set for any $0\leq\mathcal{X}<\mathcal{Y}\leq1$.
3824\end{cor}
3825 
3826\begin{proof}
3827    An open set is disconnected if it is the union of two disjoint, non-empty open sets.  By Lemma \ref{thm:opee}, $\mathbb{R}^{\mathcal{X}}_0$ is open, and it is obvious that such sets are non-empty.  It follows from Theorem \ref{thm:f3444xx} that they are disjoint, \textit{i.e.}:
3828    \begin{equation}
3829    \mathbb{R}^{\mathcal{X}}_0\cap\mathbb{R}^{\mathcal{Y}}_0=\varnothing~~.\nonumber
3830    \end{equation}
3831   
3832    \noindent The union  $\mathbb{R}^{\mathcal{X}}_0\cup\mathbb{R}^{\mathcal{Y}}_0$ satisfies the definition of a disconnected set.  
3833\end{proof}
3834 
3835 
3836 
3837 
3838\begin{rem}
3839    During the development of the intermediate neighborhoods of infinity, we found it useful to separate the $\mathcal{X}=0$ and $\mathcal{X}=1$ cases from the intermediate neighborhoods $\{\mathbb{R}_\aleph^\mathcal{X}\}$.  For efficacy of notation, now we will combine all the different neighborhoods into a streamlined, unified notation.  The following definitions supplement Definitions \ref{def:gtg34535335} and \ref{def:ffmdmmdd3md} to include the cases of $\mathcal{X}=0$ and $\mathcal{X}=1$.
3840\end{rem}
3841 
3842 
3843\begin{defin}\label{def:77777yy}
3844    To streamline notation, define
3845    \begin{align*}
3846    \mathbb{R}_\aleph^\cup&= \bigcup\limits_{0\leq\mathcal{X}\leq1}\! \mathbb{R}^\mathcal{X}_\aleph=\mathbb{R}_\aleph^0\cup\big\{\mathbb{R}^\mathcal{X}_\aleph\big\}\cup\mathbb{R}_\aleph^1\\
3847    \mathbb{R}_0^\cup&=  \bigcup\limits_{0\leq\mathcal{X}\leq1} \!\mathbb{R}^\mathcal{X}_0=\mathbb{R}_0^0\cup\big\{\mathbb{R}^\mathcal{X}_0\big\}\cup\mathbb{R}_0^1~~.
3848    \end{align*}
3849\end{defin}
3850 
3851\begin{defin}\label{def:ugu2g2g2g222}
3852    The complement of $\mathbb{R}_0^\mathcal{X}$ in $\mathbb{R}^\mathcal{X}_\aleph$ is $\mathbb{R}^\mathcal{X}_C$:
3853    \begin{equation}
3854    \mathbb{R}^\mathcal{X}_C=\mathbb{R}^\mathcal{X}_\aleph\setminus \mathbb{R}_0^\mathcal{X}~~.\nonumber
3855    \end{equation}
3856\end{defin}
3857 
3858 
3859\begin{thm}\label{thm:nosdx}
3860    There exist more positive real numbers than are in $\mathbb{R}^\cup_0$.  In other words,
3861    \begin{equation*}
3862    \mathbb{R}^+\setminus\mathbb{R}^\cup_0\neq\varnothing~~.
3863    \end{equation*}
3864\end{thm}
3865 
3866 
3867 
3868\begin{proof}
3869    By the definition of an interval, and through Axiom \ref{ax:mainR} overtly granting the connectedness of $\mathbb{R}=(-\infty,\infty)$, the interval $\mathbb{R}^+=(0,\infty)$ is connected.  To prove the present theorem, it will suffice to show that $\mathbb{R}_0^\cup$ is disconnected.  Disconnection follows from Corollary \ref{cor:et3443454}.
3870\end{proof}
3871 
3872\begin{rem}
3873    It was already expected that there may be real numbers not contained in the natural neighborhoods.  It was for this reason that we defined distinct whole neighborhoods $\mathbb{R}^\mathcal{X}_\aleph\supseteq\mathbb{R}_0^\mathcal{X}$.  In Section \ref{sec:bbbhh}, we will conjecture $\mathbb{R}_C^\mathcal{X}=\varnothing$ but first we will prove another result, one far more interesting.
3874\end{rem}
3875 
3876\begin{mainthm}\label{mthm:y7958t787555}
3877    There exist more positive real numbers than are in $\mathbb{R}^\cup_\aleph$.  In other words,
3878    \begin{equation*}
3879    \mathbb{R}^+\setminus\mathbb{R}^\cup_\aleph\neq\varnothing~~.
3880    \end{equation*}
3881\end{mainthm}
3882 
3883 
3884 
3885\begin{proof}
3886    $\mathbb{R}^+$ is a connected interval.  $\mathbb{R}^\cup_\aleph\setminus\{0\}$ is a disjoint union of open subsets of $\mathbb{R}^+$.  A connected interval cannot be covered with such a disconnected set.  The theorem is proven.
3887\end{proof}
3888 
3889 
3890 
3891 
3892 
3893 
3894\subsection{Cantor-like Sets of Real Numbers}\label{sec:bbbhhv}
3895 
3896In this section, we will continue to develop the properties of $\mathbb{R}$ by comparing the properties of $\mathbb{R}^+\setminus\mathbb{R}^\cup_\aleph$ and $\mathbb{R}^+\setminus\mathbb{R}^\cup_0$ to the well-known properties of Cantor sets.  
3897 
3898 
3899\begin{defin}\label{def:cantordddd}
3900    Munkres constructs a Cantor set $C$ as follows \cite{MUNK}.  
3901   
3902    \begin{quote}
3903        ``Let $A_0$ be the closed interval $[0,1]$ in $\mathbb{R}$.  Let $A_1$ be the set obtained from $A_0$ by deleting its `middle third' $(\frac{1}{3},\frac{2}{3})$.  Let $A_2$ be the set obtained from $A_1$ by deleting its `middle thirds' $(\frac{1}{9},\frac{2}{9})$ and $(\frac{7}{9},\frac{8}{9})$.  In general, define $A_n$ by the equation
3904        \begin{equation*}
3905        A_n=A_{n-1}-\bigcup\limits_{k=0}^\infty\left(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\right)~~.
3906        \end{equation*}
3907       
3908        \noindent The intersection
3909        \begin{equation*}
3910        C=\bigcap\limits_{n\in\mathbb{Z}^+}A_n~~,
3911        \end{equation*}
3912       
3913        \noindent is called the [\textit{ternary}] Cantor set; it is a subspace of $[0,1]$.''
3914    \end{quote}
3915\end{defin}
3916 
3917 
3918\begin{rem}
3919    The interval $[0,1]$ is the image of $\mathbf{AB}$ under the fractional distance map.  This likeness will serve as the basis for the analytical direction of the present section.
3920\end{rem}
3921 
3922 
3923 
3924\begin{defin}\label{def:v8v87wtugiddd}
3925    Define two Cantor-like sets
3926    \begin{equation*}
3927    \mathbb{F}_0=[0,\infty]\setminus\mathbb{R}^\cup_0~~,\quad\text{and}\qquad \mathbb{F}_\aleph=[0,\infty]\setminus\mathbb{R}^\cup_\aleph~~.
3928    \end{equation*}
3929\end{defin}
3930 
3931\begin{cor}
3932    Neither $\mathbb{F}_0$ nor $\mathbb{F}_\aleph$ is the empty set.
3933\end{cor}
3934 
3935\begin{proof}
3936    Proof follows from Theorem \ref{thm:nosdx} and Main Theorem \ref{mthm:y7958t787555}: there exist more positive real numbers than are in $\mathbb{R}^\cup$.  The interval $[0,\infty]$ is connected by Axiom \ref{ax:mainR}.  A connected set cannot be covered by a disjoint union of its open subsets.  $\mathbb{R}^\cup_\aleph$ and $\mathbb{R}^\cup_0$ are both disjoint unions of open subsets of $[0,\infty]$.  (We say $\mathbb{R}^0_0$ and $\mathbb{R}_\aleph^0$ are open sets in the subspace topology of $[0,\infty]$ even though they are not strictly open in $\mathbb{R}$.)  The corollary is proven.
3937\end{proof}
3938 
3939\begin{rem}
3940    To construct $\mathbb{F}_0$ and $\mathbb{F}_\aleph$, we have subtracted from $[0,\infty]$ the non-empty open neighborhood of $\aleph_\mathcal{X}$ for every infinite decimal number $0\leq\mathcal{X}\leq1$.  Whatever remains is a ``dust'' of some sort.  For this reason, we call $\mathbb{F}_0$ and $\mathbb{F}_\aleph$ Cantor-like sets.
3941\end{rem}
3942 
3943 
3944\begin{lem}\label{thm:utt2u2g211}
3945    $\mathbb{F}_\aleph$ is a subset of $\mathbb{F}_0$ or it is exactly equal to $\mathbb{F}_0$, \textit{i.e.}: $\mathbb{F}_\aleph\subseteq\mathbb{F}_0$.
3946\end{lem}
3947 
3948\begin{proof}
3949    Proof follows from Definition \ref{def:v8v87wtugiddd}.  $\mathbb{F}_\aleph$ is constructed by deleting open intervals whose lengths are at least as great as those deleted in the construction of $\mathbb{F}_0$.  Each variant of deleted interval is centered about $\aleph_\mathcal{X}$.  $\mathbb{F}_\aleph\subseteq\mathbb{F}_0$ because $\mathbb{R}_0^\mathcal{X}\subseteq\mathbb{R}^\mathcal{X}_\aleph$.  If $\mathbb{R}_C^\mathcal{X}=\varnothing$, then $\mathbb{F}_\aleph=\mathbb{F}_0$ (which is what we will choose in Section \ref{sec:bbbhh}.)
3950\end{proof}
3951 
3952 
3953 
3954\begin{thm}\label{thm:i8758755757}
3955    $\mathbb{F}_0$ and $\mathbb{F}_\aleph$ are closed subsets of $[0,\infty]$.
3956\end{thm}
3957 
3958\begin{proof}
3959    A subset $S\subset T$ is closed in $T$ if and only if its complement in $T$ is open.  The complements of $\mathbb{F}_0$ and $\mathbb{F}_\aleph$ in $[0,\infty]$ are $\mathbb{R}^\cup_0$ and $\mathbb{R}^\cup_\aleph$ respectively, both of which are disjoint unions of open sets.  $\mathbb{F}_0$ and $\mathbb{F}_\aleph$ are closed in $[0,\infty]$.
3960\end{proof}
3961 
3962\begin{rem}
3963    When constructing the ternary Cantor set (Definition \ref{def:cantordddd}), the least element of the final result of iterative deletions is zero.  By construction, the endpoints of the intervals left after each deletion of a middle third will remain forever so it is already given at the $A_1$ step that the least number in the parent interval $[0,1]$, which is zero, will be the least element of the resultant Cantor set.  When defining $\mathbb{F}$ in either variant, it is not immediately apparent what will be the least element because $0\in\mathbb{R}^0_\aleph$ or $0\in\mathbb{R}_0^0$ is deleted at the first step.  However, since $\mathbb{F}$ is closed, we know it does have a least element.
3964\end{rem}
3965 
3966\begin{defin}\label{def:7t8787xx}
3967    The connected elements of $\mathbb{F}_0$ are provisionally labeled $\mathbb{F}_0(n)$ and the connected elements of $\mathbb{F}_\aleph$ are labeled $\mathbb{F}_\aleph(n)$.  The labeling the convention in either variant is such that
3968    \begin{equation*}
3969    \forall x\in \mathbb{F}(n)\quad \forall y\in \mathbb{F}(m)\quad\text{s.t.}\quad n>m\quad\implies\quad x >y~~.
3970    \end{equation*}
3971   
3972    \noindent Each $\mathbb{F}(n)$ is connected and every two $\mathbb{F}(n),\mathbb{F}(m)$ are disconnected whenever $n\neq m$.
3973\end{defin}
3974 
3975\begin{rem}
3976    We have deleted an uncountable infinity of $\mathbb{R}^\mathcal{X}$ neighborhoods to construct $\mathbb{F}$.  The elements of $\mathbb{F}$ separate these neighborhoods so the cardinality of the disconnected elements $\mathbb{F}$ must be uncountably infinite.  Such elements cannot be enumerated with $n\in\mathbb{N}$.  To the contrary, the set $\mathbb{N}_\infty$ (Definition \ref{def:y98t9tuigjaaaa}) has a countably infinite number of elements $\aleph_\mathcal{X}-n$ and a similar number of $\aleph_\mathcal{X}+n$ for each of an uncountably infinite number of $\mathcal{X}$.  It is guaranteed that $n\in\mathbb{N}_\infty$ will provide a sufficient labeling scheme for $\mathbb{F}(n)$.  
3977\end{rem}
3978 
3979\begin{pro}\label{thm:2323353ddd}
3980    For every $\mathbb{F}_0(n)$ or $\mathbb{F}_\aleph(n)$, the respective subset of $\mathbb{R}^\cup_0$ or $\mathbb{R}^\cup_\aleph$ whose elements are less than any $x$ in $\mathbb{F}_0(n)$ or $\mathbb{F}_\aleph(n)$ has a supremum and the subset of $\mathbb{R}^\cup_0$ or $\mathbb{R}^\cup_\aleph$ whose elements are greater has an infimum.
3981\end{pro}
3982 
3983 
3984\begin{just}
3985	We will neglect the subscripts $0$ and $\aleph$ in this proof.  This proposition regards the extrema of a set of sets so those extrema will be sets themselves ordered by the big parts of the nested elements $x\in\mathbb{R}^\mathcal{X}\subset\mathbb{R}^\cup$.  Call $\mathbb{R}^\cup_-$ the set of all $\mathbb{R}^\mathcal{X}$ whose elements are less than any $x\in \mathbb{F}(n)$ and call the greater set $\mathbb{R}^\cup_+$.  By Definition \ref{def:7t8787xx}, $\mathbb{F}(n)$ is a connected interval and every two $\mathbb{F}(j),\mathbb{F}(k)$ are disconnected whenever $j\neq k$.  Furthermore, Corollary \ref{cor:et3443454} proves that every two $\mathbb{R}^{\mathcal{X}}\neq\mathbb{R}^{\mathcal{Y}}$ are disconnected.  Since $[0,\infty]$ is a connected union of $\mathbb{F}(n)$ and $\mathbb{R}^\cup$, with the former being closed intervals and the latter being open, it follows that the structure of $\mathbb{R}^+$ is an ordered union
3986    \begin{equation*}
3987    \mathbb{R}^+=...\, \mathbb{F}(n)\cup\mathbb{R}^{\mathcal{X}}\cup \mathbb{F}(n+1)  \cup\mathbb{R}^{\mathcal{Y}}\cup \mathbb{F}(n+2)\,...~~.
3988    \end{equation*}
3989   
3990    \noindent This contradicts Theorem \ref{thm:f3444xx}, however.  If there was an $\mathbb{R}^{\mathcal{Z}}$ between $\mathbb{R}^{\mathcal{X}}$ and $\mathbb{R}^{\mathcal{Y}}$, then it would necessarily be $\mathbb{R}^{\mathcal{Z}}\subset \mathbb{F}(n+1)$ contradicting the definition of $\mathbb{F}$ (Definition \ref{def:v8v87wtugiddd}).  The connected property of $\mathbb{R}$ requires, therefore, that we introduce an alternative labeling scheme before continuing.
3991   
3992    \begin{defin}\label{def:jhu777}
3993        For $n\in\mathbb{N}_\infty$, the connected elements of $ \mathbb{R }^\cup_0$ are labeled $ \mathbb{R}_0(n)$ and the connected elements of $ \mathbb{R}^\cup_\aleph$ are labeled $ \mathbb{R}_\aleph(n)$.  The labeling convention in either variant is such that
3994    \begin{equation*}
3995        \forall x\in \mathbb{R}(n)\quad \forall y\in \mathbb{R}(m)\quad\text{s.t.}\quad n>m\quad\implies\quad x >y~~.
3996    \end{equation*}
3997   
3998    \noindent It follows that $\mathbb{R}_0^0=\mathbb{R}_0(1)$ and $\mathbb{R}_\aleph^0=\mathbb{R}_\aleph(1)$.  We say that $\mathbb{R}_0(n)$ is the natural neighborhood of $\aleph(n)$ and $\mathbb{R}_\aleph(n)$ is its whole neighborhood.  Specifically, $\aleph(1)=\aleph_0=0$.
3999    \end{defin}
4000 
4001    Continuing with the justification of Proposition \ref{thm:2323353ddd}, we may now infer that $[0,\infty]$ is constructed from an ordered union of the form
4002    \begin{equation*}
4003    [0,\infty]=\mathbb{R}(1)\cup \mathbb{F}(1) \,... \, \mathbb{R}(n)\cup \mathbb{F}(n) \cup\mathbb{R}(n+1)\cup \mathbb{F}(n+1) \,...~~.
4004    \end{equation*}
4005   
4006    \noindent Since the connectedness of $\mathbb{R}$ requires the sequential alternation of the disconnected $\mathbb{R}(n)$ and $\mathbb{F}(n)$ in the total ordered union, it follows that $\mathbb{R}(k)$ is the supremum of $\mathbb{R}^\cup_-$ whose elements are less than any $x\in \mathbb{F}(k)$, and $\mathbb{R}^{\{k+1\}}$ is the infimum of $\mathbb{R}^\cup_+$ whose elements are greater than any $x\in \mathbb{F}(k)$.  This concludes the justification of Proposition \ref{thm:2323353ddd}.
