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55	\fancyhead[OC]{\textsc{Jonathan W. Tooker}}
56	\fancyhead[EC]{\textsc{Fractional Distance: The Topology of the Real Number Line}}
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97\numberwithin{equation}{section}
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99\title{{ \textbf{Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis}}}
100
101\author{{\textsc{Jonathan W. Tooker}}}
102\begin{document}
103	
104
105	
106\begin{minipage}{\textwidth}
107	\maketitle
108	\thispagestyle{empty}
109	\begin{abstract}
110		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.
111	\end{abstract}
112\end{minipage}
113	
114\raggedbottom
115  
116
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122\tableofcontents
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127\large{
128	
129	
130
131\section{Introduction}
132
133The 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.'' 
134
135The 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}}.
136
137
138Treatment 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.  
139
140In 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.
141
142The paper is structured as follows.
143\begin{itemize}
144	\item Section Two: We give a simple Euclidean definition for real numbers.  These geometric considerations set the stage for the algebraic considerations which follow.
145	\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.
146	\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.
147	\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.
148	\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.
149	\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.
150	
151	\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. 
152\end{itemize}
153 
154\section{Mathematical Preliminary}\label{sec:RN1ddd}
155	
156 
157\subsection{Real Numbers}\label{sec:RN1}
158
159
160In 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.  
161
162Generally, 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}.
163 
164
165\begin{defin}\label{def:lineXXX}
166	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.
167\end{defin}
168
169\begin{defin}\label{def:metspa}
170	A number line is a line equipped with a chart $x$ and the Euclidean metric
171	\begin{equation}
172	d(x,y)=
173	\big|y-x\big|~~.\nonumber
174	\end{equation}
175\end{defin}
176
177\begin{defin}\label{def:2412b24}
178	The real number line is a unique number line given the label ``real.''
179\end{defin}
180 
181\begin{defin}\label{def:cuts}
182	If $x$ is a cut in a line, then 
183	\begin{equation}
184	(-\infty,\infty)=(-\infty,x]\cup(x,\infty)~~.\nonumber
185	\end{equation} 
186\end{defin}
187
188
189\begin{defin}\label{def:real}
190	A real number $x\in\mathbb{R}$ is a cut in the real number line.
191\end{defin}
192
193\begin{axio}\label{ax:97977080}
194	Real numbers are such that 
195	\begin{equation*}
196	\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}~~.
197	\end{equation*}
198	
199	\noindent In other words, neither infinitesimals nor numbers having infinitesimal parts are real numbers.  
200\end{axio}
201
202
203
204\begin{axio} \label{ax:mainR}
205	Real numbers are represented in algebraic interval notation as 
206	\begin{equation}
207	\mathbb{R}=(-\infty,\infty)~~.\nonumber
208	\end{equation}
209	
210	\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.
211\end{axio}
212
213\begin{rem}
214	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}.   
215\end{rem}
216
217\begin{defin}\label{def:orig}
218	$\mathbb{R}_0$ is a subset of all real numbers
219	\begin{equation}
220	\mathbb{R}_0=\left\{  x\in\mathbb{R}~\big|~(  \exists n\in\mathbb{N})[ -n<x<n ]  \right\}     ~~.\nonumber
221	\end{equation} 
222	
223	\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.
224\end{defin}
225
226
227\begin{defin}\label{def:Rinf}
228	$\mathbb{R}_\infty$ is a subset of all real numbers with the property
229	\begin{equation}
230	\mathbb{R}_\infty=\mathbb{R}\,\backslash\,\mathbb{R}_0~~.\nonumber
231	\end{equation} 
232\end{defin}
233
234
235\subsection{Affinely Extended Real Numbers}\label{sec:33333}
236
237
238
239To 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 $\overline{\mathbb{R}}$.   
240
241
242
243\begin{defin}  \label{def:RRRinf9999}
244	For $x\in\mathbb{R}$ and $n,k\in\mathbb{N}$, we have the properties
245	\begin{equation}
246	\lim\limits_{x\to0^\pm}\dfrac{1}{x}=\text{diverges}~~,\qquad\text{and}\qquad \lim\limits_{n\to\infty}\sum_{k=1}^{n}k=\text{diverges}~~. \nonumber
247	\end{equation} 
248\end{defin}
249
250
251
252\begin{defin} \label{def:RRRinf}
253	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
254	\begin{equation}
255	\lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\infty~~, \qquad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
256	\end{equation} 
257	
258	\noindent The limit as $x$ approaches zero shall be referred to as the limit definition of infinity.
259\end{defin}
260
261
262\begin{axio} \label{ax:undefin7666y}
263	The infinite element $\infty$ is such that
264	\begin{align}
265	\infty-\infty=\text{undefined}~~,\qquad\text{and}\qquad\frac{\infty}{\infty}=\text{undefined}~~.\nonumber
266	\end{align} 
267\end{axio}
268
269
270\begin{defin} \label{def:overlineR}
271	The set of all affinely extended real numbers is 
272	\begin{equation}
273	\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}~~.\nonumber
274	\end{equation} 
275
276	\noindent This set is defined in interval notation as 
277	\begin{equation}
278	\overline{\mathbb{R}}=[-\infty,\infty]~~.\nonumber
279	\end{equation} 
280\end{defin}
281
282
283\begin{rem}\label{def:12ccc2}
284	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
285	\begin{equation*}
286	\lim\limits_{n\to\infty}x_n=\text{diverges}~~,
287	\end{equation*}
288	
289	\noindent then for the same $x_n\in\overline{\mathbb{R}}$ we have
290	\begin{equation}
291	\lim\limits_{n\to\infty}x_n=\infty~~.\nonumber
292	\end{equation}
293\end{rem}
294
295
296
297\begin{defin}\label{def:AFF2real}
298	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:
299	\begin{equation}
300	[-\infty,\infty]=[-\infty,x]\cup(x,\infty]~~.\nonumber
301	\end{equation} 
302\end{defin}
303
304
305
306
307\begin{thm} \label{thm:RnotR}
308	If $x\in\overline{\mathbb{R}}$ and $x\neq\pm\infty$, then $x\in\mathbb{R}$.
309\end{thm}
310
311
312
313\begin{proof}
314	Proof follows from Definition \ref{def:overlineR}.
315\end{proof}
316  
317
318\subsection{Line Segments}\label{sec:LineSegs}
319
320
321In 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
322\begin{equation}
323(-\infty,\infty)=(-\infty,x]\cup(x,\infty)~~.\nonumber
324\end{equation} 
325
326\noindent If $x$ is a point in a line segment, then we have a tentative, preliminary understanding that
327\begin{equation}
328[a,b]=[a,x)\cup\{x\}\cup(x,b]~~.\nonumber
329\end{equation}
330
331\begin{defin}\label{def:lineseg}
332	A line segment $AB$ is a line together with two different endpoints $A\neq B$.   
333\end{defin}
334
335
336\begin{defin}\label{def:lineseg2}
337	$AB$ is a real line segment if and only if the endpoints $A$ and $B$ bound some subset of the real line $\mathbb{R}=(-\infty,\infty)$.
338\end{defin}
339
340
341\begin{defin}
342	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.
343\end{defin}
344
345
346
347\begin{defin}\label{def:li5t35y5y5y2}
348	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 $a<b$.
349\end{defin}
350
351\begin{defin}\label{def:eucdef9}
352	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.
353\end{defin}
354
355\begin{axio}\label{def:2points}
356Line segments have the property that
357\begin{equation*}
358AB=AC\quad\iff\quad B=C~~. 
359\end{equation*}
360\end{axio}
361
362
363\begin{axio}\label{def:li5t3f5y5y5y2}
364	Two line segments $AB$ and $CD$ are equal, meaning $AB=CD$, if and only if 
365	\begin{equation} 
366	\cfrac{AB}{CD}=\cfrac{CD}{AB}=1~~.\nonumber
367	\end{equation} 
368\end{axio}
369
370
371\begin{defin}
372	$\mathbf{AB}$ is a special label given to the unique real line segment $AB\equiv[0,\infty]$.  We have
373	\begin{equation}
374	AB=\mathbf{AB}\quad\iff\quad AB\equiv[0,\infty]~~.\nonumber
375	\end{equation}
376\end{defin}
377
378
379
380
381\begin{defin}
382	$X$ is an interior point of $AB$ if and only if
383	\begin{equation}
384	X\neq A~~,~~X\neq B~~,\qquad\text{and}\qquad   X\in AB  ~~.\nonumber
385	\end{equation} 
386\end{defin}
387
388\begin{axio}\label{def:5y5dddy2}
389	If $X$ is an interior point of $AB$, then
390	\begin{equation}
391	AB=AX+XB~~.\nonumber
392	\end{equation} 
393\end{axio}
394
395
396\begin{axio}\label{ax:779hzz}
397	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 $[a,b]\equiv AB$.
398\end{axio}
399
400\begin{defin}
401	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 $a<x<b$.
402\end{defin}
403
404
405\begin{thm} 
406	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.
407\end{thm}
408
409\begin{proof}
410	$X$ is an interior point of $AB$ so, by Axiom \ref{def:5y5dddy2}, we have
411	\begin{equation}
412	AB=AX+XB~~.\nonumber
413	\end{equation}
414	
415	\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
416	\begin{equation}
417	x\in\mathscr{X}\quad\implies\quad a<x<b~~.\nonumber
418	\end{equation}
419	
420	\noindent For $(a,b)\subset\mathbb{R}$, this inequality is only satisfied by $x\in\mathbb{R}$.  The theorem is proven.
421\end{proof}
422
423\begin{rem}
424	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 $\text{len}(AB)$.
425\end{rem}
426
427
428\begin{defin}\label{def:XrepR}
429	The algebraic representation $\mathscr{X}$ of a geometric point $X$ lying along a real line segment $AB$ is
430	\begin{equation}
431	\mathscr{X}=[x_1,x_2]~~,\qquad\text{where}\qquad x_1,x_2\in\overline{\mathbb{R}}~~. \nonumber
432	\end{equation} 
433	
434	\noindent  The special (intuitive) case of $x_1=x_2=x$ gives
435	\begin{equation}
436	\mathscr{X}=[x,x]=\{x\}=x~~. \nonumber
437	\end{equation} 
438	
439	\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.
440\end{defin}
441
442
443\begin{rem}
444	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.
445\end{rem} 
446
447
448\begin{defin}
449	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
450	\begin{equation}
451	x\in\mathscr{X}=[x_1,x_2]\equiv X~~.\nonumber
452	\end{equation}
453	
454	\noindent  This statement may be abbreviated as $x\in X$ while $x\equiv X$ specifies the case of $x_1=x_2$.
455\end{defin}
456
457
458
459
460
461\begin{defin}\label{def:fet7}
462	A point $C$ is called a midpoint of a line segment $AB$ if and only if
463	\begin{equation}
464	\cfrac{AC}{ AB} =\cfrac{CB}{ AB} =\frac{1}{2}~~.\nonumber
465	\end{equation} 
466	
467	\noindent Alternatively, $C$ is a midpoint of $AB$ if and only if 
468	\begin{equation}
469	AC=CB~~,\qquad\text{and}\qquad AC+CB=AB~~.\nonumber
470	\end{equation} 
471\end{defin}
472
473
474\begin{defin}
475	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 $D$.
476\end{defin}
477
478\begin{rem}
479	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.  
480\end{rem}
481
482
483\begin{thm}\label{def:fet17}
484	All line segments have at least one midpoint.
485\end{thm}
486
487\begin{proof}
488	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
489	\begin{equation}
490	AS=AT=BS=BT~~.\nonumber
491	\end{equation} 
492	
493	\noindent   Let the line segment $ST$ intersect $AB$ at $C$.  By the Pythagorean theorem, $C$ is a midpoint of $AB$ because
494	\begin{equation}
495	AC^2+CS^2=AS^2~~,\qquad\text{and}\qquad BC^2+CS^2=BS^2~~,\nonumber
496	\end{equation} 
497	
498	\noindent together yield
499	\begin{equation}
500	AC=BC~~.\nonumber
501	\end{equation} 
502	
503	\noindent  $C$ separates $AB$ into two line segments so
504	\begin{equation}
505	AC+CB=AB~~.\nonumber
506	\end{equation} 
507	
508	\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 $AB$.
509\end{proof}
510
511\begin{figure}[t]
512	\makebox[\textwidth][c]{
513		\includegraphics[scale=.25]{twocirc2.png}}
514	\caption{This figure proves that every line segment $AB$ has one and only one midpoint.} 
515	\label{fig:twocirc}
516\end{figure} 
517
518\begin{exa}\label{ex:confrmal}
519	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
520	\begin{equation}
521	x=\tan(x')~~,\nonumber
522	\end{equation}
523	
524	\noindent so that $x$ and $x'$ are two charts related by a conformal transformation.  Using
525	\begin{equation}
526	\tan(0)=0~~,\qquad\text{and}\qquad\tan\left(\cfrac{\pi}{2}\right)=\infty~~,\nonumber
527	\end{equation}
528	
529	\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.   
530	
531	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	
532	\begin{equation}
533	x'=\cfrac{\pi}{ 6}\in\mathscr{X}_1'~~,\qquad\text{and}\qquad x'= \cfrac{\pi}{ 3}\in\mathscr{X}_2'~~.\nonumber
534	\end{equation}
535	
536	\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 $AB=\mathbf{AB}$.
537\end{exa}
538
539
540
541
542\begin{thm}\label{thm:onemid}
543	All line segments have one and only one midpoint.
544\end{thm}
545
546\begin{proof}
547	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} 
548	\begin{equation}
549	AC=CB=\cfrac{AB}{2}~~,\qquad\text{and}\qquad AD=DB=\cfrac{AB}{2}~~.\nonumber
550	\end{equation} 
551	
552	\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.  
553\end{proof}
554
555
556\section{Fractional Distance}
557
558\subsection{Fractional Distance Functions}\label{sec:FD}
559
560
561If 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.
562
563
564
565\begin{defin}\label{def:gfdf}
566	For any point $X$ on a real line segment $AB$, the geometric fractional distance function $\mathcal{D}_{\!AB}$ is a continuous bijective map 
567	\begin{equation}
568	\mathcal{D}_{\!AB}(AX):AB\to[0,1]  ~~.\nonumber
569	\end{equation} 
570	
571	\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
572	\begin{equation}
573	\mathcal{D}_{\!AB}(AX)=\begin{cases}
574	~~1\qquad\quad\text{for}\quad X=B\\[8pt]
575	\cfrac{AX}
576	{AB}~\qquad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
577	~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~.\nonumber
578	\end{equation} 
579	
580	\noindent The quotient of two real line segments is defined as a real number.
581\end{defin}
582
583\begin{rem}
584	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. 
585\end{rem}
586
587 
588
589\begin{thm}\label{thm:35353g0}
590	For any point $X\in AB$, the bijective geometric fractional distance function $\mathcal{D}_{\!AB}(AX):AB\to R$ has range $R=[0,1]$.
591\end{thm}
592
593
594\begin{proof}
595	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 $R=[0,1]$.  
596\end{proof}
597
598
599\begin{cor} 
600	All line segments have at least one midpoint.  
601\end{cor}
602
603\begin{proof}
604	(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 $AB$.
605\end{proof}
606
607
608\begin{thm}
609	Every midpoint of a line segment $AB$ is an interior point of $AB$.
610\end{thm}
611
612\begin{proof}
613	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	
614	\begin{equation}
615	\mathcal{D}_{\!AB}(AA)=0~~,\qquad\text{or}\qquad \mathcal{D}_{\!AB}(AB)=1~~.\nonumber
616	\end{equation} 
617	
618	\noindent A point $C$ is a midpoint of $AB$ if and only if 	
619	\begin{equation}
620	\mathcal{D}_{\!AB}(AC)=0.5~~.\nonumber
621	\end{equation} 
622	
623	\noindent No midpoint can be an endpoint.  
624\end{proof} 
625
626\begin{rem}
627	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.  
628\end{rem}
629
630\begin{defin}\label{def:iuuuttgrr488}
631	$\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	
632	\begin{equation}
633	\mathcal{D}^\dagger_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
634	\end{equation}
635	
636	\noindent for every point $X\in AB$.
637\end{defin}
638
639
640\begin{rem}
641	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} $.  
642\end{rem} 
643
644\begin{defin}\label{def:algfracdis}
645	The algebraic FDF of the first kind  
646	\begin{equation}
647	\mathcal{D}'_{\!AB}(AX):AB\to[0,1] ~~,\nonumber
648	\end{equation} 
649	
650	\noindent   is a map on subsets of real line segments 
651	\begin{equation}
652	\mathcal{D}'_{\!AB}(AX)=\begin{cases}
653	~~~1\qquad\quad\text{for}\quad X=B\\[8pt]
654	\cfrac{\|AX\|}{ \|AB\|}\,~~\quad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
655	~~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~,\nonumber
656	\end{equation} 
657	
658	\noindent where	
659	\begin{equation}
660	\cfrac{\|AX\|}{ \|AB\|}  =   \cfrac{\text{len}[a,x]}{\text{len}[a,b]}    ~~,\nonumber
661	\end{equation} 
662	
663	\noindent and $[a,x]$ and $[a,b]$ are the line segments $AX$ and $AB$ expressed in interval notation.   
664\end{defin}	
665
666
667\begin{defin} \label{def:agree}
668	The norm $\|AX\|=\text{len}[a,x]$ which appears in  $\mathcal{D}'_{\!AB}(AX)$ is defined so that
669	\begin{equation}
670	\mathcal{D}'_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
671	\end{equation} 
672	
673	\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
674	\begin{equation*}
675	\text{len}[a,b]=d(a,b)=\big|b-a\big|~~.
676	\end{equation*}
677\end{defin}	
678
679
680
681\begin{defin}\label{def:algfracdisq1}
682	An algebraic fractional distance function of the second kind  
683	\begin{equation}
684	\mathcal{D}''_{\!AB}(AX):[a,b]\to[0,1] ~~,\nonumber
685	\end{equation} 
686	
687	\noindent  is a map on intervals of the form
688	\begin{equation}
689	\mathcal{D}''_{\!AB}(AX)=\begin{cases}
690	~~~~~1\qquad&\text{for}\quad X=B\\[8pt]
691	\cfrac{\text{len}[a,x]}{\text{len}[a,b]}\,~~&\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
692	~~~~~0 \qquad&\text{for}\quad X=A \end{cases}~~.\nonumber
693	\end{equation}  
694\end{defin}	
695
696\begin{rem}
697	 Take note of the main difference between the two algebraic FDFs.  The first kind has a geometric domain
698	\begin{equation*}
699	\mathcal{D}'_{\!AB}(AX):AB\to\mathbb{R}~~,
700	\end{equation*}
701	
702	\noindent but the second kind has an algebraic domain
703	\begin{equation*}
704	\mathcal{D}''_{\!AB}(AX):[a,b]\to\mathbb{R}~~.
705	\end{equation*}
706	
707	\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.  