4007\end{just}
4008 
4009 
4010 
4011\begin{rem}
4012    In Example \ref{exa:f42tl2444}, we considered the limits
4013    \begin{equation*}
4014    \lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=-\infty~~,\quad\text{and}\qquad\lim\limits_{ b\to\widehat\infty} \aleph_\mathcal{X}-b=\aleph_\mathcal{X}-\widehat\infty=-\aleph_{(1-\mathcal{X})}~~,
4015    \end{equation*}
4016   
4017    \noindent as two desirable modes of limit behavior.  Now the $\mathbb{F}(n)$ notation suggests a third desirable behavior such that
4018    \begin{equation*}
4019    \lim\limits_{ b\to\aleph(2)} \aleph(n)+b=\aleph(n+1)~~.
4020    \end{equation*}  
4021   
4022    \noindent It may or may not be possible to accommodate this limiting mode.  It might be that any sequence which does not converge its own local neighborhood of fractional distance must diverge all the way to infinity in one variety or another.  Indeed, the property $\frac{d}{dx}\aleph_x=\widehat\infty$ (Theorem \ref{thm:ijdsvoiydt97c}) suggests in some sense that once a sequence fails to converge in its local $\aleph_\mathcal{X}$-neighborhood, it has to keep diverging to some maximal value.  
4023   
4024    The likely issue with such a limiting mode as $ b\to\aleph(2)$, something which may even amount to a flaw in the justification of Proposition \ref{thm:2323353ddd}, is that $\aleph(2)=\aleph_{\mathcal{X}_{\text{min}}}$ is such that $\mathcal{X}_{\text{min}}$ is the smallest positive real number.  It is generally understood that no such number exists.  We have developed a requirement for such a number in the course of supporting Proposition \ref{thm:2323353ddd} but the lack of a smallest positive real number is so well-established that we might suppose no such number exists.  Contrary to our conjuring of $\aleph_\mathcal{X}$ by requirement, there exists a large body of demonstrations that no such $\mathcal{X}_{\text{min}}$ can exist while $\aleph_\mathcal{X}$ has only been supposed not to exist.  If such a number as $\mathcal{X}_{\text{min}}$ can be derived from the ordered union given in Proposition \ref{thm:2323353ddd}, then that would be very exciting.  However, there are many problems associated with such a line of reasoning.  We will present a few of them in Section \ref{sec:pdoxes} and then we will not use the $(n)$ notations in a formal way moving forward.  
4025\end{rem}
4026 
4027\begin{defin}\label{def:9y7986yccx}
4028    To avoid the problematic $(n)$ notation, label each connected element of $\mathbb{F}$ as $\mathbb{F}^\mathcal{X}$.  For every $\mathbb{R}^\mathcal{X}$, there exists a unique $\mathbb{F}^\mathcal{X}$ such that
4029    \begin{equation*}
4030    x\in\mathbb{R}^\mathcal{X}~~,~~ z\in\mathbb{F}^\mathcal{X}\quad\implies\quad x<z~~,
4031    \end{equation*}
4032   
4033    \noindent and  
4034    \begin{equation*}
4035     y\in\mathbb{R}^\mathcal{Y}~~,~~\mathcal{Y}>\mathcal{X}\quad\implies\quad y>z~~.
4036    \end{equation*}
4037   
4038    \noindent In other words, there is a closed interval $\mathbb{F}^\mathcal{X}$ right-adjacent to every $\mathbb{R}^\mathcal{X}$ whenever $0\leq\mathcal{X}<1$.  With this definition, we move the elements of $\mathbb{F}$ into the non-problematic superscript $\mathcal{X}$ labeling scheme as opposed to moving the $\mathbb{R}^\mathcal{X}$ into the $(n)$ scheme as in Definition \ref{def:jhu777}.
4039\end{defin}
4040
4041 
4042 
4043 
4044 
4045 
4046 
4047\subsection{Paradoxes Related to Infinitesimals}\label{sec:pdoxes}
4048 
4049In this section, we demonstrate a few paradoxes, or contradictions, invoked by the $\mathbb{R}_0(n)$ enumeration scheme and its corollary concept of a least positive real number.  We solve some of paradoxes in this section with the superscript $\mathcal{X}$ label (Definition \ref{def:9y7986yccx}) and the other paradoxes are resolved in Section \ref{sec:r3r23r23r3r}.
4050 
4051 
4052\begin{defin}\label{def:iiieieie}
4053    $\mathcal{F}(n)\in\mathbb{R}$ is the unique real number in the center of $\mathbb{F}_0(n)$ and $\mathbb{F}_\aleph(n)\subseteq\mathbb{F}_0(n)$ in the sense that for every $\mathcal{F}(n)+b\in\mathbb{F}(n)$ there exists a $\mathcal{F}(n)-b\in\mathbb{F}(n)$ (in either variant of $\mathbb{F}$.)  In the alternative labeling scheme, $\mathcal{F}_\mathcal{X}$ is the number in the center of $\mathbb{F}_0^\mathcal{X}$ and $\mathbb{F}_\aleph^\mathcal{X}$.  In either label, the number has the property
4054    \begin{equation*}
4055    \text{Big}(\mathcal{F})=\mathcal{F}~~,\quad\text{and}\qquad \text{Lit}(\mathcal{F})=0~~.
4056    \end{equation*}
4057\end{defin}
4058 
4059 
4060\begin{thm}
4061    The number $\mathcal{F}(1)=\mathcal{F}_0$ has infinitesimal fractional magnitude with respect to $\mathbf{AB}$.
4062\end{thm}
4063 
4064\begin{proof}
4065    We will use Robinson's standard non-standard definition of a hyperreal infinitesimal \cite{ARNS,GOLDBLATT}: A number $\varepsilon$ is a positive infinitesimal number if and only if
4066    \begin{equation*}
4067    \forall x\in\mathbb{R}^+ \quad\exists \varepsilon\not\in\mathbb{R}\quad\text{s.t.}\quad 0<\varepsilon<x~~.
4068    \end{equation*}
4069   
4070    \noindent By construction, $\mathcal{F}(1)$ is the number in the center of the gap between $\mathbb{R}_\aleph(1)=\mathbb{R}_\aleph^0$ and $\mathbb{R}_\aleph(2)\in\{\mathbb{R}_\aleph^\mathcal{X}\}$.  Since $\mathcal{F}(1)$ is not in $\mathbb{R}_\aleph(1)=\mathbb{R}^0_\aleph$, it cannot have zero fractional magnitude;  $\mathbb{R}_\aleph(1)$ is the set of all numbers having zero fractional magnitude along $\mathbf{AB}$.  If it had non-zero real fractional magnitude, then it would be $\mathcal{F}(1)\in\{\mathbb{R}_0^\mathcal{X}\}$, an obvious contradiction because $\mathcal{F}(1)$ has less fractional magnitude than any nested element in that set of sets.  If we denote the fractional magnitude of $\mathcal{F}(1)$ with the symbol $\varepsilon$, the properties of this magnitude are exactly those given above in the definition of an infinitesimal.  The theorem is proven.
4071\end{proof}
4072 
4073\begin{defin}
4074    A number is said to be a measurable number if it can exist within the algebraic representation of some $X\in\mathbf{AB}$.  If a number is not measurable, then it is immeasurable.
4075\end{defin}
4076 
4077 
4078\begin{thm}
4079    Every $x\in\mathbb{F}$ is an immeasurable number.
4080\end{thm}
4081 
4082\begin{proof}
4083    The FDFs are bijective between their domain $AB$ and the range $[0,1]\subset\mathbb{R}$.  The range is a real interval containing no numbers with infinitesimal parts so this tells us that $\mathcal{F}(n)$ is not in the algebraic representation of any geometric $X\in \mathbf{AB}$.  In the $\mathcal{X}$ notation, the numbers in each $\mathbb{F}_\mathcal{X}$ have infinitesimally more fractional distance along $\mathbf{AB}$ than the numbers in each $\mathbb{R}^\mathcal{X}$.
4084\end{proof}
4085 
4086 
4087\begin{rem}\label{rem:87585zzzz}
4088      We have granted that every geometric point $X$ has an algebraic representation (Axiom \ref{ax:779hzz}) but we have not required the opposite.  Therefore, there is no problem with an infinitesimal fractional magnitude for $\mathcal{F}(1)$ because there is no corresponding $X\in\mathbf{AB}$ that is required to have infinitesimal fractional distance along the real Euclidean line segment $\mathbf{AB}$.  Although $\mathcal{F}(1)$ has infinitesimal fractional magnitude, the number itself is very large.  It is greater than any natural number.
4089\end{rem}
4090 
4091 
4092 
4093 
4094 
4095 
4096\begin{pdox}\label{pdox:332323}
4097    Every $\mathcal{F}(n)$ has the property
4098    \begin{equation}\label{pdox:t875874rsss}
4099    \mathcal{F}(n)=\cfrac{\aleph(n)+\aleph(n+1)}{2}~~.
4100    \end{equation}
4101   
4102    \noindent Every $\mathbb{R}_0^\mathcal{X}$ can be obtained by a translation operation on another element of $\{\mathbb{R}^\mathcal{X}_0\}$.  Doing set-wise arithmetic, we may write, for instance
4103    \begin{equation*}
4104    \hat{T}(\aleph_\delta)\mathbb{R}_0^{(\mathcal{X}-\delta)}=\aleph_\delta+\mathbb{R}_0^{(\mathcal{X}-\delta)}=\mathbb{R}_0^\mathcal{X}~~.
4105    \end{equation*}
4106   
4107    \noindent Letting $AB\equiv [\aleph(n),\aleph(n+1)]$ for some $n\geq 2$, the translational invariance requires
4108    \begin{equation*}
4109    \text{len} \big( \mathbb{R}_0(n)\cap AB \big)=\text{len} \big( \mathbb{R}_0(n+1)\cap AB\big)~~.
4110    \end{equation*}
4111   
4112    \noindent Since
4113    \begin{equation*}
4114    AB=\big\{\mathbb{R}_0(n)\cap AB\big\}\cup\mathbb{F}_0(n)\cup\big\{\mathbb{R}_0(n+1)\cap AB\big\}~~,
4115    \end{equation*}
4116   
4117    \noindent and since $\mathcal{F}(k)$ is the number in the center of the closed interval $\mathbb{F}_0(k)$, it is obvious that $\mathcal{F}(n)$ is the unique number in the center of the line segment $AB$.  In the Euclidean metric, this number is always the average of the least and greatest numbers in the algebraic representations of $A$ and $B$ respectively.  However, if the $\mathbb{R}_0(n)$ notation is a label for $\mathbb{R}_0^\mathcal{X}$ where $\mathcal{X}$ is strictly a real number, then, using the original labeling scheme without $(n)$, we find
4118    \begin{equation*}
4119    \mathcal{F}(n)=\cfrac{\aleph_\mathcal{X}+\aleph_\mathcal{Y}}{2}=\aleph_{\left(\frac{\mathcal{X}+\mathcal{Y}}{2}\right)}~~.
4120    \end{equation*}
4121   
4122    \noindent This number is most obviously an element of $\mathbb{R}_0^{\left(\frac{\mathcal{X}+\mathcal{Y}}{2}\right)}$.  This contradicts the definition $\mathcal{F}(n)\in\mathbb{F}_0(n)$.  It is paradoxical that $\aleph(n+1)$ cannot have any corresponding $\aleph_\mathcal{X}$.
4123\end{pdox}
4124 
4125\begin{pres}
4126    Paradox \ref{pdox:332323} does not exist in the $\mathcal{F}_\mathcal{X}$ notation.  If we never suppose the existence of $\aleph(n+1)$, then there is no starting point in Equation (\ref{pdox:t875874rsss}) and the paradox cannot be demonstrated, \textit{i.e.}:
4127    \begin{equation*}
4128    \mathcal{F}_\mathcal{X}\neq\cfrac{\aleph^\mathcal{X}+\aleph^\mathcal{Y}}{2}~~.
4129    \end{equation*}
4130\end{pres}
4131 
4132\begin{pdox}\label{pdox:3323v3}
4133    The neighborhood of the origin contains numbers of the form $\aleph_\mathcal{X}+b$ for $b$ strictly non-negative (and $\mathcal{X}=0$) but every intermediate neighborhood allows both signs for $b$. It follows that
4134    \begin{equation*}
4135    \text{len}\big(\mathbb{R}_0(1)\big)=\frac{1}{2}\,\text{len}\big(\mathbb{R}_0(2)\big)~~,\quad\text{and}\qquad \mathcal{F}(1)=\frac{1}{3}\,\mathcal{F}(2)~~.
4136    \end{equation*}
4137   
4138    \noindent Every element of $\mathbb{R}_0(2)$ has positive real fractional magnitude because $\mathbb{R}_0(2)\subset\{\mathbb{R}_0^\mathcal{X}\}$ but if $\mathcal{F}(1)$ has infinitesimal fractional magnitude $\varepsilon$, then $3\varepsilon$ is also less than any real number (according to Robinson's arithmetic for hyperreal numbers \cite{ARNS,GOLDBLATT}.)  If $3\varepsilon$ is infinitesimal, then that contradicts the ordering
4139    \begin{equation*}
4140    x\in\mathbb{R}_0(n)\quad\implies\quad x<\mathcal{F}(n)~~.
4141    \end{equation*}
4142\end{pdox}
4143 
4144 
4145\begin{pres}
4146    Paradox \ref{pdox:3323v3} is resolved in the $\mathcal{F}_\mathcal{X}$ formalism.  We can uniquely associate $\mathcal{F}(1)=\mathcal{F}_0$ but there is no $\mathcal{F}_{\mathcal{X}_\text{min}}$ that we might associate with $\mathcal{F}(2)$.
4147\end{pres}
4148 
4149 
4150\begin{pdox}\label{pdox:33232vv3}
4151     Each $\mathbb{F}_0(n)\subset \mathbb{F}_0$ is a closed, connected interval.  It is required, then, that
4152    \begin{equation*}
4153    \mathbb{F}_0(1)=[a,b]~~,\quad\text{and}\qquad \mathcal{F}(1)=\frac{b-a}{2}~~.
4154    \end{equation*}
4155   
4156    \noindent Assuming the normal arithmetic for $x\in\mathbb{F}$, it follows that
4157    \begin{equation*}
4158    \sup\mathbb{R}_0=b-2\mathcal{F}(1)=a~~.
4159    \end{equation*}
4160   
4161    \noindent This is paradoxical for the reasons presented in Proposition \ref{mthm:u9999979y}: $\mathbb{R}_0$ ought not have a supremum.
4162\end{pdox}
4163 
4164 
4165\begin{pdox}\label{pdox:33232vv4}
4166    If $\mathcal{F}(1)$ is a real number centered in the closed interval $\mathbb{F}(1)$, then, assuming the normal arithmetic for $x\in\mathbb{F}$, we find that $2\mathcal{F}(1)=\aleph_{\mathcal{X}_\text{min}}$ with $\mathcal{X}_\text{min}$ being the least positive real number.  This number does not exist.  Therefore, the implied identity
4167    \begin{equation*}
4168    \mathcal{F}(1)=\frac{\aleph_{\mathcal{X}_\text{min}}}{2}~~,
4169    \end{equation*}
4170   
4171    \noindent is inadmissibly paradoxical.
4172\end{pdox}
4173 
4174 
4175 
4176\subsection{Complements of Natural Neighborhoods }\label{sec:bbbhh}
4177 
4178In this section, we take many of the facts established in the previous sections and begin to put them together to form a coherent picture of $\mathbb{F}^\mathcal{X}$, $\mathbb{R}_0^\mathcal{X}$, $\mathbb{R}_\aleph^\mathcal{X}$, and $\mathbb{R}_C^\mathcal{X}$.  This is what we know so far:
4179\begin{itemize}
4180    \item  We have defined $\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}=\mathbb{R}_\aleph^\mathcal{X}$.
4181    \item We do not know whether or not $\mathbb{R}_C^\mathcal{X}=\varnothing$.  This will be the main issue decided in the present section.  
4182    \item We have not yet defined any arithmetic operations for $x\in\mathbb{F}^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}$
4183    \item We have not yet given an algebraic construction for $x\in\mathbb{F}^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}$.
4184\end{itemize}
4185 
4186\begin{rem}
4187    In this section, we will use $\mathcal{F}(1)=\mathcal{F}_0$ to refer to a real number which is an upper bound on $\mathbb{R}_0$ without assuming an attendant problematic $(n)$ enumeration scheme.
4188\end{rem}
4189 
4190\begin{thm}\label{thm:85yfhhfeee}
4191    If we assume the usual arithmetic for $\mathcal{F}_\mathcal{X}$, then the set $\mathbb{R}^0_\aleph$ lies within the left endpoint $A$ of the line segment $AB=[0,\mathcal{F}_0]$.  In other words, every element of $\mathbb{R}_\aleph^0$ has zero fractional magnitude even with respect to $\text{len}[0,\mathcal{F}_0]\lll\text{len}\,\mathbf{AB}$.