708\end{rem}
709
710
711
712\begin{axio}\label{def:order}
713	The ordering of $\mathbb{R}$ is such that for any $x,y\in\mathbb{R}$, if	
714	\begin{equation} 
715	x\in[x_1,x_2]=\mathscr{X}\equiv X~~,\qquad\text{and}\qquad y\in[y_1,y_2]=\mathscr{Y}\equiv Y~~,\nonumber
716	\end{equation} 
717	
718	\noindent then	
719	\begin{equation}
720	\mathcal{D}_{\!AB}(AX)>\mathcal{D}_{\!AB}(AY)\quad\implies\quad x>y ~~.\nonumber
721	\end{equation} 
722\end{axio}
723
724\begin{thm}\label{thm:injjj}
725	The geometric fractional distance function $\mathcal{D}_{\!AB}$ is injective (one-to-one) on all real line segments.
726\end{thm}
727
728\begin{proof}
729	By Definition \ref{def:gfdf}, the geometric FDF is 	
730	\begin{equation}
731	\mathcal{D}_{\!AB}(AX)=\begin{cases}
732	~~1\qquad\quad\text{for}\quad X=B\\[8pt]
733	\cfrac{AX}
734	{AB}~\qquad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
735	~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~.\nonumber
736	\end{equation} 
737	
738	
739	\noindent For proof by contradiction, assume $\mathcal{D}_{\!AB}$ is not always injective.  Then there exists some $X_1\neq X_2$ such that 	
740	\begin{equation}
741	\cfrac{AX_1}{AB}=\cfrac{AX_2}{AB}~~.\nonumber
742	\end{equation} 
743	
744	\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	
745	\begin{equation}
746	0=\cfrac{AX_2}{AB}-\cfrac{AX_1}{AB}=\cfrac{AX_2-AX_1}{AB} \quad\iff\quad AX_2=AX_1~~.\nonumber
747	\end{equation} 
748	
749	\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. 
750\end{proof}
751
752\begin{rem} 
753	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.}:	
754	\begin{equation}\label{eq:r344nnmm}
755	0=\cfrac{5-3}{\infty}\quad\centernot\iff\quad 5=3~~.
756	\end{equation}
757	
758	
759	\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.)   
760	
761	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
762	\begin{equation*}
763	\cfrac{AX}{AB}=\cfrac{cAB}{AB}=c~~.
764	\end{equation*}
765	
766	\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.
767\end{rem}
768
769\begin{thm}\label{thm:surjjj}
770	The geometric fractional distance function $\mathcal{D}_{\!AB}$ is surjective (onto) on all real line segments.
771\end{thm}
772
773\begin{proof}
774	Given the range $R=[0,1]$ proven in Theorem \ref{thm:35353g0}, proof follows from the notion of geometric fractional distance.
775\end{proof}
776
777\begin{rem}
778	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.
779\end{rem}
780
781\begin{conj}\label{conj:dv2324} 
782	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}.)
783\end{conj}
784
785
786\begin{thm}\label{thm:injjj2222}
787	The algebraic fractional distance function of the second kind $\mathcal{D}''_{\!AB}$ is not injective (one-to-one) on all real line segments.
788\end{thm}
789
790
791\begin{proof} 
792	Recall that Definition \ref{def:algfracdisq1} gives $\mathcal{D}''_{\!AB}:[a,b]\to [0,1]$ as	
793	\begin{equation}
794	\mathcal{D}''_{\!AB}(AX)=\begin{cases}
795	~~~~~1\qquad&\text{for}\quad X=B\\[8pt]
796	\cfrac{\text{len}[a,x]}{\text{len}[a,b]}\,~~&\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
797	~~~~~0 \qquad&\text{for}\quad X=A \end{cases}~~.\nonumber
798	\end{equation} 
799	
800	\noindent Injectivity requires that	
801	\begin{equation}
802	\mathcal{D}''_{\!AB}(AX)=\mathcal{D}''_{\!AB}(AY)\quad\iff\quad [a,x]=[a,y]\quad\iff\quad x=y~~.\nonumber
803	\end{equation}
804	
805	\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	
806	\begin{equation}
807	\mathcal{D}''_{\!\mathbf{AB}}(AN)=\cfrac{\text{len}[0,n]}{\text{len}[0,\infty]}=0~~,\qquad\text{and}\qquad \mathcal{D}''_{\!\mathbf{AB}}(AM)=\cfrac{\text{len}[0,m]}{\text{len}[0,\infty]}=0~~.\nonumber
808	\end{equation}
809	
810	\noindent 	Therefore, the algebraic FDF of the second kind is not injective on all real line segments because 	
811	\begin{equation}
812	\mathcal{D}''_{\!AB}(AN)=\mathcal{D}''_{\!AB}(AM)\quad\centernot\iff\quad n=m~~.\nonumber
813	\end{equation}
814\end{proof}
815
816
817\begin{rem}
818	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	
819	\begin{equation}
820	\mathcal{D}'_{\!AB}(AX):AB\to[0,1] ~~,\qquad\text{and}\qquad\mathcal{D}''_{\!AB}(AX):[a,b]\to[0,1]  ~~. \nonumber
821	\end{equation} 
822	
823	\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}}. 
824\end{rem}
825
826
827
828 
829
830\begin{thm}
831	The geometric fractional distance function $\mathcal{D}_{\!AB}$ is continuous everywhere on the domain $\mathbf{AB}$.
832\end{thm}
833
834\begin{proof}
835	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. 
836	
837	
838	$~$ 
839	
840	\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
841	\begin{equation}
842	\lim\limits_{x\to x_0}f(x)=f(x_0)~~.\nonumber
843	\end{equation} 
844	
845	\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
846	\begin{equation}
847	\lim\limits_{X\to X_0}\mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{D}_{\!\mathbf{AB}}(AX_0)~~.\nonumber
848	\end{equation} 
849	
850	\noindent Obviously, $\mathcal{D}_{\!\mathbf{AB}}$ satisfies the definition of continuity on the interior of $\mathbf{AB}$.
851	
852	$~$
853	
854	\noindent  $\bullet$ (\textit{Endpoint A}) A function $f(x)$ is continuous at the endpoint $a$ of its domain $[a,b]$ if and only if	
855	\begin{equation}
856	\lim\limits_{x\to a^+}f(x)=f(a)~~.\nonumber
857	\end{equation} 
858	
859	\noindent We conform to this definition of continuity with	
860	\begin{equation}
861	\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
862	\end{equation} 
863	
864	
865	$~$ 
866	
867	\noindent   $\bullet$ (\textit{Endpoint B})  A function $f(x)$ is continuous at the endpoint $b$ of its domain $[a,b]$ if and only if	
868	\begin{equation}
869	\lim\limits_{x\to b^-}f(x)=f(b)~~.\nonumber
870	\end{equation} 
871	
872	\noindent We conform to this definition with	
873	\begin{equation}
874	\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
875	\end{equation} 
876	
877	\noindent The geometric FDF is continuous everywhere on its domain.
878\end{proof}
879
880
881\begin{thm}\label{thm:algfracdisnotcont}
882	The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ is not continuous everywhere on the domain $\mathbf{AB}$.
883\end{thm}
884
885
886\begin{proof}
887	A function $f(x)$ with domain $x\in[a,b]$ is continuous at $b$ if 
888	\begin{equation}
889	\lim\limits_{x\to b^-}f(x) =f(b)~~,  \nonumber
890	\end{equation} 
891	
892	
893	\noindent In terms of $\mathcal{D}'_{\!AB}$, the statement that $\mathcal{D}'_{\!\mathbf{AB}}$ is continuous at $B$ becomes
894	\begin{equation}
895	\lim\limits_{X\to B}\mathcal{D}'_{\!\mathbf{AB}}(AX)=\mathcal{D}'_{\!\mathbf{AB}}(\mathbf{AB})=1~~.\nonumber
896	\end{equation} 
897	
898	\noindent Evaluation yields	
899	\begin{equation}
900	\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
901	\end{equation} 
902	
903	\noindent The algebraic FDF of the first kind is not continuous everywhere on all real line segments.	
904\end{proof}
905
906\begin{rem}\label{rem:kdvjuur84}
907	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 	
908	\begin{equation}
909	\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
910	\end{equation}
911	
912	\noindent  Perhaps, then, it would be better to write simply	
913	\begin{equation}
914	\lim\limits_{x\to \infty}\frac{x}{\infty} = \frac{\infty}{\infty} = \text{undefined}\neq 1~~.\nonumber
915	\end{equation}
916	
917	\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.   
918\end{rem}
919
920
921\begin{thm}\label{thm:algfracdisnotcont3}
922	The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ does not converge to a Cauchy limit at infinity.
923\end{thm}
924
925
926\begin{proof}
927	According to the Cauchy definition of the limit of $f:D\to R$ at infinity, we say that 	
928	\begin{equation}
929	\lim\limits_{x\to \infty} f(x) = l~~,\nonumber
930	\end{equation}
931	
932	\noindent if and only if	
933	\begin{equation}
934	\forall\varepsilon>0\quad\exists\delta>0\quad\text{s.t}\quad\forall x\in D~~,\nonumber 
935	\end{equation}
936	
937	\noindent we have	
938	\begin{equation}
939	0<|x-\infty|<\delta\quad\implies\quad|f(x)-l|<\varepsilon~~.\nonumber
940	\end{equation}
941	
942	\noindent There is no $\delta>\infty$ so $\mathcal{D}'_{\!AB}(AX)$ fails the Cauchy criterion for convergence to a limit at infinity.  
943\end{proof}
944
945\begin{rem}
946	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}.
947\end{rem}
948
949
950\begin{rem}
951	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}.
952\end{rem}
953
954
955\begin{thm}\label{thm:4f134134t34t}
956	If $x$ is a real number in the algebraic representations of both $X\in AB$ and $Y\in AB$, then $X=Y$.
957\end{thm}
958
959\begin{proof}
960	If $X\neq Y$, then  	
961	\begin{equation}
962	\mathcal{D}^\dagger_{\!AB}(AX)\neq\mathcal{D}^\dagger_{\!AB}(AY)~~.\nonumber
963	\end{equation}
964	
965	\noindent If $x\in X$ and $x\in Y$, then it is possible to make cuts at $X$ and $Y$ such that 	
966	\begin{equation}
967	\mathcal{D}^\dagger_{\!AB}(AX)=\cfrac{\text{len}[a,x]}{\text{len}[a,b]}=\mathcal{D}^\dagger_{\!AB}(AY)  ~~.\nonumber
968	\end{equation} 
969	
970	\noindent This contradiction requires $X=Y$.	
971\end{proof} 
972
973
974
975\subsection{Comparison of Real and Natural Numbers}\label{sec:RN} 
976
977The 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.
978
979
980\begin{defin}\label{def:3533553}
981	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 $(a,b)$.
982\end{defin} 
983
984
985\begin{thm}\label{thm:halfway}
986	There exists a unique real number halfway between zero and infinity.
987\end{thm}
988
989\begin{proof}
990	By Theorem \ref{thm:onemid} and by Definition \ref{def:fet7}, there exists one midpoint $C$ of every line segment $AB$ such that 	
991	\begin{equation}
992	\mathcal{D}_{\!AB}(AC)=0.5~~.\nonumber
993	\end{equation} 
994	
995	\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
996	\begin{equation}
997	\mathcal{D}^\dagger_{\!\mathbf{AB}}(AC)=0.5~~.\nonumber
998	\end{equation} 
999	
1000	\noindent Using $C\equiv\mathscr{C}=[c_1,c_2]$, Axiom \ref{def:5y5dddy2} and Definition \ref{def:XrepR} require	
1001	\begin{equation}
1002	\mathbf{AB}=AC+CB\quad\iff\quad[0,\infty]=[0,c_1)\cup\mathscr{C}\cup(c_2,\infty]~~.\nonumber
1003	\end{equation} 
1004	
1005	\noindent It follows that	
1006	\begin{equation}
1007	\mathscr{C}\subset\mathbb{R}~~.\nonumber
1008	\end{equation} 
1009	
1010	\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.
1011\end{proof}
1012
1013
1014\begin{rem}
1015	How can $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AC)=0.5$ when Definition \ref{def:algfracdis} gives
1016	\begin{equation}
1017	\mathcal{D}'_{\!\mathbf{AB}}(AC)=\frac{\text{len}[0,c]}{\infty}~~?\nonumber
1018	\end{equation} 
1019	
1020	\noindent The prevailing assumption about infinity is
1021	\begin{equation}\label{imp:f24f4f}
1022	x\in\mathbb{R}\quad \implies \quad\frac{x}{\infty}=0~~.
1023	\end{equation} 
1024	
1025	\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 $AB$.  
1026	
1027	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.  
1028	
1029	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
1030	\begin{equation}
1031	\mathcal{D}'_{\!\mathbf{AB}}(AC)=\frac{\text{len}[0,c]}{\infty}~~?\text{''}\nonumber
1032	\end{equation} 
1033	
1034	\noindent The answer is that Equation (\ref{imp:f24f4f}) must be reformulated as 	
1035	\begin{equation*} 
1036	x\in\mathbb{R}^0_\aleph  \quad \implies \quad     \cfrac{x}{\infty}=0 ~~. 
1037	\end{equation*} 
1038	
1039	\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
1040	\begin{equation}
1041	x\in\mathbb{R}^{0.5}_\aleph\quad \implies \quad\cfrac{x}{\infty}=0.5~~.\nonumber
1042	\end{equation} 
1043	
1044	\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 
1045	\begin{equation*}
1046	\cfrac{x+n}{\infty}=\cfrac{x}{\infty}+\cfrac{n}{\infty}=0.5+0~~.
1047	\end{equation*}
1048	
1049	\noindent Obviously, $x\in\mathbb{R}^{0.5}_\aleph$ is not a unique number though the midpoint $C$ is a unique point.
1050\end{rem}
1051
1052\begin{defin}
1053	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 $\aleph_\mathcal{X}-n\in\mathbb{R}_\aleph^\mathcal{X}$.
1054\end{defin}
1055
1056\begin{rem}
1057	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 $\tfrac{i}{2}\sim\aleph_{0.5}$.   
1058\end{rem}
1059
1060\begin{mainthm} \label{thm:ef2424t24cc}
1061	Some elements of $\mathbb{R}$ are greater than every element of $\mathbb{N}$.
1062\end{mainthm}
1063
1064\begin{proof}
1065	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 $y\in Y$
1066	\begin{equation}
1067	\mathcal{D}_{\!AB}(AX)>\mathcal{D}_{\!AB}(AY)\quad\implies\quad x>y ~~,\nonumber
1068	\end{equation} 
1069	
1070	\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 $\mathcal{X}>0$.
1071\end{proof}
1072 
1073\begin{cor} 
1074	$\mathbb{R}_\infty=\mathbb{R}\setminus\mathbb{R}_0$ is not the empty set.
1075\end{cor}
1076
1077\begin{proof}
1078	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 	
1079	\begin{equation}
1080	\mathbb{R}_\infty\neq\varnothing~~,\qquad\text{because}\qquad \mathbb{R}_\infty= \mathbb{R}\setminus\mathbb{R}_0~~.\nonumber
1081	\end{equation} 
1082\end{proof}  
1083
1084
1085\subsection{Comparison of Cuts in Lines and Points in Line Segments}\label{sec:3322442}
1086 
1087
1088In 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 $x_1\neq x_2$.
1089
1090\begin{thm}\label{thm:finlinseggg}
1091	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.
1092\end{thm}
1093
1094\begin{proof}
1095	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 
1096	\begin{equation}
1097	X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
1098	\end{equation}
1099	
1100	\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
1101	
1102	\begin{equation}
1103	\min[\mathcal{D}^\dagger_{\!AB}(AX)]=\cfrac{\text{len}[a,x_1]}{\text{len}[a,b]} =\frac{x_1-a}{b-a}~~,\nonumber
1104	\end{equation} 
1105	
1106	\noindent and 
1107	\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
1108	\end{equation} 
1109	
1110	\noindent The one-to-one property of $\mathcal{D}^\dagger_{\!AB}$ requires that 
1111	\begin{equation} \frac{x_1-a}{b-a}=\frac{x_2-a}{b-a}\quad\iff\quad x_1=x_2~~.\nonumber
1112	\end{equation} 
1113	
1114	\noindent This contradicts the assumption $x_1\neq x_2$.  The theorem is proven.
1115\end{proof}
1116
1117\begin{thm}\label{thm:finlinseggg2}
1118	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.
1119\end{thm}
1120
1121\begin{proof}
1122	By Definition \ref{def:XrepR}, every point in a line segment has an algebraic representation
1123	\begin{equation}
1124	X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
1125	\end{equation}
1126	
1127	\noindent It follows that 
1128	\begin{equation}
1129	\min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\cfrac{\text{len}[0,x_1]}{\text{len}[0,\infty]} =\frac{x_1}{\infty}~~,\nonumber
1130	\end{equation} 
1131	 
1132	
1133	\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
1134	\begin{equation}
1135	\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
1136	\end{equation} 
1137	
1138	\noindent Invoking the single-valuedness of bijective functions, we find that
1139	\begin{equation*}
1140	\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~~.
1141	\end{equation*}
1142	
1143	\noindent Therefore $x_1\neq x_2$ and the theorem is proven.
1144\end{proof}
1145
1146\begin{exa}\label{ex:333}
1147	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
1148	\begin{equation}
1149	[a,b]=\{a\}\cup(a,b]~~.\nonumber
1150	\end{equation}
1151	
1152	\noindent To separate an endpoint from a line segment we write
1153	\begin{equation}
1154	AB=A+AB~~.\nonumber
1155	\end{equation}
1156	
1157	\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
1158	\begin{equation}
1159	\|AB\|-\text{len}\,\mathscr{A}=\|AB\|\quad\iff\quad	\|AB\|=\infty~~.\nonumber
1160	\end{equation}
1161\end{exa}
1162
1163\begin{rem}\label{rem:f32232323}
1164	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
1165	\begin{itemize}
1166		\item $L\in\mathbb{R}_0$
1167		\item $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$ ($L$ larger than any $n\in\mathbb{N}$ yet not so large that $\frac{L}{\infty}\neq0$.)
1168		\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})
1169		\item $L=\infty$~~.
1170	\end{itemize}
1171
1172	\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}. 
1173\end{rem}
1174 
1175 
1176 
1177
1178 
1179\section{The Neighborhood of Infinity}\label{sec:rg8hr8y8hh}
1180 
1181
1182\subsection{Intermediate Neighborhoods of Infinity}\label{sec:Rneighb} 
1183
1184In this section, we will develop notation useful for describing real numbers whose fractional magnitude with respect to infinity is greater than zero.
1185
1186\begin{defin}\label{def:hh4hh4h4h4}
1187	The number $\aleph_\mathcal{X}$ is defined to have the property
1188	\begin{equation*}
1189	\frac{\aleph_{\mathcal{X}}}{\infty}=\mathcal{X}~~.  
1190	\end{equation*}
1191	
1192	\noindent  Equivalently, if $\aleph_\mathcal{X}\in \mathscr{X}\equiv X$, then
1193	\begin{equation*}
1194	\mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}~~.  
1195	\end{equation*} 
1196\end{defin}
1197
1198\begin{rem}\label{rem:kkkekk33}
1199	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 $\mathscr{B}\equiv B\in\mathbf{AB}$.