4192\end{thm}
4193 
4194 
4195 
4196\begin{figure}[t]
4197    \begin{center}
4198        \includegraphics[scale=0.18]{numline.png}
4199        \caption{This figure (\textit{not to scale!}) shows the neighborhood of the origin $\mathbb{R}^0_\aleph$, the substructure of that neighborhood, and the associated structure in the Cantor-like sets.  $\mathbb{F}^-$ refers to the subset of $\mathbb{F}$ which is less than or equal to $\mathcal{F}(1)=\mathcal{F}_0$.}\label{fig:f9t99}
4200    \end{center}
4201\end{figure}
4202 
4203 
4204\begin{proof}
4205    Every interval has a unique number at its center.  For $AB=[0,\mathcal{F}_0]$, this number is $c=\frac{1}{2}\mathcal{F}_0$, as in Figure \ref{fig:f9t99}.  If $c\in\mathbb{R}^0_\aleph$, meaning that the fractional magnitude with respect to $\mathbf{AB}$ was zero, then $2c=\mathcal{F}_0$ would also have $2\times0=0$ fractional magnitude with respect to $\mathbf{AB}$.  This is contradictory because it would require $\mathcal{F}_0\in\mathbb{R}^0_\aleph$.  Continuing the argument, we find that for any $n\in\mathbb{N}$, the number $\frac{1}{n}\mathcal{F}_0$ must not be an element of $\mathbb{R}_\aleph^0$.  Now assume $\frac{1}{n}\mathcal{F}_0\in X\in AB\equiv[0,\mathcal{F}_0]$ and $X\neq A$.  Since the quotient of two line segments is defined as a real number (Definition \ref{def:gfdf}), and since the difference of two real numbers is always greater than some inverse natural number (Axiom \ref{ax:97977080}), we may write for some $m\in\mathbb{N}$
4206    \begin{equation*}
4207    \cfrac{AX}{AB}-\cfrac{AA}{AB}>\frac{1}{m}~~.
4208    \end{equation*}
4209   
4210    \noindent This is satisfied for any $X\neq A$ so $\frac{1}{n}\mathcal{F}_0$ can be a number in the algebraic representation of any $X\neq A$.  Since $\frac{1}{n}\mathcal{F}_0\not\in\mathbb{R}^0_\aleph$,  $\mathbb{R}^0_\aleph$ must be inside the algebraic representation of the left endpoint $A$ of $AB\equiv[0,\mathcal{F}_0]$.
4211\end{proof}
4212 
4213 
4214\begin{cor}\label{thm:rgrrggr3rg}
4215    If we assume the usual arithmetic for $\mathcal{F}_\mathcal{X}$, then for any $x\in\mathbb{R}_0$ such that $x\in X$, and for $X\in AB$ such that $AB\equiv[0,\mathcal{F}_0]$, we have
4216    \begin{equation}
4217    \mathcal{D}_{\!AB}(AX)=0~~.\nonumber
4218    \end{equation}
4219\end{cor}
4220 
4221 
4222\begin{proof}
4223    By the property $\mathbb{R}_0^0\subseteq\mathbb{R}_\aleph^0$, proof follows from Theorem \ref{thm:85yfhhfeee}.
4224   
4225    $~$
4226   
4227    Alternatively, $\mathcal{D}_{\!AB}$ is such that   
4228    \begin{equation}
4229    \mathcal{D}_{\!AB}(AX)=\mathcal{D}^\dagger_{\!AB}(AX)=\cfrac{x}{\mathcal{F}_0}~~.\nonumber
4230    \end{equation}
4231   
4232    \noindent The case of $x=0$ is trivial.  To prove the other cases by contradiction, suppose $z>0$ and that  
4233    \begin{equation}
4234    \cfrac{x}{\mathcal{F}_0}=z~~.\nonumber
4235    \end{equation}
4236   
4237    \noindent Since $\|x\|<\|\mathcal{F}_0\|$ and $x,\mathcal{F}_0\in\mathbb{R}^+$, it follows that $0< z<1$.  All such $z\in\mathbb{R}_0$ have a multiplicative inverse $z^{-1}\in\mathbb{R}_0$ so
4238    \begin{equation}
4239    \cfrac{x}{z\mathcal{F}_0}=1\quad\iff\quad z^{-1}x=\mathcal{F}_0~~.\nonumber
4240    \end{equation}
4241   
4242    \noindent This is a contradiction because $z^{-1}x\in\mathbb{R}_0$ but $\mathcal{F}_0$ is greater than any element of $\mathbb{R}_0$. 
4243\end{proof}
4244 
4245\begin{rem}
4246    Suppose we define $\digamma_{\!\mathcal{X}}=\mathcal{X}\cdot\mathcal{F}_0$ so that it mirrors the structure of $\aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty$.  Since $\mathbb{R}^0_\aleph$ has zero fractional distance even along $AB\equiv[0,\mathcal{F}_0]$, we could define a set of whole neighborhoods along $AB$
4247    \begin{equation*}
4248    \mathbb{R}^\mathcal{X}_\mathcal{F}=\big\{ \digamma_{\!\mathcal{X}}+b~\big|~ b\in\mathbb{R}^0_\aleph  \big\}~~,
4249    \end{equation*}
4250   
4251    \noindent exactly dual to the elements of $\{\mathbb{R}^\mathcal{X}_\aleph\}$ spaced along $\mathbf{AB}\equiv[0,\infty]$.  By subtracting every $\mathbb{R}^\mathcal{X}_\mathcal{F}$ from the interval $[0,\mathcal{F}_0]$ we would create another Cantor-like set.  Following the prescription given in Section \ref{sec:bbbhhv}, we would invoke the connectedness of the interval to label the disconnected elements of the new Cantor-like set, and we would label the numbers in the centers of each of those disconnected intervals.  Call those number $\mathcal{G}(n)$ labeled with $n\in\mathbb{N}_\infty$ so that they are dual to the $\mathcal{F}(n)$ in the duality transformation $[0,\infty]\to[0,\mathcal{F}_0]$, and so that they have a non-problematic labeling scheme as $\mathcal{G_X}$ with $\mathcal{X}$ measuring fractional distance along $AB\equiv[0,\mathcal{F}_0]\centernot\equiv\mathbf{AB}$.  By replicating the present course of analysis, we could show that no element of $\mathbb{R}^0_\aleph$ has non-zero fractional magnitude even with respect to $\text{len}[0,\mathcal{G}(1)]\lll\text{len}[0,\mathcal{F}_0]\lll\text{len}\,\mathbf{AB}$.  We could do this forever---defining more and more, tinier and tinier Cantor-like sets---and $\mathbb{R}^0_\aleph$ would never leave the left endpoint $A$ of the line segment whose algebraic representation is $[0,\Gamma_0]$ with $\Gamma_0$ being the number in the center of the leftmost connected element of the umpteenth Cantor-like set.  
4252   
4253    To accommodate the interpretation of the positive branch of $\mathbb{R}$ as a Euclidean line segment, we were forced to introduce numbers in the form $\aleph_\mathcal{X}$.  As a consequence, we were forced to introduce numbers of the form $\mathcal{F}(n)$ to describe the numbers in the Cantor-like sets whose elements are $\mathbb{F}(n)$.  If we tried to define $\mathbb{F}(n)$ as a neighborhood of the form
4254    \begin{equation*}
4255    \mathbb{F}(n)\stackrel{?}{=}  \big\{  a\mathcal{F}(n)+b  ~\big|~ a,|b|\in\mathbb{R}_\aleph^0  \big\}~~,
4256    \end{equation*}  
4257   
4258    \noindent then we would immediately encounter problem.  This set is clearly open while we have already proven that it must be closed (Theorem \ref{thm:i8758755757}.)  Therefore, we are left with a mystery set $\mathbb{F}_0$ whose elements are not easily decided.  Since this is the second such set we have, $\mathbb{R}_C^\mathcal{X}$ being the first, we ought to combine them into a single mystery set.  We have not proven that $\mathbb{R}_\aleph^\mathcal{X}\setminus\mathbb{R}_0^\mathcal{X}=\varnothing$ but neither have we proven the existence of such numbers ($\mathbb{R}_\aleph^\mathcal{X}\setminus\mathbb{R}_0^\mathcal{X}\neq\varnothing$) as we have with $\aleph_\mathcal{X}$ and $\mathcal{F}_\mathcal{X}$.  
4259   
4260    We required with Axiom \ref{ax:constaxcjco} that every $x\in\mathbb{R}$ may be constructed algebraically as a Cartesian product of Cauchy equivalence classes of rational numbers, or a partition of such products, but so far we have only found such constructions for those few numbers in the natural neighborhoods.  To avoid needless complication, therefore, we will conjecture that $\mathbb{R}_C(n)$ is the empty set and that, consequently, $\mathbb{F}_0=\mathbb{F}_\aleph$.  Then we will have all of the $\mathbb{R}^\mathcal{X}_0=\mathbb{R}^\mathcal{X}_0=\mathbb{R}^\mathcal{X}$ neighborhoods defined cleanly as ordered pairs of subsets of $C_\mathbb{Q}$ and we will move everything else into the Cantor-like set $\mathbb{F}$.  By the following conjecture, we will have given algebraic constructions and arithmetic axioms for every number in $\mathbb{R}^\cup_0=\mathbb{R}^\cup_\aleph$.  Everything which remains to be completed is transferred by Conjecture \ref{conj:ZZjjj3j3333} into $\mathbb{F}.$
4261\end{rem}
4262  
4263 
4264\begin{conj}\label{conj:ZZjjj3j3333}
4265    Every number having zero fractional magnitude with respect to $\mathbf{AB}$ is an element of $\mathbb{R}_0$.  Most generally,
4266    \begin{equation*}
4267    \mathbb{R}^\mathcal{X}_\aleph=\mathbb{R}^\mathcal{X}_0~~,\quad\text{and}\qquad \mathbb{R}_C^\mathcal{X}=\varnothing~~.
4268    \end{equation*}
4269\end{conj}
4270 
4271\begin{rem}
4272    One would also want to conjecture the countercase to Conjecture \ref{conj:ZZjjj3j3333} and examine the requirements for establishing naturally numbered tiers of increasing large numbers, larger than any natural number, but still having zero fractional distance with respect to $\mathbf{AB}$.  However, we will take the opposite tack here.  Now that we have conjectured that the whole and natural neighborhoods are the same, we will drop the $0$ and $\aleph$ subscripts from their respective objects.
4273\end{rem}
4274
4275
4276 
4277 
4278 \subsection{Dedekind Cuts and The Least Upper Bound Problem}\label{sec:r3r23r23r3r} 
4279 
4280 
4281Axiom \ref{ax:constaxcjco} gave $x\in\mathbb{R}$ as a Cartesian product of Cauchy equivalence classes of rational numbers, or as a partition of all such products.  In this section, we invoke the partition clause to define $x\in\mathbb{F}$ as modified Dedekind partitions of $C_\mathbb{Q}^\mathbf{AB}$.  We have defined the arithmetic of the equivalence classes themselves in Section \ref{sec:consXXX}, but we have not proven that partitions obey the arithmetic axioms in the way that the direct equivalence classes $[x]\subset C^\mathbf{AB}_\mathbb{Q}=C_\mathbb{Q}\times C_\mathbb{Q}$.  We will prove that they do not, and cannot, obey the arithmetic axioms without the introduction of further algebraic (multiplectic) structure.
4282We will prove that $\mathcal{F}^\cup=\mathbb{F}$ which means that the successive $\mathbb{R}^\mathcal{X}$ share an extremum.  Then we will suggest that the $x\in\mathbb{F}$ can be used as the natural numbers on a chart conformally related to the Euclidean chart by a conformal parameter greater than or equal to $\aleph_1$.  Based on the regularly spaced $\mathcal{F}(n)$, we would define a transfinite version of $\mathbb{N}$, call it $\mathbb{N_T}$, whose unit increments are such that real numbers have vanishing fractional distance with respect to them.  Then we would infer $\mathbb{Q}$ and define zero where a curious thing occurs.  To the extent that the identity of a real number is its Euclidean distance from the origin of $\mathbb{R}$, if $2\in\mathbb{N_T}$ is one unit of conformally transfinite distance to the right of $1\in\mathbb{N_T}$, then the origin of $\mathbb{N_T}$ cannot be be the simultaneous origin of $\mathbb{N}$ and $\mathbb{R}$.  The radius of the neighborhood of the origin is one half the length of any element of $\{\mathbb{R}^\mathcal{X}\}$.  Interestingly, $\infty$ not being affected in any way, ever, by any conformal parameter so from a geometric perspective we can expect the conformally transfinite version of $\mathbb{R}$ to be $\mathbb{R}$ itself identically.  It would only remain to define a Euclidean distance function who domain is in the transfinite regime.  Again, since $n\in\mathbb{N_T}$ looks like $n\in\mathbb{N}$ and should be define to behave accordingly, we can expect that the metric of the transfinitely rescaled version of $\mathbb{R}$ would have the ordinary Euclidean metric for its metric space definition.
4283
4284We will resolve the paradoxes of Section \ref{sec:pdoxes} by making a distinction between arithmatic (measurable) and non-arithmatic (immeasurable) numbers.  
4285
4286
4287Why should there not be islands of arithmatic numbers as subsets along $\mathbb{R}$?
4288
4289To the extent that this work in pure mathematics has developed along a pre-existing research direction in mathematical physics, there are some things should be pointed out.  Quantum mechanics has the curiously measurable half-inter spin quantization of fermions, and now we have demonstrated a half-integer interval, or ``quantum,'' of spacing inherent between the origins of the successively transfinite scaled copies of $\mathbb{R}$.  The number $\mathcal{F}(1)$ lies one third of the way down the interval $[0,\mathcal{F}_2]$.  Therefore, we also have the asymmetric fractional charges of the quarks: $-\frac{1}{3}e$ for three of them and $\frac{2}{3}e$ for the other three.  Indeed, from many physicists' point of view, the most fascinating experiment in quantum physics is not the double slit experiment, but the lab rotations of the electron's angular momentum vector.  If such a spin is isolated and measured in the lab and the apparatus is rotated through $\theta$ radians of arc, then a direct remeasurement of the spatial polarization of the angular momentum will show that it has 
4290 
4291 =====================
4292 
4293 FINISH SPINORS
4294 
4295 opposite phase of rotation can't be be done with the vector transformation law.  need to invent spinors.  Opposite phase of variation in $\hat0+b$ and $\widehat\infty-b$.
4296 
4297 ====================
4298 
4299\begin{defin}
4300    A Dedekind cut is a partition of the rationals into two sets $L$ and $R$ such that every real number is equal to some partition $x=(L,R)$ with the following properties.
4301    \begin{itemize}
4302        \item $L$ is non-empty.
4303        \item $L\neq\mathbb{Q}$
4304        \item If $x,y\in\mathbb{Q}$, if $x<y$, and if $y\in L$, then $x\in L$.
4305        \item If $x\in L$, then there exists $y\in L$ such that $y>x$.
4306    \end{itemize}
4307\end{defin}
4308  
4309 
4310 \begin{defin}
4311    An extended Dedekind cut is a partition of $C_\mathbb{Q}^\mathbf{AB}$ into two sets $L$ and $R$ such that every real number is equal to some partition $x=(L,R)$ with the following properties.
4312    \begin{itemize}
4313        \item $L$ is non-empty.
4314        \item $L\neq C_\mathbb{Q}^\mathbf{AB}$
4315        \item If $x,y\in C_\mathbb{Q}^\mathbf{AB}$, if $x<y$, and if $y\in L$, then $x\in L$.
4316        \item If $x\in L$, then there exists $y\in L$ such that $y>x$.
4317    \end{itemize}
4318 \end{defin}
4319 
4320 
4321\begin{thm}\label{thm:sdwd1cdwc}
4322    The number $\mathcal{F}_0$ is an extended Dedekind partition of $C_\mathbb{Q}^\mathbf{AB}$.
4323\end{thm}
4324 
4325\begin{proof}
4326    Let
4327    \begin{equation*}
4328    L=\big\{  [x]\subset C_\mathbb{Q}^\mathbf{AB}   ~\big|~  \text{Big}(x)=0 \big\}~~,\quad\text{and}\qquad R=\big\{  [x]\subset C_\mathbb{Q}^\mathbf{AB}   ~\big|~  \text{Big}(x)>0 \big\}~~,
4329    \end{equation*}
4330   
4331    \noindent To the extent that Dedekind himself wrote, ``In every case in which a cut $(A_1,A_2)$ is given that is not produced by a rational number $a$, which we consider to be completely defined by this cut; we will say that the number corresponds to this cut or that it produces the cut,'' we will call the present cut $\mathcal{F}_0$.  Therefore, $\mathcal{F}_0=(L,R)$ and the theorem is proven.
4332\end{proof}
4333
4334\begin{defin}\label{def:98y698t9t9}
4335	The extended Dedekind form of $\mathcal{F_X}=(L,R)\in\mathbb{R}$ is such that
4336	\begin{equation*}
4337	L=L_1\cup L_2~~,\quad\text{and}\qquad R=R_1\cup R_2~~,
4338	\end{equation*} 
4339	
4340	\noindent where
4341	\begin{align*}
4342	L_1=\big\{[x]\subset C_\mathbb{Q}^\mathbf{AB} ~\big|~  \text{Big}([x])\leq\aleph_\mathcal{X} \big\}~~,\quad&\text{and}\qquad 	L_2=\big\{ \mathbb{F}^\mathcal{Y} ~\big|~ 0\leq\mathcal{Y}<\mathcal{X}\big\}~~,\\
4343	R_1=\big\{[x]\subset C_\mathbb{Q}^\mathbf{AB} ~\big|~  \text{Big}([x])>\aleph_\mathcal{X} \big\}~~,\quad&\text{and}\qquad 	R_2=\big\{ \mathbb{F}^\mathcal{Y} ~\big|~ \mathcal{X}<\mathcal{Y}<1\big\}~~.