1200\end{rem}
1201
1202
1203\begin{defin}\label{def:gtg34535335}
1204	For $0<\mathcal{X}<1 $, $\mathbb{R}^\mathcal{X}_\aleph$ is a subset of positive real numbers $\mathbb{R}^+$  such that
1205	\begin{equation}
1206	\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
1207	\end{equation}
1208
1209	\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 $\mathbb{R}$.)
1210\end{defin}
1211
1212\begin{defin}\label{def:ffmdmmdd3md}
1213	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
1214	\begin{equation}
1215	\mathbb{R}^\mathcal{X}_0=\big\{ \aleph_\mathcal{X}+ b~\big| ~ b\in  \mathbb{R}_0    \big\} ~~.\nonumber
1216	\end{equation}
1217	
1218	\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 $0<\mathcal{X}<1$.
1219\end{defin}
1220
1221
1222
1223\begin{defin}\label{def:15525252525}
1224	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
1225	\begin{equation*}
1226	\text{Big}(\aleph_\mathcal{X}+b)=\aleph_\mathcal{X}~~,\qquad\text{and}\qquad \text{Lit}(\aleph_\mathcal{X}+b)=b~~.
1227	\end{equation*}
1228\end{defin}
1229
1230\begin{rem}	
1231	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.
1232	
1233	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.
1234\end{rem}
1235 
1236
1237\begin{defin}
1238	The whole neighborhood of the origin is 
1239	\begin{equation*}
1240	\mathbb{R}_\aleph^0=\big\{ x~\big|~x\in \mathscr{A}\equiv A\in\mathbf{AB} \big\}~~,
1241	\end{equation*}
1242	
1243	\noindent and the natural neighborhood of the origin is 
1244	\begin{equation*}
1245	\mathbb{R}_0^0=\big\{ x~\big|~x\in \mathbb{R}_0,~x\geq0 \big\}~~,
1246	\end{equation*}
1247\end{defin}
1248
1249\begin{rem}
1250	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}.
1251\end{rem}
1252
1253
1254
1255
1256\begin{defin}\label{def:defoig777}
1257	A real number $x$ is said to be in the neighborhood of the origin if and only if
1258	\begin{equation}
1259	x\in X~~,\qquad\text{and}\qquad \mathcal{D}_{\!\mathbf{AB}}(AX)=0 ~~.\nonumber
1260	\end{equation} 
1261	
1262	\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
1263	\begin{equation}
1264	x\in  X~~,\qquad\text{and}\qquad \mathcal{D}_{\!\mathbf{AB}}(AX)\neq0 ~~.\nonumber
1265	\end{equation} 
1266\end{defin}
1267
1268
1269
1270\begin{rem}\label{rem:v3444}
1271	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 $n\in\mathbb{N}$.
1272\end{rem}
1273 
1274
1275\begin{defin}\label{def:delta0x00}
1276	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
1277	\begin{equation*}
1278	\text{Ball}(x,\delta)=(x-\delta,x+\delta)~~.
1279	\end{equation*} 
1280\end{defin}
1281
1282\begin{defin}\label{def:delta000}
1283	The $\delta$-neighborhood of an interior point $X\in AB$ is a line segment $YZ$ where	
1284	\begin{equation}
1285	\big|\mathcal{D}_{\!AB}(AX)-\mathcal{D}_{\!AB}(AY)\big|=\big|\mathcal{D}_{\!AB}(AX)-\mathcal{D}_{\!AB}(AZ)\big|=\delta~~.\nonumber
1286	\end{equation}
1287\end{defin}
1288
1289\begin{rem}
1290	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.
1291\end{rem}
1292
1293
1294
1295\begin{defin}\label{def:35e44e3}
1296	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 $\mathbb{R}^0_\aleph\not\subset\{\mathbb{R}^\mathcal{X}_\aleph\}$.)
1297\end{defin}
1298 
1299
1300
1301\begin{defin} \label{def:fh93ry983y9y222}
1302	An alternative definition for $\mathbb{R}^\mathcal{X}_\aleph$ valid in the neighborhood of infinity, meaning for $0<\mathcal{X}<1$, is	
1303	\begin{equation}
1304	\mathbb{R}^\mathcal{X}_\aleph=\big\{ \aleph_\mathcal{X}\pm b~\big|~  b\in\mathbb{R}^0_\aleph  \big\} ~~.\nonumber
1305	\end{equation}
1306	 
1307	\noindent  This definition is totally equivalent to Definition \ref{def:gtg34535335}.
1308\end{defin}
1309
1310 
1311\subsection{Equivalence Classes for Intermediate Natural Neighborhoods of Infinity}\label{sec:fconst}
1312
1313
1314Euclid'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}.
1315
1316
1317\begin{defin}
1318	The rational numbers $\mathbb{Q}$ are an Archimedean number field satisfying all of the well-known field axioms given in Section \ref{sec:fieldAX}.
1319\end{defin}
1320
1321\begin{defin}\label{def:CSq}
1322	A sequence $\{x_n\}$ is a Cauchy sequence if and only if 
1323	\begin{equation*}
1324	\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~~.
1325	\end{equation*}
1326\end{defin}
1327 
1328\begin{defin}
1329	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.
1330\end{defin}
1331
1332\begin{defin}
1333	$C_\mathbb{Q}$ is the set of all Cauchy sequences of rational numbers.
1334\end{defin}
1335
1336\begin{rem}
1337	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.
1338\end{rem}
1339
1340\begin{axio}\label{ax:constaxcjco}
1341	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. 
1342\end{axio}
1343
1344\begin{axio}\label{def:lkkd33kkdzz}
1345	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.
1346\end{axio}
1347
1348
1349\begin{rem}\label{rem:hgy88877}
1350	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}. 
1351\end{rem}
1352 
1353
1354\begin{defin}\label{def:i9696969a}
1355	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
1356	\begin{equation*}
1357	0+x=x~~,\qquad\text{and}\qquad x\cdot0=0~~.
1358	\end{equation*}
1359	
1360	\noindent Expressions containing $\hat0$ are not to be simplified by either of these operations.
1361\end{defin}
1362
1363
1364
1365\begin{axio}\label{ax:y98tguh}
1366	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
1367	\begin{equation*}
1368	\big\{x_n\big\}\in[x]\quad\iff\quad\big\{\, \hat0+x_n\big\}\in[x]~~,
1369	\end{equation*}
1370	
1371	\noindent or that, equivalently, there exists an additive identity element for every $x\in\mathbb{Q}$.
1372\end{axio}
1373
1374\begin{exa}\label{ex:yy89779aaa}
1375	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
1376	\begin{equation*}
1377	\big\{\, \hat0+x_n\big\}\equiv\big\{A+x_n\big\}\in[A+x]~~.
1378	\end{equation*}
1379	
1380	\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	
1381	\begin{equation}
1382	\mathbb{R}^\mathcal{X}_0=\big\{ \aleph_\mathcal{X}+ b~\big|~ b\in  \mathbb{R}_0   \big\} ~~,\nonumber
1383	\end{equation}
1384	
1385	\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
1386	\begin{equation*}
1387	\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~~.
1388	\end{equation*}
1389	
1390	\noindent 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 $\mathcal{X}\in\mathbb{R}_0$.
1391\end{exa}
1392
1393
1394\begin{defin}\label{def:CABQ}
1395	$C_\mathbb{Q}^{\mathbf{AB}}$ is a Cartesian product of $C_\mathbb{Q}$ with the set of all $X\in\mathbf{AB}$.  Specifically,
1396	\begin{equation*}
1397	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\}~~.
1398	\end{equation*}
1399	
1400	\noindent Since it is considered desirable to give a totally algebraic construction, we may give the equivalent definition 
1401	\begin{equation*}
1402	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\}~~.
1403	\end{equation*}	
1404\end{defin}
1405
1406\begin{rem}
1407	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.  
1408\end{rem}
1409
1410\begin{defin}\label{def:kkqkqqk11}
1411	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
1412	\begin{equation*}
1413	\mathcal{D}_{\!\mathbf{AB}}(AX)=\mathcal{X}\quad\implies \quad [X]\equiv[\aleph_{\mathcal{X}}]=\aleph_{[\mathcal{X}]}=\aleph_{\mathcal{X}}~~.
1414	\end{equation*}
1415	
1416	\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}.
1417\end{defin}
1418
1419 
1420\begin{defin}\label{def:jhjjjjj}
1421	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 $[X]\equiv[\aleph_\mathcal{X}]$.
1422\end{defin}
1423
1424\begin{axio}\label{def:it759595}
1425	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 
1426	\begin{equation*}
1427	x=\big([\mathcal{X}],[b]\big)\subset C_{\mathbb{Q}}\times C_{\mathbb{Q}}~~,
1428	\end{equation*}
1429	
1430	\noindent where the Cartesian product is
1431	\begin{equation*}
1432	 C_\mathbb{Q} \times  C_\mathbb{Q} : \big([\mathcal{X}],[b]\big)\to\aleph_{[\mathcal{X}]}+[b]~~.
1433	\end{equation*}	
1434\end{axio}
1435
1436\begin{rem}
1437	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 $C_\mathbb{Q}$.
1438\end{rem}
1439
1440
1441
1442\begin{exa}
1443	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
1444	\begin{equation*}
1445	x=[x]=[\{x_n\}]=\big\{x_1,x_2,x_3,...\big\}~~.