4344	\end{align*}
4345\end{defin}
4346
4347\begin{rem}
4348	In Definition \ref{def:98y698t9t9}, we have taken it for granted that $\mathcal{F_X}$ is the only number in $\mathbb{F}^\mathcal{X}$.  Without that assumption, $L$ would have taken the form $L=L_1\cup L_2\cup L_3$ with 
4349	\begin{equation*}
4350	L_3=\big\{ z\in\mathbb{F}^\mathcal{X}~\big|~z<\mathcal{F_X} \big\}~~,
4351	\end{equation*}
4352	
4353	\noindent and $R$ likewise.  Since we have assumed $x\in\mathbb{F}^\mathcal{X}\iff x=\mathcal{F_X}$, now we will prove it.
4354\end{rem}
4355
4356
4357
4358
4359\begin{mainthm}\label{mthm:v7557}
4360	$\mathcal{F}_\mathcal{X}$ is the only number in $\mathbb{F}^\mathcal{X}\subset \mathbb{R}$.  In other words, $\mathbb{F}^\mathcal{X}=\mathcal{F_X}$ or, equivalently, $\mathbb{F}=\{\mathcal{F_X}\}$.
4361\end{mainthm}
4362  
4363
4364\begin{proof}
4365	It will suffice to prove this theorem if we show that $\mathbb{F}^\mathcal{X}$ is a closed one point set.  Definition \ref{def:9y7986yccx} gives
4366	\begin{equation*}
4367	x\in\mathbb{R}^\mathcal{X}~~,~~ z\in\mathbb{F}^\mathcal{X}\quad\implies\quad x<z~~,
4368	\end{equation*}
4369	
4370	\noindent and  
4371	\begin{equation*}
4372	y\in\mathbb{R}^\mathcal{Y}~~,~~\mathcal{Y}>\mathcal{X}\quad\implies\quad y>z~~,
4373	\end{equation*}
4374	
4375	\noindent so it is established that $\mathbb{F}^\mathcal{X}$ is an upper bound on $\mathbb{R}^\mathcal{X}$ and a lower bound on $\mathbb{R}^\mathcal{Y}$ for any $\mathcal{Y}>\mathcal{X}$.  If $x\in\mathbb{F}^\mathcal{X}$ is the least upper bound of $\mathbb{R}^\mathcal{X}$ and the simultaneous greatest lower bound of $\mathbb{R}^\mathcal{Y}$, then $x=\mathcal{F_X}$ is the unique $x\in\mathbb{F}^\mathcal{X}$ and the proof will be completed.  For proof by contradiction, assume $u\in\mathbb{R}$ is an upper bound on $\mathbb{R}^\mathcal{X}$ with the property
4376	\begin{equation*}
4377	u<\mathcal{F_X}~~.
4378	\end{equation*}
4379	
4380	\noindent By Axiom \ref{ax:constaxcjco}, $u$ must be a partition of $C^\mathbf{AB}_\mathbb{Q}$ or an equivalence class therein.  We will divide the proof into two parts.
4381	
4382	$~$
4383	
4384	\noindent $\bullet$ (Equivalence class) If it was $[u]\subset C^\mathbf{AB}_\mathbb{Q}$, then $u\in\mathbb{R}^\mathcal{Z}\subset\mathbb{R}^\cup$.  If $\mathcal{Z}>\mathcal{X}$, then $u>\mathcal{F_X}$, a contradiction.  If $\mathcal{X}\geq\mathcal{Z}$, then $u$ is not an upper bound on $\mathbb{R}^\mathcal{X}$, another contradiction.  Now it is proven that $u\neq[u]\subset C^\mathbf{AB}_\mathbb{Q}$.
4385	
4386	$~$
4387	
4388	\noindent $\bullet$ (Partition) The partition definition is
4389	\begin{equation*}
4390	u=(L_u,R_u)~~.
4391	\end{equation*}
4392	
4393	\noindent If $u<\mathcal{F_X}$, and if $\mathcal{F_X}=(L,R)$, then there exists $\Sigma\in L$ such that 
4394	\begin{equation*}
4395	L_u=L\setminus\Sigma~~,\quad\text{and}\qquad R_u=R\cup \Sigma~~.
4396	\end{equation*}
4397	
4398	\noindent From Definition \ref{def:98y698t9t9}, we have
4399	\begin{equation*}
4400	L=\big\{[x]\subset C_\mathbb{Q}^\mathbf{AB} ~\big|~  \text{Big}([x])\leq\aleph_\mathcal{X} \big\}\cup\big\{ \mathbb{F}^\mathcal{Y} ~\big|~ 0\leq\mathcal{Y}<\mathcal{X}\big\}~~.
4401	\end{equation*}
4402	
4403	\noindent We have already ruled out $[u]\subset C^\mathbf{AB}_\mathbb{Q}$.  Obviously no $\Sigma\subset\mathbb{F}^\mathcal{Y}$ can be an upper bound on $\mathbb{R}^\mathcal{X}$ when $\mathcal{X}>\mathcal{Y}$.  There is no such $\Sigma\in L$ and we contradict the supposition that it does exist.  Since there is no upper bound on $\mathbb{R}^\mathcal{X}$ less than $\mathcal{F_X}$, $\mathcal{F_X}$ must be the least upper bound of $\mathbb{R}^\mathcal{X}$.  
4404	
4405	$~$
4406	
4407	\noindent A similar demonstration proves that $\mathcal{F_X}$ must be the greatest lower bound of $\mathbb{R}^\mathcal{X}$.  It follows that $\mathbb{F}^\mathcal{X}$ is a closed one-point set.  The theorem is proven.
4408\end{proof}  
4409
4410\begin{rem}
4411	The least upper bound problem rears its comely head.  With Main Theorem \ref{mthm:v7557}, we have given $\mathcal{F}_0=\sup(\mathbb{R}_0)$ but we have already made a strong case that no such supremum can exist.  In the remainder of this section, we will conclude the development of the fractional distance approach to $\mathbb{R}$ such that the reasoning behind the least upper bound problem is carefully sidestepped.  Before we continue, we will outline exactly what it means ``to conclude the development.''  With Main Theorem \ref{mthm:v7557}, we have now given a construction by Cauchy equivalence classes for every $x\in\mathbb{R}$.  Every measurable $x\in\mathbb{R}^\cup$ is a directly subset of $C_\mathbb{Q}^\mathbf{AB}$ and the arithmetic of such numbers is given in Section \ref{sec:5}.  Every immeasurable $x\in\mathbb{F}$ is now formally constructed as an extended Dedekind partition of $C_\mathbb{Q}^\mathbf{AB}$.  Since $\mathbb{R}=\mathbb{R}^\cup\cup\mathbb{F}$ all real numbers now have a direct algebraic construction.  We assume $\mathbb{N}$, define $0$, then we construct $\mathbb{Q}$, $C_\mathbb{Q}$, and $C_\mathbb{Q}^\mathbf{AB}$, and then we take $x\in\mathbb{R}$ as the elements and partitions of $C_\mathbb{Q}^\mathbf{AB}$.  In Section \ref{sec:consXXX}, however, it was only proven that the equivalence classes themselves obey the axioms.  We cannot simply throw $\mathcal{F}_\mathcal{X}$ in there---not in any rigorous fashion---because there is not a Cauchy equivalence class $[\mathcal{F_X}]\subset C_\mathbb{Q}^\mathbf{AB}$ for any $[\mathcal{X}]\subset C_\mathbb{Q}$.  Even if we forced the arithmetic axioms onto $\mathcal{F}^\mathcal{X}$ with more axioms, we would immediately hit the least upper bound problem developed in Proposition \ref{mthm:u9999979y}.  Now we will solve the least upper bound problem and conclude this section with the Archimedes property of $\mathbb{F}$.
4412\end{rem}
4413 
4414 
4415\begin{defin}
4416    If a real number is in $[x]\subset C^\mathbf{AB}_\mathbb{Q}$, then it is called an arithmatic number (pronounced arith$\cdot$matic.)  All measurable numbers $\mathbb{R}^\cup=C_\mathbb{Q}^\mathbf{AB}$ are arithmatic.  If a real number is a partition of $C^\mathbf{AB}_\mathbb{Q}$ not given by any subset therein, then it is called a non-arithmatic number.  All immeasurable numbers are non-arithmatic.
4417\end{defin}
4418 
4419\begin{axio}\label{ax:f9tw9ef5}
4420    Arithmetic operations are not defined among arithmatic and non-arithmetic numbers.
4421\end{axio}
4422
4423\begin{defin} 
4424	If $\mathbb{R}$ has the least upper bound property, then every nonempty subset $S\subset\mathbb{R}$ that has an upper bound must have a least upper bound $u\in\mathbb{R}$ such that $u=\sup(S)$.  
4425\end{defin}
4426  
4427  
4428\begin{pro}
4429    (Restatement of the least upper bound problem.)  $\mathbb{R}$ does not have the least upper bound property because $\mathbb{R}_0$ cannot have a supremum if arithmetic is defined for $x\in\mathbb{F}$ in the usual way.  If $x\in\mathbb{F}$ was to obey the arithmetic axioms, then $\mathcal{F}_0-1\in\mathbb{R}_0$.  By the closure of $\mathbb{R}_0$, $\mathcal{F}_0-1+2$ is also an element of $\mathbb{R}_0$.  This contradicts the identity $\mathcal{F}_0=\sup(\mathbb{R}_0)$.
4430\end{pro}
4431 
4432\begin{refut}
4433    Let $x$ be an arithmatic real number.  Axiom \ref{ax:f9tw9ef5} is such that
4434    \begin{equation*}
4435    \mathcal{F}=\sup(\mathbb{R}_0)\quad\implies\quad\sup(\mathbb{R}_0)\pm x=\text{undefined}~~.
4436    \end{equation*}
4437   
4438    \noindent The arithmetic can not be used to demonstrate the condition in the justification of this proposition.  $\mathbb{R}_0$ most certainly can have a supremum.  The supremum of each open set of numbers $\mathbb{R}^\mathcal{X}$ that are $100\times\mathcal{X}\%$ of the way down the real number line is $\mathcal{F_X}$: a non-arithmatic number.
4439\end{refut}
4440
4441\begin{thm}
4442	$\mathbb{R}$ has the least upper bound property which is also called the Dedekind property or Dedekind completeness.
4443\end{thm}
4444
4445\begin{proof}
4446	The Dedekind property requires that if $L$ and $R$ are two non-empty subsets of $\mathbb{R}$ such that $\mathbb{R}=L\cup R$, meaning $(L,R)$ is a partition of $\mathbb{R}$, and that if 
4447	\begin{equation*}
4448	x\in L~~,~~y\in R\quad\implies\quad x<y~~,
4449	\end{equation*}
4450	
4451	 \noindent then either $L$ has a greatest member or $R$ has a least member.  This property is explicit in the connectedness of the algebraic interval.  The non-arithmatic immeasurable numbers inherit their ordering with respect to the $\leq$ relation from the total ordering of $\mathbb{R}$.  With Main Theorem \ref{mthm:v7557}, we have established the connectedness of the successive intervals in $\mathbb{R}^0\cup\{\mathbb{R}^\mathcal{X}\}$.  The supremum of one neighborhood is the infimum of the next.  The maximal neighborhood $\mathbb{R}^1$ does not have a supremum but it is exempted because it does not have an upper bound $u\in\mathbb{R}$ at all.  The upper bound of the maximal neighborhood of infinity diverges to infinity.  Since no upper bound exists, a least upper bound cannot exist.  All subsets of $\mathbb{R}$ with an upper bound also have a least upper bound.  This proves the theorem.
4452\end{proof}
4453
4454\begin{defin}\label{def:lllkf4llf}
4455	A set $S$ is totally ordered if it obeys the following order axioms.
4456	\begin{itemize}
4457		\item (O1a) Elements of $F$ have trichotomy: If $x,y\in F$, then one and only one of the following is true: $x<y$, $x=y$, or $x>y$.
4458		\item (O2a) The $<$ relation is transitive: If $x,y,z\in F$, then $x<y$ and $y<z$ together imply $x<z$.
4459		\item (O3a) If $x,y,z\in F$, then $x<y$ implies $x+z<y+z$ or at least one sum is undefined.
4460		\item (O4a) If $x,y,z \in F$, and if $z>0$, then $x<y$ implies $xz< y z$ or at least one product is undefined.
4461	\end{itemize}
4462\end{defin}
4463
4464\begin{rem}
4465	Axioms (O1a)-(O4a) are almost exactly the (O1)-(O4) ordering axioms of a complete ordered number field (Axiom \ref{ax:fieldord}.)  We have changed the two axioms involving arithmetic operations $+$ and $\times$ in addition to the order relation $\leq$.  The changes reflect the arithmetic of $x\in\mathbb{F}$ which is truncated from existence with an axiom that every real number is less than some natural number.
4466\end{rem}
4467 
4468
4469\begin{thm}
4470	$\mathbb{R}$ is a totally ordered set.
4471\end{thm}
4472
4473\begin{proof}
4474	We will prove each of $(\text{O}1)$ to $(\text{O}4)$.
4475	
4476	$~$
4477	
4478	\noindent $\bullet$ (O1a) Trichotomy is trivially inherent to the order established for $\mathbb{R}^\cup$ (Axiom \ref{ax:order}).  Trichotomy is fully satisfied in $\mathbb{R}=\mathbb{R}^\cup\cup\mathbb{F}$ by the result $\mathcal{F}_\mathcal{X}=\sup(\mathbb{R}^\mathcal{X})$.
4479	
4480	$~$
4481	
4482	\noindent $\bullet$ (O2a) Transitivity is satisfied by the given order axiom and corollary results for the extrema.
4483	
4484	$~$
4485	
4486	\noindent $\bullet$ (O3a) Since we have restricted this part of Definition \ref{def:lllkf4llf} to the arithmatic numbers, the arithmetic axioms give compliance as stated.
4487	
4488	$~$
4489	
4490	\noindent $\bullet$ (O4a) Satisfaction follows in the manner of (O3a).	
4491	
4492	$~$
4493	
4494	\noindent The ordering relation $\leq$ for $\mathbb{R}$ is such that $\mathbb{R}$ is totally ordered.
4495\end{proof}
4496
4497\begin{rem}
4498	In Definition \ref{def:lllkf4llf}, we have modified slightly the usual definition of total order (Axiom \ref{ax:fieldord}) so that (O3) and (O4) regard arithmatic and non-arithmatic numbers.  We will justify this exception to the usual definition of total order as follows.  Since we are not using a number field approach to $\mathbb{R}$, we need not state the definition of total order in the exact for of the axioms of a totally ordered number field with unified laws of arithmetic.  The lack of arithmetic definitions for the immeasurable numbers doesn't affect their well-ordering with respect to the measurable ones, so the lack of defined operations has no bearing on the concept of the ordering of the set.  Indeed, where we have spoken of the ``geometric notions of addition and multiplication'' throughout the preceding analysis of fractional distance, there should not exist geometrically identical arithmetic operations for totally algebraic numbers with geometrically immeasurable fractional distance.
4499
4500    In the development of our least upper bound problem, it was implicitly assumed that $\sup(\mathbb{R}_0)$ must be an arithmatic number in the way that \textbf{\textit{all real numbers were supposed to be algebraic until the connectedness of the interval demanded non-algebraic numbers to fill in the gaps}}.  A number is said to be canonically algebraic if it is the root of a certain polynomial.  This is not what is meant using the adjective ``algebraic'' to describe the immeasurables.  Then the word refers to the lack of a geometric picture of fractional distance for $x\in\mathbb{F}$.  The geometric picture of $x\in\mathbb{F}$ comes from the algebraic ordering with the $\leq$ relation being defined for a given chart $x$.  It is clear that $\mathcal{F_X}$ are geometric Euclidean magnitudes, or cuts, measured relative to the origin $\hat0$ of an infinitely long line $\mathbb{R}$.  However, it is not clear how this works in the metric space picture of $\mathbb{R}$ where $d(0,\mathcal{F_X})=|\mathcal{F_X}-0|=\text{undefined}$.  Interesting that we have shown in complimentary fashion (Main Theorem \ref{thm:algfracdisnotcont4}) that arithmetic in the neighborhood of infinity allows one to take all-important limits at infinity within standard analysis.  There is no need to invoke the metric space definition of $\mathbb{R}$ to take these limits.  We do it with the modernized Euclidean definition so why should it be a problem that the metric function is undefined?  In Definition \ref{def:metspa} we defined a number line as is a line equipped with a chart $x$ and the Euclidean metric
4501    	\begin{equation}
4502    	d(x,y)=
4503    	\big|y-x\big|~~.\nonumber
4504    	\end{equation}
4505
4506	\noindent Now that we have closely examined all the details, we can equally define a number line equipped with a chart $x=\pm\mathbb{R}^\cup\mathbb{F}$.  With this definition, taken \textit{a priori} as an axiom, it is possible to reproduce the entire fractional distance analysis presently presented.
4507    
4508     What he have done with the separation of the reals into arithmatic and non-arithmatic numbers mirrors the usual separation between algebraic numbers, which are the roots of non-zero polynomials with rational coefficients, and canonically non-algebraic numbers which are not.  Canonically non-algebraic numbers are supposed to exist because they are needed to fill in the gaps in $\mathbb{Q}$ which are not allowed if $\mathbb{R}$ is to satisfy the definition of a 1D connected interval.  Now we have gone one step further and shown that non-arithmatic numbers are needed to fill in the connectedness of the many neighborhoods.   
4509\end{rem}
4510 
4511 
4512 
4513 
4514 
4515 
4516 
4517\begin{pres}\label{pdox:33232vv3aaa}
4518    (Resolution of Paradox \ref{pdox:33232vv3}.)  The paradox depends on assumed usual arithmetic for non-arithmatic immeasurable numbers.  The paradox is remedied by the non-arithmatic property of $\mathcal{F^X}$.
4519\end{pres}
4520 
4521 
4522\begin{pres}\label{pdox:33232vv4qeve}
4523    (Resolution of Paradox \ref{pdox:33232vv4}.)  The paradox depends on assumed usual arithmetic for non-arithmatic immeasurable numbers.  The paradox is remedied by the non-arithmatic property of $\mathcal{F^X}$.
4524\end{pres}
4525 
4526\begin{rem}
4527    In Section \ref{sec:gduidt7t}, we examined tangentially whether or not the Pythagorean theorem is inherently an algebraic notion, or a geometric notion.  Throughout this treatise, we have spoken likewise of ``the geometric notions of addition and multiplication.''  If $\mathcal{F_X}$ is an immeasurable number $x\not\in\mathbb{R}^\cup$ such that ordinary notions of geometry cannot be applied to it, by what means might we axiomatize the arithmetic?  We have shown in Section \ref{sec:pdoxes} that the straightforward introduction of infinitesimals is probably not the correct way forward, and we have shown it for all the reasons infinitesimals are usually not allowed into real analysis.  So, to pierce the reader's veil of skepticism regarding the absurdity of ordered but non-arithmatic real numbers $\mathcal{F^X}$, we now point out these are the only real numbers not forced into the arithmetic axioms by some identification as an element of $C^\mathbf{AB}_\mathbb{Q}$.  Immeasurable real numbers $x\in\mathbb{F}$ are partitions of the big parts of $C^\mathbf{AB}_\mathbb{Q}$ only.  They are not uniquely identified with any $[x]\in C^\mathbf{AB}_\mathbb{Q}$ as are the partitions of the big little parts jointly.  Measurable numbers differ from immeasurable numbers because the Dedekind partition corresponding to any $x\in\mathbb{R}^\cup$ must specify a little part for the partition.  
4528   
4529    Measurable and immeasurable numbers are so markedly different in their qualia that it is certainly reasonable to suppose they obey different arithmetic axioms.  Since we have also demonstrated by the least upper bound problem a requirement that standard analysis not contain any Cartesian products of the form
4530    \begin{equation*}
4531    \mathbb{R}^\cup\times\mathbb{F}=\big\{ x+\mathcal{F_X}     ~\big|~  x=[x],~ [x],[\mathcal{X}]\subset C^\mathbf{AB}_\mathbb{Q},~\mathcal{F_{[\mathcal{X}]}}\in\mathbb{F} \big\}~~,
4532    \end{equation*}
4533    
4534    \noindent or    
4535    \begin{equation*}
4536     \mathbb{R}^\cup\times\mathbb{F}=\big\{ x\cdot\mathcal{F_X}    ~\big|~   x=[x],~ [x],[\mathcal{X}]\subset C^\mathbf{AB}_\mathbb{Q},~\mathcal{F_{[\mathcal{X}]}}\in\mathbb{F}\big\}~~,
4537    \end{equation*}
4538    
4539    \noindent we have developed an axiomatic framework which displays exactly the required behavior.  The axioms are such that the attendant fractional distance approach to standard analysis forbids Cartesian products of the form $\mathbb{R}^\cup\times\mathbb{F}$.  What, then, shall we do with $\mathbb{F}$?  Undefined definitions beg for definitions.  This is what we've cleverly applied to replace the notion ``diverges'' with the $\widehat\infty$ symbol; we've defined that symbol to mean the thing does not converge in $\mathbb{R}$.  Let us examine the process of algebraic construction according to Cauchy sequences.  We have assumed $\mathbb{N}$ at the outset of our algebraic constructive process to write.  
4540    
4541    \begin{equation*}
4542    \mathbb{N}\to\mathbb{N}\cup\{0\}\to\mathbb{Q}\to C_\mathbb{Q}\to C_\mathbb{Q}^\mathbf{AB}~~.
4543    \end{equation*} 
4544    
4545    \noindent If we complement the undefined definitions for the arithmetic of immeasurable numbers such that they obey the arithmetic of the natural numbers on the conformal chart whose algebraic infinity is on the scale of geometric infinity relative to a smaller chart.  Then our process of algebraic becomes 
4546    \begin{equation*}
4547    \mathbb{N}\to\mathbb{N}\cup\{0\}\to\mathbb{Q}\to C_\mathbb{Q}\to C_\mathbb{Q}^\mathbf{AB}\to\mathbb{N_T}~~,
4548    \end{equation*}
4549    
4550    \noindent then we can continue to construct infinitely bigger and bigger copies of $\mathbb{R}$ forever with conformal rescaling parameters greater than or equal to $\aleph_1$.  Due to the absorptive properties of geometric infinity, conformal parameters such as $\aleph_\mathcal{X}$ never change the Euclidean conceptual component underlying everything: $\mathbb{R}=(\infty,\infty)$. 
4551    
4552    
4553    
4554    the abstract embedding dimension can always exceed geometric infinity by virtue of its inherent abstractness.  We should take the arithmetic of $\mathcal{F}(n)$ to be
4555     
4556\end{rem}
4557   
4558   
4559 \begin{cor}
4560 	The complex neighborhood of infinity is the complement of the complex neighborhood of the origin on the Riemann sphere.
4561 \end{cor}
4562   
4563   \begin{proof}
4564   	dfbwfbwrbwrbsw
4565   	\end{proof}
4566   
4567   
4568===================
4569 
4570 
4571 
4572 AXIOM (O3): we needed to modify it with a criteria
4573 
4574
4575 
4576 \noindent and now we are left with $\mathbb{F}$ which is just like a giant sized copy of another $\mathbb{N}$
4577 
4578 For the Archimedes property of $\mathbb{F}$, we ought to add some definitions for the undefined arithmetic operations on $\mathbb{F}$
4579 
4580 Let these be the natural numbers on a higher level of infinity.  
4581 
4582 
4583 By defining axiomatic self-similarity via the axiom $\mathbb{N}=\mathbb{N_T}$ we remove the need to even assume $\mathbb{N}$ at the beginning of the algebraic constructions of 
4584 
4585 REVISIT (O3), REVISIT ARCHIMEDES
4586 
4587 Are natural numbers infinitesimal with respect to the bigger natural numbers?
4588
4589
4590
4591
4592============
4593
4594
4595
4596
4597SHOW $\mathcal{F}$ SATISFIES ARCHIMEDES
4598 
4599Let the $\mathcal{F^X}$ be the natural numbers on a higher level of $\aleph$
4600
4601
4602
4603=====================
4604
46051 and 2 point compactifications: holomorphism versus meromorphism.
4606
4607ADD COR ABOUT THE NEIGHBORHOOD OF INFINITY BEING THE COMPLEMENT OF THE NEIGHBORHOOD OF THE ORIGIN.  GEOMETRIC INFINITE POINT DENSITY
4608 
4609 
4610 \subsection{The Topology of the Real Number Line}\label{sec:topo}
4611 
4612 
4613The thesis of this paper has to been preserve the Euclidean geometric construction of $\mathbb{R}$ through an algebraic construction which does not preclude the existence of the neighborhood of infinity.  We began with Axiom \ref{ax:mainR} stating that real numbers are represented in algebraic interval notation as    
4614\begin{equation}
4615\mathbb{R}=(-\infty,\infty)~~.\nonumber
4616\end{equation}
4617 
4618\noindent  In the analysis of this requirement, we built the problem-free set $\mathbb{R}^\cup$ but we were forced into a paradoxical corner with $\mathbb{F}$.  Aside from the paradoxes related to infinitesimals, there is seemingly no direct way to define the $x\in\mathbb{F}$ in terms of subsets of $C_\mathbb{Q}$ because set $\{|a|\mathcal{F}(n)\pm b\}$ will necessarily be open even when $[n],[a],[b]\subset C_\mathbb{Q}$.  In general, everything was going smoothly until $\mathbb{F}$ jumped out of the closet. What is the problem?
4619 
4620We need to look at Axiom \ref{ax:mainR}.  If our intention is to preserve the geometric Euclidean notion that a real number is a cut in line, then why have we posed our most fundamental axiom in terms of the algebraic interval?  Would it not be much better to pose the fundamental geometric axiom in the geometric language?  We have show in Remark \ref{rem:87585zzzz} that the elements of $\mathbb{F}$ are not in the algebraic representations of any $X\in\mathbf{AB}$.  Therefore, if we reformulate the fundamental axiom just slightly, we can get rid of $\mathbb{F}$ and solve all the associated problems.  In this section, we will reformulate the language in which we posed fundamental axiom that a real number is nothing more than cut in a geometric line.  Then we will give a basis for the topology supporting this axiom and state some of the properties.
4621 
4622\begin{axio}
4623    The fundamental axiom of algebraic construction.  The basis of the topology of the real number line is the usual one.  
4624\end{axio}
4625 
4626 
4627 
4628\begin{defin}\label{def:762222}
4629    The basis $B_0$ for the usual topology on $\mathbb{R}$ is the collection of all 1D open intervals such that
4630    \begin{equation*}
4631    B_0=\big\{  (a,b)  ~\big|~  [a],[b] \subset C_\mathbb{Q} ,~ a<b\big\}~~.
4632    \end{equation*}
4633   
4634    The connectedness of $\mathbb{R}$ is implicit in the interval notation. 
4635\end{defin}
4636 
4637\begin{rem}
4638    Definition \ref{def:762222} is such that ``having the usual topology'' is exactly equal to Axiom \ref{ax:mainR} granting that $\mathbb{R}=(-\infty,\infty)$.
4639\end{rem}
4640 
4641 
4642\begin{axio} \label{ax:mainR2}
4643    The fundamental axiom of geometric construction.  Non-negative real numbers are algebraic representations of points in an infinitely long line segment, \textit{i.e.}:
4644    \begin{equation}
4645    \mathbb{R}^+=\big\{  x ~\big|~ x\in X\in\mathbf{AB} ,~x>0,~x<\widehat\infty \big\}~~.\nonumber
4646    \end{equation}
4647\end{axio}
4648 
4649 
4650 
4651 
4652\begin{defin}
4653    The basis for the fractional distance topology on $\mathbb{R}$ is $B=B_\mathcal{X}\cup B_\infty$ such that
4654    \begin{align*}
4655    B_\mathcal{X}&=\big\{     (\aleph_\mathcal{X}+a,\aleph_\mathcal{X}+b)       ~\big|~  [\mathcal{X}],[a],[b] \subset C_\mathbb{Q},~a<b ,~-1<\mathcal{X}<1 \big\}\\
4656    B_\infty&=\big\{     (\pm\widehat\infty\mp a,\pm\widehat\infty\mp b)       ~\big|~   [a],[b] \!\subset\! C_\mathbb{Q},~a\!>\!b\!>\!0~\text{if}\,+\!\widehat\infty,~0\!<\!a\!<\!b~\text{if}\,-\!\widehat\infty \big\}~.
4657    \end{align*}  
4658\end{defin}
4659 
4660 
4661\begin{thm}
4662    If every real number in the usual topology is a Cauchy equivalence class $[x]\subset C_\mathbb{Q}$, then the fractional distance topology is finer.
4663\end{thm}
4664 
4665\begin{proof}
4666    One topology is said to be finer than another if it contains more open sets.  The fractional distance topology contains an uncountable infinity of separately labeled copies of the open sets in
4667    \begin{equation*}
4668    B_{C_\mathbb{{Q}}}=\big\{    (a,b)    ~\big|~  [a],[b]\subset C_\mathbb{Q},~[a]<[b]  \big\}~~,
4669    \end{equation*}
4670   
4671    \noindent one for each of an uncountably infinite number of $[\mathcal{X}]$.
4672\end{proof}
4673 
4674 
4675\begin{thm}
4676    If every real number in the usual topology is a product of Cauchy equivalence classes, meaning that $[x]\subset C_\mathbb{Q}^\mathbf{AB}$, then fractional distance topology is more coarse than the usual topology.
4677\end{thm}
4678 
4679\begin{proof}
4680    In this case, the basis of the usual topology will contain all the open sets in $B=B_\mathcal{X}\cup B_\infty$,  and it will also contain elements of $\mathbb{F}$.  Therefore, the usual topology is finer than the fractional distance topology when numbers are defined a priori with $C_\mathbb{Q}^\mathbf{AB}$.
4681\end{proof}
4682 
4683\begin{rem}
4684    As we have sought to simplify the analysis by introducing the Archimedean property of 1D transfinitely continued real numbers as an axiom (Axiom \ref{ax:nncncncn}), now we will introduce simplifying topological axioms.
4685\end{rem}
4686 
4687\begin{axio}\label{ax:chvhvhvh1}
4688    A number is a cut in a real number line and there exist an infinite number of such lines, well-ordered.
4689\end{axio}
4690 
4691\begin{axio}\label{ax:chvhvhvh}
4692    The real numbers as topological space are called $X$.  The topological space is
4693    \begin{equation*}
4694    X=\big\{(-\aleph_1,\aleph_1);\mathbb{T}\big\}~~,
4695    \end{equation*}
4696   
4697    \noindent where
4698    \begin{equation*}
4699    (-\aleph_1,\aleph_1)=(-\widehat\infty,\widehat\infty)~~,\quad\text{and}\qquad \mathbb{T}=(-\aleph_\infty,\aleph_\infty)=(-\infty,\infty)~~.
4700    \end{equation*}
4701\end{axio}
4702 
4703\begin{rem}
4704    Is a geometric point $X$ also a topological space $X$?
4705   
4706    In Definition \ref{def:762222}, we assumed that the real line was a unique line, as per Definition \ref{def:2412b24}.  If we take it as a topological axiom that there are an infinite number of totally ordered lines, each having the usual topology, then we may generate a finer topology than the usual usual topology by considering an infinite number of usual topologies.  These are exactly the open sets in $\mathbb{R}^\cup$, and now we do not need to axiomatize $\mathbb{F}$ to demonstrate the usual topology.  Every copy of the real line has the usual topology, regardless of the ordering relation of the big part of its reals numbers.  Since we have explicitly restricted the algebraic representations to the natural neighborhoods whose numbers are all less than some natural number, we totally exhibit the usual usual topology followed by the ordered number field approach based on the Cantor and Dedekind approaches to $\mathbb{Q}$.  The ordering parameter of the many real lines is the constant big part of the numbers which are the cuts in each line, and we have restricted all such cuts to have magnitude less than some natural number by by the conjecture $\mathbb{R}_C^\mathcal{X}=\varnothing$ (Conjecture \ref{conj:ZZjjj3j3333}.)  
4707     
4708   
4709    The usual topology of the real number line doesn't have anything to do with the neighborhood of infinity and certainly it has nothing to do with non-arithmatic numbers that are the simultaneous supremum of infimum of two different neighborhoods.  Therefore, when we use Axiom \ref{ax:chvhvhvh1} suppose there are an infinite number of infinitely long interval, each being a natural neighborhood of some local origin labeled as the additive big part unique to each line.  On the line labeled $\mathcal{X}$, the cut $b$ units from the origin is $x=\aleph_\mathcal{X}+b$.  In this way, by the assumption of an infinite number of copies of the real line, the usual topology does not require $\mathbb{F}$ to exist.  The only reason we included it was to satisfy the fundamental axiom of algebraic construction.  Now the simplifying topological axiom about the fractional distance topology, Axiom \ref{ax:chvhvhvh1} that the fractional distance topology is the usual topology distributed across an infinite number of lines, is such that we do not need to connect the lines.  There is no reason to connect things with $\mathbb{F}$.
4710 
4711        Axiom \ref{ax:constaxcjco} was the following.  Every $x\in\mathbb{R}$ may be constructed algebraically as a Cartesian product of Cauchy equivalence classes of rational numbers, or as a partition of all such products.  There is no longer any need to admit partitions of $C^\mathbf{AB}_\mathbb{Q}$.  The simplifying topological axioms allows further simplification of fundamental Axiom \ref{ax:constaxcjco}.  We can restrict to formal algebraic constructions $x\in\mathbb{R}\implies[x]\subset C^\mathbf{AB}_\mathbb{Q}$ by Cauchy equivalence classes only without admitting simultaneous Dedekind cuts.
4712    \end{rem}
4713   
4714    \begin{axio}\label{ax:vaxcjdco}
4715        Every $x\in\mathbb{R}$ may be constructed algebraically as a Cartesian product of Cauchy equivalence classes of rational numbers.
4716    \end{axio}
4717   
4718    \begin{rem}
4719        The main purpose of the simplifying topological axiom is that it allows us reinterpret ``the usual topology'' in the light that obviously an infinite number of copies of the usual topology are allowed.  Therefore, an infinite number of copies of the usual topology were already included in the usual usual topology.  The usual topology should be taken as fine as possible and the usual fractional distance topology is much finer than the usual usual topology. The main new contribution presented here has been to define a total ordering over all such lines by means of fractional distance.  This is how we induce the neighborhood of infinity.  In Section \ref{sec:g54584688}, we will apply the principles of fractional distance to a famously vexatious problem in mathematics and we will find a nice solution which is deeply satisfying.