1446	\end{equation*}
1447	
1448	\noindent Then, moving the iterator into the superscript position for notation purposes, $\aleph_x$ is a Cauchy equivalence class
1449	\begin{equation*}
1450	\aleph_x=\aleph_{[x]}=[\aleph_x]=[\{\aleph_x^n\}]=[\aleph_{\{x_n\}}]=\big\{\aleph_{x_1},\aleph_{x_2},\aleph_{x_3},...\big\}
1451	\end{equation*}
1452\end{exa}
1453
1454  
1455\begin{thm}
1456	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 $x=y$.
1457\end{thm}
1458
1459\begin{proof}
1460	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 $C_\mathbb{Q}$
1461	\begin{equation*}
1462	\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]~~.
1463	\end{equation*}
1464	
1465	\noindent By the definition of the equivalence class, every element of $C_\mathbb{Q}$ is such that
1466	\begin{equation*}
1467	[x]=[y]\quad\iff\quad x=y~~,
1468	\end{equation*}
1469	
1470	\noindent so the same must be true for the ordered pairs:
1471	\begin{equation*}
1472	\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)~~.
1473	\end{equation*}
1474	
1475	\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.
1476	 
1477\end{proof}
1478    
1479
1480
1481\subsection{The Maximal Neighborhood of Infinity}\label{sec:nbinf}
1482
1483The 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 $0\not<\mathcal{X}\not<1$. 
1484
1485\begin{defin}\label{def:y969tfff22}
1486	The whole maximal neighborhood of infinity is 
1487	\begin{equation*}
1488	\mathbb{R}_\aleph^1=\big\{ \aleph_1-b ~\big|~ b\in\mathbb{R}^0_\aleph  \big\}~~.
1489	\end{equation*}
1490\end{defin}
1491
1492
1493\begin{rem}
1494	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.
1495\end{rem}
1496
1497
1498\begin{defin}
1499	$\infty$ is called geometric infinity or simply infinity.
1500\end{defin}
1501
1502
1503\begin{defin}
1504	$\widehat\infty$ is called algebraic infinity.  It shall be called infinity hat as well.
1505\end{defin}
1506
1507
1508
1509\begin{defin}\label{def:adabs}
1510	Additive absorption is a property of $\infty$ such that all $x\in\mathbb{R}$ are additive identities of $\infty$.  The additive absorptive property is
1511	\begin{equation}
1512	\infty\pm x=\infty~~. \nonumber
1513	\end{equation}
1514	
1515	\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
1516	\begin{equation}
1517	\infty\cdot x=\begin{cases}
1518	~~\infty&\text{for}\quad x>0\\
1519	-\infty&\text{for}\quad x<0
1520	\end{cases}. \nonumber
1521	\end{equation}
1522\end{defin}
1523
1524
1525\begin{rem}
1526	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.
1527\end{rem}
1528
1529
1530\begin{defin}\label{def:hat33}
1531	The symbol $\widehat\infty$ refers to an infinite element
1532	\begin{equation}
1533	\lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\big|\widehat\infty\big|~~, \qquad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \big|\widehat\infty \big| ~~,\nonumber
1534	\end{equation} 
1535	
1536	\noindent together with an instruction not to perform the additive or multiplicative operations usually imbued to infinite elements.  
1537\end{defin}
1538
1539\begin{rem}
1540	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}.
1541\end{rem}
1542
1543\begin{thm}\label{def:dvvj8yv8r}
1544	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 $\mathbb{R}=(-\widehat\infty,\widehat\infty)$.
1545\end{thm}
1546
1547\begin{proof}
1548	For $a,b\in\mathbb{R}^+$, it may be taken for granted that 
1549	\begin{equation*}
1550	(-a,b)=(-|a|,|b|)~~,
1551	\end{equation*}
1552	
1553	\noindent and it follows, therefore, that this is true for $a,b\in\overline{\mathbb{R}}^+$.  Then
1554	\begin{equation}
1555	\pm\big|\widehat\infty\big|=\pm\big|\infty\big|\quad\implies\quad \mathbb{R}=(-\widehat\infty,\widehat\infty)~~,\nonumber
1556	\end{equation}
1557	
1558	\noindent and the theorem is proven.
1559\end{proof}
1560
1561
1562 
1563
1564\begin{exa}\label{exa:588585557aaa}
1565	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
1566	\begin{equation*}
1567	a+\infty=a+\lim\limits_{x\to0 }\dfrac{1}{x}=\lim\limits_{x\to0 }\dfrac{1+ax}{x}=\text{diverges}=\infty~~.
1568	\end{equation*}
1569	
1570	\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 
1571	\begin{equation*}
1572	a\cdot\infty=a\cdot\lim\limits_{x\to0 }\dfrac{1}{x}=\lim\limits_{x\to0 }\dfrac{a}{x}=\text{diverges}=\infty~~.
1573	\end{equation*}
1574	
1575	\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.''  
1576	
1577	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
1578	\begin{equation*}
1579	\frac{\aleph_{\mathcal{X}}}{\infty}=\mathcal{X}\quad\iff\quad \aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty~~.  
1580	\end{equation*}
1581	
1582	\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.  
1583\end{exa}
1584 
1585
1586
1587\begin{defin}\label{def:ugy8re8r7777}
1588	For any $\mathcal{X}\in\mathbb{R}$, the symbol $\aleph_\mathcal{X}$ is defined as
1589	\begin{equation*}
1590	\aleph_\mathcal{X}=\mathcal{X}\cdot\widehat\infty~~.
1591	\end{equation*}
1592\end{defin}
1593 
1594
1595\begin{defin}\label{def:zzzzz86et}
1596	In terms of $\widehat\infty$, the whole maximal neighborhood of infinity is defined as
1597	\begin{equation}
1598	\mathbb{R}^1_\aleph=\big\{  \widehat{\infty}-b~\big|~b\in\mathbb{R}^0_\aleph,~b\neq0 \big\}     ~~.\nonumber
1599	\end{equation} 
1600\end{defin}
1601
1602
1603
1604\begin{defin}\label{def:dhgw86et86et}
1605	The maximal natural neighborhood of infinity is defined as
1606	\begin{equation}
1607	\mathbb{R}^1_0=\big\{  \widehat\infty-b ~\big|~b\in\mathbb{R}_0^+ \big\}     ~~.\nonumber
1608	\end{equation} 
1609\end{defin}
1610 
1611
1612\subsection{Equivalence Classes for the Maximal Natural Neighborhood of Infinity}\label{sec:fconstM}
1613
1614
1615We 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.
1616
1617In 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 $\aleph_\mathcal{X}$.
1618
1619
1620
1621\begin{axio}
1622	A Euclidean line segment $AB$ \cite{EE} is invariant under permutations of the labels of its endpoints, \textit{e.g.}: $AB=BA$.  
1623\end{axio}
1624
1625
1626\begin{defin}\label{def:NCP}
1627	Define a geometric permutation operator $\hat P$ such that
1628	\begin{equation*}
1629	\hat P (AB)=BA~~.
1630	\end{equation*}
1631\end{defin}
1632
1633
1634\begin{rem}
1635	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 
1636	\begin{equation*}
1637	 \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}~~.
1638	\end{equation*}
1639	
1640	\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 $[A]=[\aleph_0]=\aleph_{[0]}=0=\hat0$.
1641\end{rem}
1642
1643\begin{defin}\label{def:kkk3kk3111}
1644	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.
1645\end{defin}
1646
1647\begin{defin}
1648	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
1649	\begin{equation*}
1650	\hat{\mathcal{P}}_0:\hat0\times C_\mathbb{Q}\to\widehat\infty\times C_\mathbb{Q}~~,
1651	\end{equation*}
1652	
1653	\noindent where
1654	\begin{equation*}
1655	\hat0\times C_\mathbb{Q}=\big\{\hat0+[x]~\big|~[x]\subset C_\mathbb{Q}\big\}~~,\qquad\text{and}\qquad\widehat\infty\times C_\mathbb{Q}=\big\{\widehat\infty-[x]~\big|~[x]\subset C_\mathbb{Q}\big\}~~.
1656	\end{equation*}
1657\end{defin}
1658
1659\begin{exa}
1660	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 
1661	\begin{equation*}
1662	\hat P(\mathbf{AB})=\mathbf{BA}~~.
1663	\end{equation*}
1664	
1665	\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 
1666	\begin{equation*}
1667	\mathcal{D}_{\!\mathbf{AB}}(\mathbf{AB})=\mathcal{D}^\dagger_{\!\mathbf{AB}}(\mathbf{AB})=1~~,
1668	\end{equation*}
1669	
1670	\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
1671	\begin{equation*}
1672	b=\hat{\mathcal{P}}_0(\hat 0,[x])=\widehat\infty-[x]~~.
1673	\end{equation*}
1674	
1675	\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.
1676\end{exa}
1677
1678\begin{rem}
1679	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.  
1680	
1681	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
1682	\begin{equation}
1683	\lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\big|\widehat\infty\big| ~~,\nonumber
1684	\end{equation} 
1685	
1686	\noindent without $\widehat\infty$ itself being interchangeably equal with the limit expression.
1687 
1688	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 $\widehat\infty$.
1689\end{rem}
1690
1691\begin{defin}
1692	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.
1693\end{defin}
1694
1695
1696\begin{exa}\label{exa:2r233535533535}
1697	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.  
1698	
1699	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
1700	\begin{equation}
1701	d(x,y)=
1702	\big|y-x\big|~~.\nonumber
1703	\end{equation}
1704	
1705	\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 
1706	\begin{equation*}
1707	(x,y)=(x_0,y_0)~~,\qquad\text{and}\qquad(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=(\infty-x_0,\infty-y_0)~~,
1708	\end{equation*}
1709	
1710	\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
1711	\begin{equation}
1712	d(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=
1713	\big|\infty-x_0-\big(\infty-y_0\big)\big|=
1714	\big|\infty-\infty\big|=\text{undefined}~~.\nonumber
1715	\end{equation}
1716	
1717	\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.