4720    \end{rem}
4721   
4722   
4723    ============
4724   
4725    TWO WAYS OF DEFINING THE TOPOLOGICAL SPACE:
4726     metric space
4727     
4728     
4729     \begin{rem}
4730        Still, the numbers $x\in\mathbb{F}$ are interesting.  They are Dedekind cuts in the real number line.  We have every reason to think the arithmetic of immeasurable Dedekind cuts is other than the structure defined for Cauchy equivalence classes.  It is still preferable to include $\mathbb{F}$ because the well-ordering of $\mathcal{F^X}$ as the suprema of the open sets allows us to write $(\aleph_\mathcal{X},\mathcal{Y})$ as a connected interval.  If this is a connected interval, then $\mathcal{F^X}$ is in there.  It comes down to what we take the usual topology to mean.  Does the usual topology require $\mathbb{F}$, or does the fractional distance topology forego $\mathbb{F}$ by the piece-wise counting of an infinite number of neighborhoods of the origins on an infinite number of lines, each having the usual topology?
4731     \end{rem}
4732 
4733 ============================
4734 
4735\begin{thm}
4736	In the fractional distance topology, every real number belongs to a complete ordered number field.
4737\end{thm}
4738
4739\begin{proof}
4740	By Axiom \ref{ax:fieldssss33}, the numbers in the natural neighborhoods obey the axioms of a complete ordered field.  For $b\in\mathbb{R}_0$, define the its field identity operator $\hat1=1\in\mathbb{N}$ with 
4741	\begin{equation*}
4742	\hat A_\mathcal{X} b=\aleph_\mathcal{X}+b~~.
4743	\end{equation*}
4744	
4745	\noindent The meaning of this is to define the Euclidean unit of distance for a given instance of $\mathbb{R}$ as being on the scale of another Euclidean chart rescaled by a transfinite conformal parameter greater than, equal to, or less than $\aleph_1$.  If we do not define the identity in this way, the $x\in\mathbb{R}^\mathcal{X}$ do not form a closed additive group as required for compliance with the field axioms.  It is always true that
4746	\begin{equation*}
4747	\exists n\in\mathbb{N}\quad\text{s.t.}\quad \sum_{k=0}^{n}\big( \aleph_\mathcal{X} +b\big)>\aleph_1~~.
4748	\end{equation*}
4749	
4750	\noindent In the fractional distance topology, however, we may rearrange the axioms such that
4751	\begin{equation*}
4752	C_\mathbb{Q}^\mathbf{AB}=C_\mathbb{Q}\times C_\mathbb{Q}=([\mathcal{X}],[x])~~,
4753	\end{equation*}
4754	
4755	\noindent where
4756	\begin{equation*}
4757	(\mathcal{X}_0,b_0)=  A_{\mathcal{X}_0}\cdot b_0 ~~.
4758	\end{equation*}
4759	
4760	\noindent We simply redefine the reference of the ordered pair of Cauchy sequences of rationals.  The operator symbol $\hat A_\mathcal{X}$ tells us not worry about the arithmetic of the big part until later.  Then
4761	\begin{equation*}
4762	\centernot\exists n\in\mathbb{N}\quad\text{s.t.}\quad \sum_{k=0}^{n}\hat A_\mathcal{X} b=\hat A_\mathcal{X}\sum_{k=0}^{n} b>\aleph_1~~.
4763	\end{equation*} 
4764\end{proof}
4765
4766\begin{rem}
4767	$\hat A_\mathcal{X}$ is the identity operator for the disconnected open subset in the fractional distance basis, and $n,b\in\mathbb{R}_0$ imply $nb\in\mathbb{R}_0$.  Closure under arithmetic is proven.  Axiom \ref{ax:fieldssss33} grants that all $b\in\mathbb{R}_0$ obey the field axioms, so the elements of the basis sets of the fractional distance topology have associativity among the $\{\times,\div\}$ operations.  All discrepancies with the field axioms are resolved when the fractional distance topology is chosen such that all real numbers are axiomatically measurable.  This further allows us to transport $\mathbb{F}$ into some other copy of $\mathbb{R}$ as its algebraically progenitive instance of $\mathbb{N}$.
4768\end{rem}
4769 
4770 
4771 
4772\section{The Riemann Hypothesis}\label{sec:g54584688}
4773 
4774 
4775The Riemann hypothesis  dates to Riemann's 1859 paper \cite{RIEMANN}.  Since the axioms of a complete ordered field date to Dedekind's 1872 paper \cite{DEDE}, it would be improper to claim that the Riemann hypothesis is formulated in terms of the ordered field definition of $\mathbb{R}$.  Likewise, Cantor's definition of real numbers as equivalence classes of rationals dates to his own 1872 paper \cite{CANTOR} so the Riemann hypothesis cannot be understood as being phrased in the language of real numbers as Cauchy sequences.  The topological space as a mathematical concept did not exist until well into the 20th century so it would be similarly absurd to claim the Riemann's hypothesis is formulated in terms of the ``usual'' topology of $\mathbb{R}$.  While we cannot directly show what definition of $\mathbb{R}$ Riemann had in mind when formulating his hypothesis, we may glean very much from the plain fact that Riemann made no comment or nod toward any definition of $\mathbb{R}$ whatsoever.  This should be taken to mean that Riemann assumed his definition of $\mathbb{R}$ would have been absolutely, unambiguously known \textit{a priori} to his intended audience.  The only possible definition which might have been available to satisfy this condition in 1859 was Euclid's definition of real numbers as geometric magnitudes.  Indeed, Riemann's program of Riemannian geometry is a direct extension of Euclidean geometry so, to a very high degree, this qualitatively supports the notion that Riemann had in mind the cut-in-a-number-line definition of $\mathbb{R}$ given by Euclid in \textit{The Elements}.  
4776   
4777When one examines \textit{The Elements} \cite{EE}, the very many diagrams, definitions, and postulates make it exceedingly obvious that Euclid's definition of a real number as a magnitude, one having a proportion and ratio with respect to all other magnitudes, is exactly the one given here in Definition \ref{def:cuts}.  We have
4778\begin{equation*}
4779\mathbb{R}\setminus x=(-\infty,x)\cup(x,\infty)~~,
4780\end{equation*}
4781   
4782   
4783\noindent as an alternative identical formulation of the Euclidean statement
4784\begin{equation*}
4785x\in\mathbb{R}^+\quad\iff\quad(0,\infty)=(0,x]\cup(x,\infty)~~.
4786\end{equation*}
4787   
4788\noindent It is reasonable to conclude that Riemann formulated his hypothesis with it mind that any definition of $\mathbb{R}$ consistent with the Euclidean magnitude would be sufficient.  The domain of $\zeta(z)$, namely $\mathbb{C}$, would be constructed from two orthogonal copies of $\mathbb{R}$, one of them having the requisite phase factor $i$.  Rather than the underlying definition of $\mathbb{R}$, the object of relevance in the Riemann hypothesis should be the behavior of $\zeta(z)$ at various $z$.
4789 
4790 
4791 
4792\subsection{Non-trivial Zeros in the Critical Strip}
4793 
4794In this section, we will prove the negation of the Riemann hypothesis.
4795 
4796 
4797 
4798\begin{defin}
4799    The Riemann $\zeta$ function is the analytic continuation of the Dirichlet series to a meromorphic function on the entire complex plane.  In the region $\text{Re}(z)\!>\!1$, $\zeta$ has the simple form
4800    \begin{align}
4801    \zeta(z)&=\sum_{n=1}\frac{1}{n^z}~~.\nonumber
4802    \end{align}
4803     
4804    \noindent Here we will treat $\zeta$ as a holomorphic function so the domain of $\zeta$ is continued onto the entire complex plane excepting the pole at $z\!=\!1$.  This is accomplished by way of Riemann's functional equation \cite{RIEMANN,ZZZ2,ZZZ4,ZZZ13,ZZZ62,ZZZ5,ZZZ3,ZZZ11,ZZZ6,ZZZ7,ZZZ9,ZZZ10,ZZZ12,ZZZ14}
4805    \begin{equation}
4806    \zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~.\nonumber
4807    \end{equation}
4808\end{defin}
4809 
4810\begin{rem}
4811    The Riemann $\zeta$ function $\zeta:\mathbb{C}\to\mathbb{C}$ is meromorphic on $\mathbb{C}$ and holomorphic on $\mathbb{C}\setminus Z_1$ where $Z_1$ denotes the pole at $z(x,y)=z(r,\theta)=1$.  It is a well-known property of holomorphic functions that their zeros are isolated on a domain or else the function is constant on that domain.  However, this property relies on the implicit axiom that all pairs of points $z_1,z_2$ in any subdomain $D\subset\mathbb{C}$ are such that $d(z_1,z_2)\in\mathbb{R}_0$.  When we do not take this implicit axiom, further specification is required.  The property becomes: if the zeros of a holomorphic function are not isolated, then the function is constant everywhere on a disc of radius $r_0\in\mathbb{R}_0$ about any of the non-isolated zeros.  Here we will make rigorous this line of reasoning.
4812\end{rem}
4813 
4814 
4815\begin{pro}\label{prop:68wdf9t}
4816    If $f$ is a holomorphic function defined everywhere on an open connected set $D \subset \mathbb{C}$, and if there exists more than one $z_0 \in D$ such that $f(z_0) = 0$, then $f$ is constant on $D$ or the set containing all $z_0 \in D$ is totally disconnected.
4817\end{pro}
4818 
4819\begin{refut}
4820    This proposition is usually proven by a line of reasoning starting with the following.  By the holomorphism of $f$ and the property $f(z_0)=0$, we know there exists a convergent Taylor series representation of $f(z)$ for all $|z-z_0| < r_0$ with $r_0\in\mathbb{R}$.  Here the proposition fails pseudo-trivially because we can select $r_0\in\mathbb{R}^\mathcal{X}_0$ and assume
4821    \begin{equation*}
4822    \big|z-z_0\big|>\big(\aleph_\mathcal{X}+a\big)~~,
4823    \end{equation*}
4824   
4825    \noindent to show that the Taylor series does not converge (when $\mathcal{X}>0$.)  We have
4826    \begin{align*}
4827    f(z)&=f(z_0)+f'(z_0)\big(z-z_0\big)+\dfrac{f''(z_0)}{2!}\big(z-z_0\big)^2+...~~.
4828    \end{align*}
4829   
4830    \noindent The first term in the series vanishes by definition and so, therefore, we have by assumption
4831    \begin{align*}
4832    f(z)&>f'(z_0)\big(\aleph_\mathcal{X}+b\big)+\sum_{n=2}^{\infty}\dfrac{f^{(n)}(z_0)}{n!}\big(\aleph_\mathcal{X}+b\big)^n ~~.
4833    \end{align*}
4834   
4835    \noindent The Taylor series expansion of $f$ does not converge for $|z-z_0| \in \mathbb{R}^\mathcal{X}_0$.  This follows from $(\aleph_\mathcal{X}+b)^n>\aleph_1$ for all $n\geq2$, as per Axiom \ref{ax:1g1g1g1}.
4836\end{refut}
4837 
4838 
4839 
4840\begin{axio}\label{ax:yguib}
4841    If $f$ is a holomorphic function defined everywhere on an open connected set $D \subset \mathbb{C}$, if there exists more than one $z_0 \in D$ such that $f(z_0) = 0$, and if every $p \in D$ is such that $|z_0-p| \in \mathbb{R}_0$, then $f$ is constant on $D$ or the set containing all $z_0 \in D$ is totally disconnected.
4842\end{axio}
4843 
4844\begin{rem}
4845    Various proofs of Axiom \ref{ax:yguib} are well-known.  They are taken for granted.
4846\end{rem}
4847 
4848\begin{mainthm}\label{thm:11qqgg4gq}
4849    If $\{\gamma_n\}$ is an increasing sequence containing the imaginary parts of the non-trivial zeros of the Riemann $\zeta$ function in the upper complex half-plane, then
4850    \begin{equation*}
4851    \lim\limits_{n\to(\aleph_\mathcal{X}+b)}\big|\gamma_{n+1}-\gamma_n\big|=0~~.
4852    \end{equation*}
4853\end{mainthm}
4854 
4855 
4856 
4857\begin{proof}
4858    To prove the theorem, we will follow Titchmarsh's proof \cite{ZZZ2} of a theorem of Littlewood \cite{LW}.  The original theorem is as follows.
4859   
4860    \begin{quote}
4861        ``For every large $T$, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying
4862        \begin{equation*}
4863        |\gamma-T|<\dfrac{A}{\log\log\log T}~~.\text{''}
4864        \end{equation*}
4865    \end{quote}
4866   
4867    \noindent Note that $A$ is some constant $A\in\mathbb{R}_0$.  For proof by contradiction, assume
4868    \begin{equation*}
4869    \lim\limits_{n\to(\aleph_\mathcal{X}+b)}\big|\gamma_{n+1}-\gamma_n\big|\neq0~~.
4870    \end{equation*}
4871   
4872    \noindent Then there exists some $m(n)$ and some $a \in \mathbb{R}_0^+$ such that
4873    \begin{equation*}
4874    \lim\limits_{m(n)\to(\aleph_\mathcal{X}+b)}\big|\gamma_{m(n)+1}-\gamma_{m(n)}\big|>2a~~.
4875    \end{equation*}
4876   
4877    \noindent Let $T_n$ be the average of $\gamma_{m(n)+1}$ and $\gamma_{m(n)}$ so
4878    \begin{equation*}
4879    T_n=\dfrac{\gamma_{m(n)+1}+\gamma_{m(n)}}{2}~~.
4880    \end{equation*}
4881   
4882    \noindent Now we have
4883    \begin{equation*}
4884    \lim\limits_{T_n\to(\aleph_\mathcal{X}+b)}\big| \gamma-T_n  \big|>a~~,
4885    \end{equation*}
4886   
4887    \noindent because $T_n$ is centered between the next greater and next lesser $\gamma_n$.  We have shown that this pair of $\gamma_n$ are separated by more than $2a$.  This contradicts Littlewood's result
4888    \begin{equation*}
4889    |\gamma-T_n|<\dfrac{A}{\log\log\log T_n}~~,\quad\text{whenever}\qquad\dfrac{A}{\log\log\log T_n}<a~~.
4890    \end{equation*}
4891   
4892    \noindent The limit $T_n \to \aleph_\mathcal{X}+b$ is exactly such a case because
4893    \begin{equation*}
4894    \log(\aleph_\mathcal{X}+b)=\log(\mathcal{X}\widehat\infty)+\log(b)=\log(\mathcal{X})\log(\widehat\infty)+\log(b)=\aleph_{\log(\mathcal{X})}+\log(b)~~.
4895    \end{equation*}
4896   
4897    \noindent Evaluating the log a few more times and then applying $A/(\aleph_{\mathcal{X}'}+b')=0$ shows that the expression is always less than $a\in\mathbb{R}_0^+$.  Therefore, the elements of $\{\gamma_n\}$ form an unbroken line when $|\text{Im}(z)|\in\mathbb{R}_\infty$.  This proves the theorem.
4898\end{proof}
4899 
4900 
4901 
4902\begin{rem}
4903    Note that $\{\gamma_n\}$ is not such that each element can be labeled with $n\in\mathbb{N}$ because the zeros become uncountably infinite in the neighborhood of infinity.  Rather, $\{\gamma_n\}$ must be a sequence in the sense that it is an ordered set of mathematical objects whose final object in an interval.  Also note that $\{\gamma_n\}$ is proper sequence in the usual sense when we take $n\in\mathbb{N}_\infty$, as in Definition \ref{def:y98t9tuigjaaaa}.
4904\end{rem}
4905 
4906 
4907 
4908\begin{cor}\label{mthm:llool}
4909    The Riemann $\zeta$ function has zeros within the critical strip yet off the critical line.  
4910\end{cor}
4911 
4912\begin{proof}
4913    Proof follows from Axiom \ref{ax:yguib} and Main Theorem \ref{thm:11qqgg4gq}.  If the imaginary parts of the zeros form an unbroken line in the neighborhood of infinity, then the zeros are not isolated.  Since $\zeta$ is holomorphic on $\mathbb{C}\setminus Z_1$, it must have zeros everywhere on a disc of radius $r_0\in\mathbb{R}_0$ of any of the zeros on the critical line.  Some of these zeros, obviously, are within the critical strip yet off the critical line.  
4914\end{proof}
4915 
4916 
4917\subsection{Non-trivial Zeros in the Neighborhood of Minus Infinity}
4918 
4919 
4920The trivial zeros of the Riemann $\zeta$ function are the negative even integers $z=-2,-4,-6...$ \cite{RHDEF}.  In this section, we prove that $\zeta$ has non-trivial zeros outside of the critical strip.  The theorem of Hadamard and de la Vall\'ee-Poussin \cite{HAD,DLVP} is usually taken to rule out the existence of such zeros and here we will conjecture that the theorem fails in the neighborhood of infinity. Indeed, it follows from Corollary \ref{mthm:llool} that $\zeta$ has zeros on the line $\text{Re}(z)=1$ and this is something else contradicts the theorem of Hadamard and de la Vall\'ee-Poussin.  We will conjecture that the result of Hadamard and de la Vall\'ee-Poussin fails in the neighborhood of infinity.