1718	
1719	Algebraic infinity is a number which avoids all of the problems here listed.  Under permutation, we have 
1720	\begin{equation*}
1721	(x,y)=(x_0,y_0)~~,\qquad\text{and}\qquad(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=(\widehat\infty-x_0,\widehat\infty-y_0)~~.
1722	\end{equation*}
1723	
1724	\noindent Jumping ahead to the arithmetic of such numbers axiomatized in Section \ref{sec:aritAX}, we find
1725	\begin{equation}
1726	d(\hat{\mathcal{P}}_0(x_0),\hat{\mathcal{P}}_0(y_0))=
1727	\big|\widehat\infty-x_0-\big(\widehat\infty-y_0\big)\big|=
1728	\big|y_0-x_0\big|=d(x_0,y_0)~~,\nonumber
1729	\end{equation}	
1730	
1731	\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 $\widehat\infty<\infty$.
1732\end{exa}
1733
1734
1735
1736
1737\begin{rem}
1738	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
1739	\begin{equation}
1740	\tan\left(\cfrac{\pi}{2}\right)=\infty~~,\nonumber
1741	\end{equation}
1742	
1743	\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
1744	\begin{equation*}
1745	\lim\limits_{\theta\to\frac{\pi}{2}}\tan(
1746	\theta)=\lim\limits_{\theta\to\frac{\pi}{2}}\frac{\sin(
1747		\theta)}{\cos(\theta)}=\lim\limits_{\substack{x\to0\\y\to1}}\frac{y}{x}=\infty~~.
1748	\end{equation*}
1749	
1750	\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
1751	\begin{equation*}
1752	\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}~~.
1753	\end{equation*}
1754	
1755	\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.
1756\end{rem}
1757
1758\begin{defin}
1759	The symbol $\aleph_1$ is an alternative notation for algebraic infinity.  We have
1760	\begin{equation*}
1761	\aleph_1=\widehat\infty~~,\qquad\text{and}\qquad\aleph_1\neq\infty~~.
1762	\end{equation*}
1763\end{defin}
1764
1765\begin{rem}
1766	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.
1767\end{rem}
1768
1769
1770\begin{axio} \label{ax:undefinby}
1771	$\aleph_1$ is such that
1772	\begin{align}
1773	\aleph_1-\aleph_1=0~~,\qquad\text{and}\qquad\cfrac{\aleph_1}{\aleph_1}=1~~.\nonumber
1774	\end{align} 
1775\end{axio}
1776
1777
1778 
1779
1780\begin{thm}
1781	The maximal whole neighborhood of infinity is a subset of the real numbers.
1782\end{thm}
1783
1784\begin{proof}
1785	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 
1786	\begin{equation}
1787	\mathbb{R}^1_\aleph=\big\{  \widehat{\infty}-b~\big|~b\in\mathbb{R}^0_\aleph,~b\neq0 \big\}     ~~.\nonumber
1788	\end{equation} 
1789	
1790	\noindent We clearly have
1791	\begin{equation*}
1792	(-\infty,\infty)=(-\infty,\aleph_\mathcal{X}+b]\cup(\aleph_\mathcal{X}+b,\infty)~~.
1793	\end{equation*} 
1794	
1795	
1796	\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}.)
1797\end{proof} 
1798
1799
1800\begin{cor}
1801	All numbers $x\in\{\mathbb{R}^\mathcal{X}_\aleph\}$ are real numbers.
1802\end{cor}
1803
1804\begin{proof}
1805	The ordering of $\mathbb{R}$ given by Axiom \ref{def:order} is such that $0<\mathcal{X}<1$ guarantees
1806	\begin{equation*}
1807	(0,\infty)=(0,\aleph_\mathcal{X}\pm b]\cup(\aleph_\mathcal{X}\pm b,\infty)~~.
1808	\end{equation*}
1809	
1810	\noindent Definition \ref{def:real} is satisfied trivially and the theorem is proven.
1811\end{proof}
1812
1813\begin{rem}
1814	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.
1815\end{rem}
1816 
1817\section{Arithmetic} \label{sec:5}
1818
1819
1820 
1821
1822\subsection{Operations for Infinity} 
1823
1824Here 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}.
1825
1826\begin{axio}\label{def:infOPSa}
1827	The operations for $\widehat\infty=\aleph_1$ with $b\in\mathbb{R}_0^+$ are
1828	\begin{align}
1829	\widehat\infty\pm b&= \pm b+\widehat\infty\nonumber\\ 
1830	\widehat\infty\pm\big( -b \big)&=\widehat\infty\mp b\nonumber\\
1831	-\big(\pm\widehat\infty\big)&=\mp\widehat\infty\nonumber\\
1832	\widehat\infty\cdot b=b\cdot\widehat\infty&=\aleph_{b}\nonumber\\
1833	\cfrac{\widehat\infty}{ b}&=\aleph_{(b^{-1})}\nonumber\\
1834	\cfrac{b}{\vphantom{ \widehat{A} }\widehat\infty}&=0\nonumber~~.
1835	\end{align}
1836\end{axio}
1837
1838
1839
1840\begin{axio}\label{ax:opo2p2}
1841	We give the following supplemental axioms for zero and $ \widehat\infty$. 
1842	\begin{align}
1843	\widehat\infty+0=0+\widehat\infty&=\widehat\infty\nonumber\\ 
1844	\widehat\infty\cdot0=0\cdot\widehat\infty&=0\nonumber\\
1845	\cfrac{0}{\vphantom{ \widehat{A} }\widehat\infty}&=0\nonumber\\
1846	\cfrac{\vphantom{ \widehat{A} } \widehat\infty}{0}&=\text{undefined}\nonumber~~.
1847	\end{align} 
1848\end{axio}
1849
1850\begin{rem}
1851	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}.
1852\end{rem}
1853
1854\begin{exa}\label{ex:788788m}
1855	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 
1856	\begin{equation*}
1857	\cfrac{c}{\widehat\infty}=0~~.
1858	\end{equation*}
1859	
1860	\noindent Now suppose $0\cdot\widehat\infty$ is a defined operation so that
1861	\begin{equation*}
1862	z=0\cdot\widehat\infty~~.
1863	\end{equation*}
1864	
1865	\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
1866	\begin{equation*} 
1867	z=0\cdot\widehat\infty=\cfrac{c}{\widehat\infty}\cdot\widehat\infty=c\cdot \cfrac{\widehat\infty}{\widehat\infty}=c~~.
1868	\end{equation*}
1869	
1870	\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 $0\cdot\widehat\infty=0$.
1871\end{exa}
1872
1873
1874
1875
1876\subsection{Arithmetic Axioms for Real Numbers in Natural Neighborhoods}\label{sec:aritAX}
1877
1878When 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.  
1879
1880The 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
1881\begin{equation*}
1882\mathbb{R}\subset\mathbb{T}=\big\{  x~\big|~-\aleph_\infty<x<\aleph_\infty \big\}~~.
1883\end{equation*}
1884
1885\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 $\{\mathbb{T};+\}$.  
1886
1887
1888
1889\begin{axio}\label{ax:fieldssss33} 
1890	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}.
1891\end{axio}
1892
1893\begin{rem}
1894	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 $\mathcal{X}=1$.
1895\end{rem}
1896
1897\begin{axio}\label{ax:plus}
1898	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 $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.
1899		\begin{center}{\normalsize 
1900			\begin{longtable}{| c || c | c | c | c |  }
1901				\hline
1902				  +& 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]
1903				\hline\hline
1904				 $0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $&0& $y$  &$\aleph_\mathcal{Y}+ a$ & $\widehat\infty- |a|$ \\ [4pt]
1905				\hline
1906				 $~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]
1907				\hline
1908				 $~\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]
1909				\hline
1910				  $~\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]
1911				\hline
1912			\end{longtable}		} 
1913	\end{center}
1914\end{axio}  
1915
1916\begin{rem}
1917	The most important property given by Axiom \ref{ax:plus} is 
1918	\begin{equation*}
1919	\aleph_\mathcal{X}+\aleph_\mathcal{Y}=\aleph_{(\mathcal{X}+\mathcal{Y})}~~.
1920	\end{equation*}
1921	
1922	\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 $x\in\mathbb{R}$.  
1923\end{rem} 
1924
1925
1926\begin{axio}\label{ax:1g1g1g1}
1927	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 $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$.
1928		\begin{center}{\normalsize 
1929			\begin{longtable}{| c || c | c | c | c | c | }
1930				\hline
1931				$\times$& 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]
1932				\hline\hline
1933				$0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $&0& $0$  &  $0$  &$0$ & $0$\\ [4pt]
1934				\hline
1935				$\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]
1936				\hline
1937				$~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $0$ & $\mp x$&$xy$ &$\aleph_{(x\mathcal{Y})}+ ax$&$\aleph_x-|a|x$ \\[4pt]
1938				\hline
1939				$~\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]
1940				\hline
1941				$~\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]
1942				\hline
1943			\end{longtable}	} 
1944		\end{center}
1945\end{axio}
1946
1947
1948\begin{rem}
1949	The most important property given in Axiom \ref{ax:1g1g1g1} is
1950	\begin{equation*}
1951	\pm\aleph_{\mathcal{X}}=\aleph_{(\pm\mathcal{X})}~~.
1952	\end{equation*}
1953	
1954	\noindent This operation follows from
1955	\begin{equation*}
1956	\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})}~~.