4921 
4922 
4923 
4924 
4925\begin{thm}\label{thm:5y74757}
4926    The Riemann $\zeta$ function is equal to one for any $\text{Re}(z_0)\in\mathbb{R}^\mathcal{X}_\aleph$ such that $0<\mathcal{X}\leq1$.
4927\end{thm}
4928 
4929\begin{proof}
4930    Observe that the Dirichlet sum form of $\zeta$
4931    \begin{equation}
4932    \zeta(z)=\sum_{n\in\mathbb{N}}\cfrac{1}{n^z}~~,\nonumber
4933    \end{equation}
4934   
4935    \noindent  takes $z_0=\big(\aleph_\mathcal{X}+b\big)+iy_0$ as  
4936    \begin{align}
4937    \zeta(z_0)&=\sum_{n=1}\cfrac{1}{n^{(\aleph_\mathcal{X}+b)+iy_0}}\nonumber\\
4938    &=\sum_{n=1}\cfrac{n^{-b}n^{-iy_0}}{n^{\aleph_\mathcal{X}}}\nonumber\\
4939    &=\sum_{n=1}\cfrac{n^{-b}}{\big(n^{\mathcal{X}}\big)^{\!\widehat\infty}}\bigg(\cos(y_0\ln n)-i\sin(y_0\ln n)\bigg)\nonumber\\
4940    &=1+\sum_{n=2}\cfrac{n^{-b}}{\widehat\infty}\bigg(\cos(y_0\ln n)-i\sin(y_0\ln n)\bigg)=1~~.\nonumber
4941    \end{align}
4942\end{proof}
4943 
4944\begin{mainthm}\label{thm:5rg3r3rf757}
4945    The Riemann $\zeta$ function has non-trivial zeros $z_0$ such that $-\normalfont{\text{Re}}(z_0)\in\mathbb{R}^\mathcal{X}_\aleph$ for $0<\mathcal{X}\leq1$.  In other words, $\zeta$ has non-trivial zeros in the neighborhood of minus real infinity.
4946\end{mainthm}
4947 
4948\begin{proof}
4949    Riemann's functional form of $\zeta$ \cite{RIEMANN} is
4950    \begin{equation}
4951    \zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~.\nonumber
4952    \end{equation}
4953   
4954    \noindent Theorem \ref{thm:5y74757} gives $\zeta(\aleph_\mathcal{X}+b)=1$ when we set $y_0=0$ so we will use Riemann's equation to prove this theorem by computing $\zeta(z)$ at $z_0=-(\aleph_\mathcal{X}+b)+1$.  (This value for $z_0$ follows from $1-z_0=\aleph_\mathcal{X}+b$.)   We have
4955    \begin{align}
4956    \zeta\big[-\!(\aleph_\mathcal{X}+b)+1\big]&=\lim\limits_{z\to-(\aleph_\mathcal{X}+b)+1}\bigg(\dfrac{(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\bigg)\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg(\Gamma(z)\zeta(z)\bigg)~\nonumber\\
4957    &=\lim\limits_{z\to-(\aleph_\mathcal{X}+b)+1}\bigg(2\sin\left(\dfrac{\pi z}{2}\right)\bigg)\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg((2\pi)^{-z}\Gamma(z)\zeta(z)\bigg)~~.\nonumber
4958    \end{align}
4959   
4960    \noindent For the limit involving $\Gamma$, we will compute the limit as a product of two limits.  We separate terms as
4961    \begin{align}
4962    \lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg((2\pi)^{-z}\Gamma(z)\zeta(z)\bigg)=\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg((2\pi)^{-z}\Gamma(z)\bigg)\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\zeta(z)~~.\nonumber
4963    \end{align}
4964   
4965    \noindent From Theorem \ref{thm:5y74757}, we know the limit involving $\zeta$ is equal to one.  For the remaining limit, we will insert the identity and again compute it as the product of two limits.  If $z$ approaches $(\aleph_\mathcal{X}+b)$ along the real axis, then it follows from Axiom \ref{ax:div1g1g1g1} that
4966    \begin{equation}
4967    1=\cfrac{z-(\aleph_\mathcal{X}+b)}{z-(\aleph_\mathcal{X}+b)}~~.\nonumber
4968    \end{equation}
4969   
4970   
4971   
4972    \noindent Inserting the identity yields
4973    \begin{align}
4974    \lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg((2\pi)^{-z}\Gamma(z)\bigg)&=\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\bigg((2\pi)^{-z}\Gamma(z)\cfrac{z-(\aleph_\mathcal{X}+b)}{z-(\aleph_\mathcal{X}+b)}\bigg)~~.\nonumber
4975    \end{align}
4976   
4977    \noindent Let
4978    \begin{equation}
4979    A=\Gamma(z)\bigg(z-(\aleph_\mathcal{X}+b)\bigg)~~,\quad\text{and}\qquad B=\cfrac{(2\pi)^{-z}}{z-(\aleph_\mathcal{X}+b)}~~.\nonumber
4980    \end{equation}
4981   
4982    \noindent  To get the limit of $A$ into workable form, we will use the property $\Gamma(z)=z^{-1}\Gamma(z+1)$ to derive an expression for $\Gamma[z-(\aleph_\mathcal{X}+b)+1]$.  If we can write $\Gamma(z)$ in terms of $\Gamma[z-(\aleph_\mathcal{X}+b)+1]$, then the limit as $z$ approaches $(\aleph_\mathcal{X}+b)$ will be very easy to compute.  Observe that
4983    \begin{align}
4984    \Gamma\big[z-(\aleph_\mathcal{X}+b)+1\big]&=\Gamma\big[z-(\aleph_\mathcal{X}+b)+2\big]\bigg(z-(\aleph_\mathcal{X}+b)+1\bigg)^{\!-1}~~.\nonumber
4985    \end{align}
4986   
4987    \noindent On the RHS, we see that $\Gamma$'s argument is increased by one with respect to the $\Gamma$ function that appears on the LHS.  The purpose of inserting the identity $z-(\aleph_\mathcal{X}+b)[z-(\aleph_\mathcal{X}+b)]^{-1}=1$ was precisely to exploit this self-referential identity of the $\Gamma$ function which is most generally expressed as
4988    \begin{align*}
4989    \Gamma\big(z\big)&=\Gamma\big(z+1\big)z^{-1}~~.
4990    \end{align*}
4991   
4992    \noindent By taking a limit of recursion, we will let $z$ approach a number in the neighborhood of infinity. Then through the axiomatized addition of such numbers (Axiom \ref{ax:plus}), we will cast the argument of $\Gamma$ into the neighborhood of the origin where $\Gamma$'s properties are well-known.  The limit is
4993    \begin{align}
4994    \Gamma\big[z-(\aleph_\mathcal{X}+b)+1\big]&=\Gamma(z)\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\prod_{k=1}^{n}\bigg(z-(\aleph_\mathcal{X}+b)+k\bigg)^{\!-1}\ ~~.\nonumber
4995    \end{align}
4996   
4997    \noindent Moving the infinite product to the other side yields
4998    \begin{align}
4999    \Gamma(z)&=\Gamma\big[z-(\aleph_\mathcal{X}+b)+1\big]\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\prod_{k=1}^{n}\bigg(z-(\aleph_\mathcal{X}+b)+k\bigg)~~.\nonumber
5000    \end{align}
5001   
5002    \noindent We have let $A=\Gamma(z)(z-(\aleph_\mathcal{X}+b))$ where the coefficient $z-(\aleph_\mathcal{X}+b)$ can be expressed as the $k=0$ term in the infinite product.  It follows that    
5003   
5004    \begin{equation}
5005    A=\Gamma\big[z-(\aleph_\mathcal{X}+b)+1\big]\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\prod_{k=0}^{n}\bigg(z-(\aleph_\mathcal{X}+b)+k\bigg)~~.\nonumber
5006    \end{equation}
5007   
5008    \noindent To evaluate the limit of AB, we will take the limits of $A$ and $B$ separately.  The limit of $A$ is
5009    \begin{align*}
5010    \lim\limits_{z\to(\aleph_\mathcal{X}+b)}A&=\Gamma\big[(\aleph_\mathcal{X}+b)-(\aleph_\mathcal{X}+b)+1\big]\times\\
5011    &\qquad\qquad\qquad\qquad\times\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\prod_{k=0}^{n}\!\bigg(\!(\aleph_\mathcal{X}+b)-(\aleph_\mathcal{X}+b)+k\bigg)~~.
5012    \end{align*}
5013   
5014   
5015    \noindent Axiom \ref{ax:plus} gives $(\aleph_\mathcal{X}+b)-(\aleph_\mathcal{X}+b)=0$ so
5016    \begin{align}
5017    \lim\limits_{z\to(\aleph_\mathcal{X}+b)}A&=\Gamma(1)\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\prod_{k=0}^{n}k=0~~.\nonumber
5018    \end{align}
5019   
5020    \noindent Direct evaluation of the $z\to(\aleph_\mathcal{X}+b)$ limit of $B=(2\pi)^{-z}(z-(\aleph_\mathcal{X}+b))^{-1}$ gives $\frac{0}{0}$ so we need to use L'H\^opital's rule.  Evaluation yields
5021    \begin{align}
5022    \lim\limits_{z\to(\aleph_\mathcal{X}+b)}B&\stackrel{*}{=}\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\left(\cfrac{\dfrac{d}{dz}(2\pi)^{-z}}{\dfrac{d}{dz}\bigg(z-(\aleph_\mathcal{X}+b)\bigg)}\right)\nonumber\\
5023    &=\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\dfrac{d}{dz} e^{-z\ln (2\pi)}\nonumber\\
5024    &=-\ln (2\pi)\ e^{-(\aleph_\mathcal{X}+b)\ln (2\pi)}\nonumber\\
5025    &=-\ln (2\pi)\ \cfrac{e^{-b\ln (2\pi)}}{\big(e^{\mathcal{X}}\big)^{\!\widehat\infty}}\nonumber\\
5026    &= -\ln(2\pi)\cfrac{e^{-b\ln (2\pi)}}{\widehat\infty}\nonumber\\
5027    &=0~~.\nonumber
5028    \end{align}
5029   
5030    \noindent Therefore, we find that the limit of $AB$ is $0$.  It follows that
5031    \begin{align}
5032    \zeta\big[-\!(\aleph_\mathcal{X}+b)+1\big]&=\lim\limits_{z\to-(\aleph_\mathcal{X}+b)+1} 2\sin\left(\dfrac{\pi z}{2}\right) \cdot0=0~~.\nonumber
5033    \end{align}
5034\end{proof}
5035 
5036 
5037 
5038 
5039 
5040\begin{rem}
5041    To demonstrate that Riemann's functional form of $\zeta$ is robust, we should check for consistency by reversing the sign of $z$ and $1-z$ to show that there is no contradiction.  What this means is that we have computed value in the left complex half-plane using a known value in the right complex half-plane, and now we should use the known value on the left to see what it says about the value on the right.  We have    
5042    \begin{align}
5043    \Gamma(-\aleph_\mathcal{X}+1)&=\cfrac{1}{-\aleph_\mathcal{X}+1}\prod_{n=1}^{\widehat\infty}\left(1-\aleph_{\left(\frac{\mathcal{X}}{n}\right)}+\cfrac{\vphantom{\hat\aleph}1}{n}\right)^{-1}\left(  1+\cfrac{1}{n}  \right)^{ -\aleph_\mathcal{X}+1}=0\nonumber~~,\nonumber
5044    \end{align}
5045   
5046    \noindent and we have shown in Main Theorem \ref{thm:5rg3r3rf757} that $\zeta(-\aleph_\mathcal{X}+1)=0$.  Using Riemann's formula to derive the relationship between $\zeta(z)$ and $\zeta(1-z)$
5047    \begin{equation}
5048    \zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~,\nonumber
5049    \end{equation}
5050   
5051   
5052    \noindent we will compute $\zeta(\aleph_\mathcal{X})$.  Evaluation yields
5053    \begin{align}
5054    \zeta(\aleph_\mathcal{X})&=2(2\pi)^{\aleph_\mathcal{X}-1}\sin \left(\dfrac{\pi \aleph_\mathcal{X}}{2}\right)\Gamma(-\aleph_\mathcal{X}+1)\zeta(-\aleph_\mathcal{X}+1)=(\widehat\infty)(0)(0)~~.\nonumber
5055    \end{align}
5056   
5057    \noindent This equation is undefined and we cannot obtain a contradiction. 
5058\end{rem}
5059 
5060 
5061 
5062\begin{rem}
5063    Patterson writes the following in reference \cite{ZZZ4}.  
5064   
5065    \begin{quote}
5066        ``There is a second representation of $\zeta$ due to Euler in 1749 which is perhaps more fundamental and which is the reason for the significance of the zeta-function.  This is
5067        \begin{equation*}
5068        \zeta(s)=\prod_{p \in\text{primes}}\big(1-p^{-s}\big)^{-1}~~.
5069        \end{equation*}
5070       
5071        \noindent where the product is taken over all prime numbers $p$.  This is called the Euler Product representation of the zeta-function and gives analytic expression to the fundamental theorem of arithmetic.''
5072    \end{quote}
5073   
5074    The fundamental theorem of arithmetic is given in Euclid's \textit{Elements} \cite{EE} Book 7, Propositions 30, 31, and 32.  A modern statement of the fundamental theorem of arithmetic is that every natural number greater than one is a prime number or it is a product of prime numbers.  The ultimate goal of all of number theory being concerned with the distribution of the prime numbers, now we will demonstrate as a corollary result that the Euler product form of $\zeta$ \cite{ZZZ4,EULER2} shares at least some zeros with the the Riemann $\zeta$ function in the left complex half-plane where the convergence of the Euler product to the Riemann $\zeta$ function is not proven.  
5075\end{rem}
5076 
5077 
5078\begin{cor}\label{thm:5y74f757}
5079    The Euler product from of $\zeta$ has non-trivial zeros with negative real parts in $\mathbb{R}_\infty$.
5080\end{cor}
5081
5082
5083
5084
5085\begin{proof}
5086    Consider a number $z_0\in\mathbb{C}$ such that 
5087    \begin{equation}
5088    z_0=-(\aleph_\mathcal{X}+b)+iy_0~~,\quad\text{where}\qquad b,y_0\in\mathbb{R}_0~~.\nonumber
5089    \end{equation}
5090   
5091    \noindent Observe that the Euler product form of $\zeta$ \cite{EULER2} takes $z_0$ as  
5092    \begin{align}
5093    \zeta(z_0)&=\prod_{p}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy_0}}\nonumber\\
5094    &=  \left(\cfrac{1}{1-P^{(\mathcal{X}\aleph_1+b)-iy_0}} \right) \prod_{p\neq  P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy_0}}    \nonumber\\
5095    &=\left(\cfrac{1}{1-\cfrac{\vphantom{\hat1}1}{P^b}\big(P^{\mathcal{X}}\big)^{\!\widehat\infty}\left[     \cos(y_0\ln P)-i\sin(y_0\ln P)\right]}  \right)  \prod_{p\neq  P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy_0}}     \nonumber~~.
5096    \end{align}
5097   
5098    \noindent Let $y_0\ln P=2n\pi$ for some prime $P$ and $n\in\mathbb{N}$ or $n=0$.  Then
5099    \begin{align}
5100    \zeta(z_0)&=\left(\cfrac{1}{1-\widehat\infty}\right) \prod_{p\neq  P}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy_0}}=0~~.\nonumber
5101    \end{align}
5102\end{proof}
5103 
5104 
5105\begin{conj}
5106    The theorem of Hadamard and de la Vall\'ee-Poussin \cite{HAD,DLVP} showing that $\zeta$ never vanishes on the line $\text{Re}(z)=1$ should fail along the portions of that line lying in the neighborhood of infinity.  Likewise, the result proving that $\zeta$ cannot have any zeros beyond the critical strip other than the negative even integers should fail in the neighborhood of infinity.
5107\end{conj}
5108 
5109 
5110 
5111 
5112\newpage
5113\appendix
5114 
5115 
5116\numberwithin{thm}{section}
5117 
5118\section{Developing Mathematical Systems Historically}
5119 
5120 
5121Because this treatise so concisely follows a \textit{very} long trail of preexisting philosophical pursuits in mathematics, we present here as an appendix a concise summary of some of the important questions which motivated the modernist approach to complementing Euclid as the foundation of real analysis.  In the article \textit{The Real Numbers: From Stevin to Hilbert}, O'Connor and Robertson write the following \cite{OCR2}.
5122 
5123\begin{quote}
5124    ``By the time Stevin proposed the use of decimal fractions in 1585, the concept of a real number had developed little from the that of Euclid's \textit{Elements}.  Details of the earlier contributions are examined in some detail in our article \textit{The real numbers: from Pythagoras to Stevin}.''  
5125\end{quote}
5126 
5127This appendix summarizes two articles by O'Connor and Robertson which outline the history of what are today called the real numbers \cite{OCR1,OCR2}.  This appendix essentializes the trail of facts supporting the present axiom-constructive fractional distance approach to the real number system.  Setting the stage for the theme, O'Connor and Robertson write the following.
5128 
5129\begin{quote}
5130    ``By the beginning of the 20th century then, the concept of a real number had moved away completely from the concept of a number which had existed from the most ancient times to the beginning of the 19th century, namely its connection with measurement and quantity.''
5131\end{quote}
5132 
5133\noindent O'Connor and Robertson cite Wallis as writing the following.
5134 
5135\begin{quote}
5136    ``[S]uch proportion is not to be expressed in the commonly received ways of notation.''
5137\end{quote}
5138 
5139Wallis makes a wholehearted declaration of the mathematical matter contended by fractional distance.  Sometimes it is necessary to introduce new notations such as $\aleph_\mathcal{X}$, $\widehat\infty$, and $\mathbb{F}(n)$.  Therefore, should it be claimed that one may not simply declare a thing such as $\aleph_\mathcal{X}$, Wallis is cited as evidence that one may and that at times one must.  Further emphasizing the importance of the influx of new notations into contemporary mathematics, O'Connor and Robertson write the following.
5140 
5141\begin{quote}
5142    ``A major advance was made by Stevin in 1585 in \textit{La Thiende} when he introduced decimal fractions.  One has to understand here that in fact it was in a sense fortuitous that his invention led to a much deeper understanding of numbers for he certainly did not introduce the notation with that in mind.  Only finite decimals were allowed, so with his notation only certain rationals [\textit{were}] to be represented exactly.  Other rationals could be represented approximately and Stevin saw the system as a means to calculate with approximate rational values.  His notation was to be taken up by Clavius and Napier but others resisted using it since they saw it as a backwards step to adopt a system which could not even represent $\frac{1}{3}$ exactly.''
5143\end{quote}
5144 
5145 
5146\noindent Still yet further emphasizing the rightful place of new notation in mathematics, O'Connor and Robertson write the following.
5147 
5148 
5149\begin{quote}
5150    ``Strictly speaking, only that which is logically impossible (i.e.: which contradicts itself) counts as impossible for the mathematician.''
5151\end{quote}
5152 
5153\noindent All progress in mathematics, therefore, must be predicated from time to time upon the introduction of new notations such as $\aleph_\mathcal{X}$ and $\widehat\infty$.  
5154 
5155We have shown the aesthetic likeness of the present course to the previous course.  Stevin introduced decimal fractions and now we have introduced infinity hat.  Leibniz gave us the integral symbol and now there exists a real number $\aleph_{0.5}$ (which was already known as long as ago Euler who wrote $\frac{i}{2}$.)  Now we will emphasize that the course in question has always been the means by which to unify algebra and geometry.  O'Connor and Robertson write the following.
5156 
5157 
5158\begin{quote}
5159    ``Similarly Cantor realized that if he wants the line to represent the real numbers then he has to introduce an axiom to recover the connection between the way real numbers are now being defined and the old concept of measurement.''
5160\end{quote}
5161 
5162\noindent O'Connor and Robertson specifically identify Cantor's motivations \cite{OCR2} as the same given here.  How can we best preserve the geometric notion of an infinite line in the algebraic arena?  If one supposes that ``infinity is not allowed,'' and lets that be the end of the inquiry into the preservation of the notion of infinite geometric extent, then it is unlikely that the resulting mathematical system will make sufficient provisions for that fundamental notion.  Indeed, the entire theme of this paper has been to change existing mathematical systems so as to better accommodate the notion of infinite geometric extent.   Cantor himself wrote the following.
5163 
5164\begin{quote}
5165    ``[\textit{O}]ne may add an axiom which simply says that every numerical quantity also has a determined point on the straight line whose coordinate is equal to that quantity.''
5166\end{quote}
5167 
5168In the present treatise, we have extended Cantor to separately consider the ``determined'' geometric point from the numbers in the algebraic representation of that point.  Indeed, this is the main distinction between our own approach and Cantor's approach.  This issue fairly well represents the issue cited earlier as the source of much pathology in modern analysis: Cantor's presumption of a one-to-one correspondence between numbers and points is a fair proxy one's choice to distinguish algebraic FDFs of the first and second kinds.  Cantor's concept of fractional distance seems to favor $\mathcal{D}^\dagger_{\!AB}=\mathcal{D}''_{\!AB}$ whereas we have demonstrated the philosophical superiority of $\mathcal{D}^\dagger_{\!AB}=\mathcal{D}'_{\!AB}$.  We can glean from Cantor's words that he likely associated only one number with each point but we have shown that this is only best when the line segment is of finite length.  If the determined point is in an infinitely long line segment such as $X\in\mathbf{AB}$, then we have proven that the determined point does not have one uniquely associated real number.
5169 
5170In this treatise, we have restated the Archimedes property in English and Latin mathematical symbols.  We have also given a modern restatement of the Archimedes property as the Archimedes property of 1D transfinitely continued numbers (Axiom \ref{ax:nncncncn}.)  Similarly, Hilbert gave his own modernized restatement of that property when giving his geometry axioms.  O'Connor and Robertson write the following.
5171 
5172\begin{quote}
5173    ``[\textit{Hilbert's statement of the Archimedes property was}] that given positive numbers $a$ and $b$ then it is possible to add $a$ to itself a finite number of times so that the sum exceeds $b$.''
5174\end{quote}
5175 
5176\noindent What Hilbert wrote specifically was the following.
5177 
5178\begin{quote}
5179    ``If AB and CD are any segments then there exists a number $n$ such that $n$ segments $CD$ constructed contiguously from $A$, along the ray from $A$ through $B$, will pass beyond the point $B$.''
5180\end{quote}
5181 
5182\noindent Hilbert's original reliance on the $AB$ notation to give a statement of the Archimedes property for a Euclidean line segment very strongly highlights the historical similitude of the present approach to a modernizing algebraic capstone on Euclidean geometry.  Hilbert's axioms of geometry applied to Dedekind cuts give us the field axioms, more or less, so it is remarkable that we were likewise called, while working to the same ends as Hilbert, to give a restatement of what Euclid meant when he said he had it on good authority that Archimedes had heard from Eudoxus that such and such was the real Archimedes property of real numbers.  In the case of Hilbert's statement of the Archimedes property, we see that Hilbert gave a finite multiplier but did not explicitly require $n\in\mathbb{N}$.  The extended natural numbers $n\in\mathbb{N}_\infty$ provide the multipliers needed to preserve Hilbert's statement of the property in the fractional distance framework of real analysis.
5183 
5184Regarding the very ancient history, O'Connor and Robertson write the following.
5185 
5186\begin{quote}
5187    ``It seems clear that Pythagoras would have thought of $1,2,3,4\dots$ (the natural numbers in the terminology of today) in a geometrical way, not as lengths of a line as we do, but rather in the form of discrete points.  Addition, subtraction, and multiplication of integers are natural concepts with this type of representation but there seems to have been no notion of division.''
5188\end{quote}
5189 
5190\noindent Even as long as ago as Pythagoras, the open question of the separation of algebraic numbers from geometric magnitudes was already one of import.  Also, we have a distinct similitude here with the possibility that we might give the regularly-spaced, disconnected immeasurable numbers $\mathcal{F^X}\in\mathbb{F}$ an arithmetic axiom such that they are the real numbers on an infinitely bigger copy of the real number line.
5191 
5192In the present treatise, like Hilbert very recently, we have sought to build a hybrid constructive framework for mathematical analysis which maximizes the synergy between algebra and geometry.  O'Connor and Robertson write the following.
5193 
5194\begin{quote}
5195    ``[\textit{I}]t should be mentioned at this stage that the Egyptians and the Babylonians had a different notion of a number to that of the ancient Greeks.  The Babylonians looked at reciprocals and also at approximations to irrational numbers, such as $\sqrt{2}$, long before Greek mathematicians considered approximations.  The Egyptians also looked at approximating irrational numbers.\newline $~~~~$``Let us now look at the position as it occurs in Euclid's \textit{Elements}.  This is an important stage since it would remain the state of play for nearly the next 2000 years. In Book V Euclid considers magnitudes and the theory of proportion of magnitudes. It is probable (and claimed in a later version of \textit{The Elements}) that this was the work of Eudoxus. Usually when Euclid wants to illustrate a theorem about magnitudes he gives a diagram representing the magnitude by a line segment. However magnitude is an abstract concept to Euclid and applies to lines, surfaces and solids. Also, more generally, Euclid also knows that his theory applies to time and angles.\newline $~~~~$``Given that Euclid is famous for an axiomatic approach to mathematics, one might expect him to begin with a definition of magnitude and state some unproved axioms. However he leaves the concept of magnitude undefined and his first two definitions refer to the part of a magnitude and a multiple of a magnitude.''
5196\end{quote}
5197 
5198 
5199 
5200O'Connor and Robertson proceed to break down Euclid's Book V as we have when examining the Archimedes property in Section \ref{sec:archim}.  Therefore, we will list the properties and comments again in expanded form.  We consolidate the comments on Euclid's original text with Fitzpatrick's labeled (RF), our own comments labeled (JT), and the comments of O'Connor and Robertson labeled (OR).
5201 
5202$~$
5203 
5204\noindent \textbf{Book V, Definition 1} A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.
5205 
5206\begin{quote}
5207    (RF) In other words, $\alpha$ is said to be a part of $\beta$ if $\beta=m\alpha$.
5208   
5209    (JT) The first definition makes it obvious that the multiplier is not meant to be a natural number.  If the magnitude of ten units of geometric length is to be greater than one of nine, then there must exist non-integer multipliers.
5210   
5211    (OR) Again the term ``measures'' here is undefined but clearly Euclid means that (in modern symbols) the smaller magnitude $x$ is a part of the greater magnitude $y$ if $nx=y$ for some natural numbers $n>1$.
5212\end{quote}
5213 
5214When O'Connor and Robertson write $n\in\mathbb{N}$, they do not take into consideration numbers having non-integer quotients, \textit{e.g.}: $10\!:\!9$, or else they are only giving a subcase of what is meant in the original context.  Using the natural numbers to demonstrate the property makes sense if one takes the auxiliary axiom that there are no real numbers greater than every natural number.  In that case, the Archimedes property is irreducibly represented in the natural number statement of the multiplier.  The supposition that every real number is in the neighborhood of the origin was a normal axiom at the time of the publication of References \cite{OCR1,OCR2}.  The main difference between the present approach and the historical approaches to merging geometry and algebra is that we have not tried to squeeze the notion of geometric infinity into the algebraic sector.  In the present conventions, $\widehat\infty$ is such that the algebraic structure is totally subordinate to geometric structure.  The primary theme of the past few centuries has been one of attempting to subordinate geometry to algebra.
5215 
5216Many historical approaches have assumed some algebraic axioms and then tried to fit everything inside those axioms by ignoring geometric infinity and making a rule that one must never mention it.  Note the equal weighting of the gravity of the matters in the choice to suppose one of two axioms.
5217 
5218 
5219\begin{axio}\label{ax:fhf924792491}
5220    There exists a non-empty set $\{\mathbb{R}^\mathcal{X}\}$ of real numbers greater than any natural number.
5221\end{axio}
5222 
5223\begin{axio}\label{ax:fhf924792492}
5224    There does not exist any real number greater than every natural number so, therefore, $\mathbb{R}\setminus\mathbb{R}_0=\varnothing$.
5225\end{axio}
5226 
5227 
5228An assigned superiority in the algebraic sector might make Axiom \ref{ax:fhf924792492} the more attractive axiom because it allows everything to be written with the field axioms.  By assigning the superior quality as the historical geometric conception of numbers, we are drawn to Axiom \ref{ax:fhf924792491} as the preferable axiom.  Additionally, we have proven multiply that Axiom \ref{ax:fhf924792492} causes undesirable contradictions with the geometric notion of fractional distance.  Even when algebraic considerations are chosen as superior to geometric ones, the superior axiom must not contradict its inferior complement.  The neighborhood of infinity does exist; fractional distance requires it.  The question is only whether we should adopt an algebraic convention which reflects the geometric reality.
5229 
5230$~$
5231 
5232 
5233 
5234 
5235\noindent \textbf{Book V, Definition 2} And the greater is a multiple of the lesser whenever it is measured by the lesser.
5236 
5237\begin{quote}
5238    (JT) This definition makes it explicitly clear that the manner in which one magnitude may measure another is such that, for example, nine can measure ten by $10\!:\!9$.
5239   
5240    (OR) Then comes the definition of ratio.
5241\end{quote}
5242 
5243 
5244$~$
5245 
5246\noindent \textbf{Book V, Definition 3} A ratio is a certain type of condition with respect to size of two magnitudes of the same kind.
5247 
5248\begin{quote}
5249    (RF) In modern notation, the ratio of two magnitudes, $\alpha$ and $\beta$, is denoted $\alpha\,:\,\beta$.
5250   
5251    (JT) This definition tells us that $\mathbb{R}$ is equipped with $\leq$ relation.  The specification of two magnitudes of the same kind tells us, essentially, that Euclid does not want his reader to compare lengths with areas, volumes, or hypervolumes.
5252   
5253    (OR) This is an exceptionally vague definition of ratio which basically fails to define it at all.  [\textit{Euclid}] then defines when magnitudes have a ratio, which according to the definition is when there is a multiple (by a natural number) of the first which exceeds the second and a multiple of the second which exceeds the first.
5254\end{quote}
5255 
5256$~$
5257 
5258 
5259\noindent \textbf{Book V, Definition 4} (Those) magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.
5260 
5261\begin{quote}
5262    (RF) In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.
5263   
5264    (JT) The Archimedes property of real numbers requires that for every real number, there is a greater real number.  In other words and in a general way, there is no largest real number because $\aleph_1\not\in\mathbb{R}$.  Usually it is said that a smallest real number is also precluded by the inverse of the unbounded large number.  Surprisingly, the usual topology requirement of the fundamental axiom of algebraic construction seems to indicate that a smallest real number must exist.  This is the $\mathcal{X}$ value of $\aleph(2)=\aleph_{\mathcal{X}}$.  The issue of a smallest positive real number has been a historically vexing contention in the intuitive sense.  If every interior point in a connected interval $(-1,1)$ is left- and right-adjacent to another point, then writing
5265    \begin{equation*}
5266    (-1,1)=(-1,0]\cup(0,1)~~,
5267    \end{equation*}
5268   
5269    \noindent suggests, in an intuitive way at least, that zero must be left-adjacent to the smallest positive real number.  However, the protocols of mathematics override intuition and it is said that zero is not left-adjacent to any element of $(0,1)$ because every element of $(0,1)$ has a $\delta$-neighborhood lying totally within $(0,1)$.  So, if some way is found to claw a least positive real number of the precepts of fractional distance, then the concept of no greatest real number would also have to be done away with due to the invariance of $\mathbf{AB}$ under permutations of the labels of its endpoints.  Infinity minus the least positive real number would be the greatest real number.
5270   
5271    (OR) The Archimedean axiom stated that given positive numbers $a$ and $b$ then it is possible to add $a$ to itself a finite number of times so that the sum exceed $b$.
5272\end{quote}
5273 
5274$~$
5275 
5276 
5277\noindent \textbf{Book V, Definition 5} Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when equal multiples of the first and third both exceed, are both equal to, or are both less than, equal multiples of the second and fourth, respectively, being taken in corresponding order, according to any kind of multiplication whatever.
5278 
5279\begin{quote}
5280    (RF) In other words, $\alpha\,:\,\beta\,::\,\gamma\,:\,\delta$ if and only if $m\alpha>n\beta$ whenever $m\gamma>n\delta$, $m\alpha=n\beta$ whenever $m\gamma=n\delta$, and $m\alpha<n\beta$ whenever $m\gamma<n\delta$, for all $m$ and $n$.  This definition is the kernel of Eudoxus' theory of proportion, and is valid even if $\alpha$, $\beta$, \textit{etc.}, are irrational.
5281   
5282    (JT) This definition gives the trichotomy of the $\leq$ relation.  Also note that the ratio of ratios is like the ratio of two fractional distances.
5283   
5284    (OR) Then comes the vital definition of when two magnitudes are in the same ratio as a second pair of magnitudes. As it is quite hard to understand in Euclid's language, let us translate it into modern notation. It says that $a : b = c : d$ if given any natural numbers $n$ and $m$ we have
5285    \begin{align*}
5286    na > mb \quad&\text{if and only if}\quad nc > md\\
5287    na = mb \quad&\text{if and only if}\quad nc = md\\
5288    na < mb  \quad&\text{if and only if}\quad nc < md~~.
5289    \end{align*}
5290 
5291    Euclid then goes on to prove theorems which look to a modern mathematician as if magnitudes are vectors, integers are scalars, and he is proving the vector space axioms.
5292\end{quote}
5293 
5294The main hurdle in the vector space conception of $\mathbb{R}$ is that the product of two vectors is a scalar but the product of two real numbers in another real number.  Even in the transfinite continuation beyond algebraic infinity, even when the product of two things in the line always remains within the geometrically infinite line as if it were a vector space, the problem remains that the product of two 1D transfinitely continued extended real numbers will be another 1D transfinitely continued extended real number, and there is no distinguishing a vector from a scalar.  However, one easily imagines $\widehat\infty$ as an anchor point for 1D vectors $x\in\mathbb{R}$ different than the anchor point at the origin.  Vectors anchored in the neighborhood of the origin look like $\hat0+\vec b$ and those anchored in the neighborhood of positive infinity look like $\widehat\infty-\vec b$.  The 1D vector space picture is very becoming the notion of a 1D geometric space but the lack of distinction among vector and scalars forbids any approach to the commonly stated modern vector space axioms.  
5295 
5296\newpage
5297 
5298\bibliographystyle{unsrt}
5299\bibliography{TOIC}{}
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5301\end{document}