1957	\end{equation*}
1958	
1959	\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
1960	\begin{align*}
1961	\big(\widehat\infty- |b|\big)\big(\widehat\infty- |a|\big)&=\widehat\infty\cdot\widehat\infty-|b|\widehat\infty-|a|\widehat\infty+|ba|\\
1962	&=\aleph_1\cdot\aleph_1-|a|\aleph_1-|b|\aleph_1+|ba|\\
1963	&=\aleph_{(\aleph_1)}-\aleph_{|a|}-\aleph_{|b|}+|ba|\\
1964	&=\aleph_{(\aleph_1)}-\big(\aleph_{|a|}+\aleph_{|b|}\big)+|ba|\\
1965	&=\aleph_{\widehat\infty}+ \aleph_{(-|a|-|b|)}+|ba|\\
1966	&=\aleph_{(\widehat\infty-|a|-|b|)} +|ba|~~.
1967	\end{align*}
1968\end{rem}
1969
1970
1971
1972\begin{rem}
1973	It follows from Axioms \ref{ax:plus} and \ref{ax:1g1g1g1} that
1974	\begin{align*}
1975	\big(\widehat\infty-b\big)-\big(\widehat\infty-a\big)&=a-b ~~.
1976	\end{align*}
1977	
1978	\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.
1979\end{rem} 
1980
1981 
1982
1983\begin{exa}
1984	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
1985	\begin{equation*}
1986	ax^2+bx+c=0~~,
1987	\end{equation*}
1988	
1989	\noindent having roots
1990	\begin{equation*}
1991	x=\frac{-b\pm\sqrt{\vphantom{\hat{B}}b^2-4ac}}{2a}~~.
1992	\end{equation*}
1993	
1994	\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 $B$.  
1995\end{exa} 
1996
1997\begin{rem}
1998	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}
1999\end{rem}
2000
2001\begin{axio}\label{ax:dssfgss33}
2002	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 $x\not\in\mathbb{R}_0$.
2003\end{axio}
2004
2005\begin{axio}\label{ax:div1g1g1g1}
2006	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.
2007		\begin{center}{\normalsize 
2008			\begin{longtable}{| c || c | c | c | c | }
2009				\hline
2010				$\div$& 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]
2011				\hline\hline
2012				$0 \vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $& nan  & $0$  & $0$  &   $0$\\ [4pt]
2013				\hline
2014				$~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ &  nan &  $\frac{x}{y}$ & 0  &  0 \\[4pt]
2015				\hline
2016				$~\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]
2017				\hline
2018				$~\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]
2019				\hline
2020			\end{longtable}	 }
2021		\end{center}
2022\end{axio}
2023
2024
2025
2026\begin{exa}
2027	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
2028	\begin{equation*}
2029	\cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=\cfrac{\mathcal{X}}{\mathcal{Y}}~~.
2030	\end{equation*}
2031	
2032	\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
2033	
2034	\begin{align*}
2035	\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)\\
2036	\aleph_\mathcal{X}+b&= \aleph_\mathcal{X}+\cfrac{\mathcal{X}a}{\mathcal{Y}}~~.
2037	\end{align*}
2038	
2039	\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 $\{\times,\div\}$.
2040\end{exa}
2041
2042\begin{exa}
2043	This example demonstrates another immediate contradiction should we assume associativity among multiplication and division.  Axiom \ref{ax:div1g1g1g1} gives
2044	\begin{equation*}
2045	\cfrac{\aleph_\mathcal{Y}}{\aleph_\mathcal{X}}=\cfrac{\mathcal{Y}}{\mathcal{X}}~~,\qquad\text{and}\qquad\cfrac{1}{\aleph_\mathcal{X}}=0~~.
2046	\end{equation*}
2047	
2048	\noindent If we bestow the associativity, then
2049	\begin{equation*}
2050	\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}}~~.
2051	\end{equation*}	
2052\end{exa}
2053 
2054
2055
2056\begin{axio}\label{ax:order}
2057	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 
2058	\begin{align*}
2059	a&>b\\
2060	c&>d>0\\
2061	x&>y\\
2062	\mathcal{X}&>\mathcal{Y}.
2063	\end{align*}
2064	
2065	 
2066	\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.
2067	 
2068	\begin{center}{\normalsize 
2069			\begin{longtable}{| c || c | c | c | c | c | }
2070				\hline
2071				$\leq$ & $~y\in\mathbb{R}_0~$ & $ \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]
2072				\hline\hline  
2073				$~x~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$<$ & $<$ &$<$  &$<$   \\[4pt]
2074				\hline
2075				$~\big(\aleph_\mathcal{X}+a\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$>$ & $>$ & $<$ &$<$    \\[4pt]
2076				\hline
2077				$~\big(\widehat\infty-|c|\big)~\vphantom{\widehat{\overline{\mathbb{R}_0^\mathcal{X}}}} $ & $>$ &$>$ & $>$ & $<$ &$<$   \\[4pt]
2078				\hline
2079		\end{longtable}		} 
2080	\end{center}
2081\end{axio}  
2082
2083
2084\begin{thm}\label{thm:noinv}
2085	Real numbers in the intermediate natural neighborhoods of infinity $x\in\{\mathbb{R}^\mathcal{X}_0\}$ do not have a multiplicative inverse.
2086\end{thm}
2087
2088\begin{proof}	
2089	The number $x^{-1}$ is the multiplicative inverse of $x\in\mathbb{R}$ if and only if 
2090	\begin{equation*}
2091	x\cdot x^{-1}=x^{-1}\cdot x=1~~.
2092	\end{equation*}
2093	
2094	\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
2095	\begin{equation*}
2096	\big(\aleph_\mathcal{X}+b\big)x^{-1}=\aleph_{(\mathcal{X}x^{-1})}+bx^{-1}=1~~.
2097	\end{equation*} 
2098	
2099	\noindent Equating the big and little parts of this expression, we obtain two requirements 
2100	\begin{equation*}
2101	\aleph_{(\mathcal{X}x^{-1})}=\aleph_0\quad\iff\quad\mathcal{X}x^{-1}=0\quad\iff\quad x^{-1}=0~~,
2102	\end{equation*}
2103	
2104	\noindent and 
2105	\begin{equation*}
2106	bx^{-1}=1\quad\iff\quad x^{-1}=\frac{1}{b}~~.
2107	\end{equation*}
2108	
2109	\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.
2110\end{proof}
2111
2112
2113\begin{thm}\label{thm:noinvplus}
2114	All real numbers $x\in\{\mathbb{R}^\mathcal{X}_0\}$ have an additive inverse.
2115\end{thm}
2116
2117\begin{proof}
2118	The number $x^{-1}$ is the additive inverse of $x$ if and only if 
2119	\begin{equation*}
2120	x+x^{-1}=x^{-1}+ x=0~~.
2121	\end{equation*}
2122	
2123	\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
2124	\begin{equation*}
2125	1=\big(\aleph_{\mathcal{X}}+b\big)+\big(\aleph_{(\mathcal{X}^{-1})}+b^{-1}\big)=\aleph_{(\mathcal{X}+\mathcal{X}^{-1})}+(b+b^{-1})~~.
2126	\end{equation*}
2127	
2128	\noindent Equating the big and little parts of this expression, we obtain two requirements 
2129	\begin{equation*}
2130	\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}~~,
2131	\end{equation*}
2132	
2133	\noindent and 
2134	\begin{equation*}
2135	b+b^{-1}=1\quad\iff\quad b^{-1}=-b~~.
2136	\end{equation*}
2137	
2138	\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.
2139\end{proof}
2140
2141\begin{defin}
2142	A divisive identity is a number $e$ satisfying $x\div e=x$.  The divisive identity element of $\mathbb{R}$ is $1\in\mathbb{R}_0$.
2143\end{defin}
2144
2145
2146\begin{thm}\label{thm:noinvdiv}
2147	All real numbers $x\in\{\mathbb{R}^\mathcal{X}_0\}$ have a non-unique divisive inverse.
2148\end{thm}
2149
2150\begin{proof}
2151	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.
2152\end{proof}
2153
2154
2155 
2156\subsection{Limit Considerations Regarding the Arithmetic Axioms}\label{sec:cons} 
2157
2158We 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.
2159
2160So, 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.
2161
2162\begin{exa}\label{ex:ewee}
2163	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
2164	\begin{equation*}
2165	\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}~~.
2166	\end{equation*}
2167	
2168	\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
2169	\begin{equation*}
2170	\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 ~~.
2171	\end{equation*}
2172	
2173	\noindent This contradicts Axiom \ref{ax:undefin7666y} which gives 
2174	\begin{align*} 
2175	\infty-\infty=\text{undefined}~~. 
2176	\end{align*}
2177	
2178	\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
2179	\begin{equation*}
2180	\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 ~~.
2181	\end{equation*}
2182	
2183	\noindent This is exactly what is given in Axiom \ref{ax:undefinby} so the ansatz is borne out.  
2184\end{exa}
2185
2186\begin{rem}
2187	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.
2188\end{rem}
2189
2190  
2191 
2192\begin{thm}\label{thm:kksjdj1}
2193	The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1} giving for $a,b\in\mathbb{R}_0^+$
2194	\begin{equation*}
2195	\big(\widehat\infty-b\big)-\big(\widehat\infty-a\big)=a-b~~,
2196	\end{equation*}
2197	
2198	\noindent follows from the limit definition of infinity.
2199\end{thm}
2200
2201\begin{proof}
2202	Proof follows from direct substitution of the limit definition of infinity (Definition \ref{def:RRRinf}.)  We have
2203	\begin{align*}
2204	\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]\\
2205	&=\lim\limits_{x\to0}\left(\frac{1}{x}-b-\frac{1}{x}+a\right)\\
2206	&=\lim\limits_{x\to0}\big(-b+a\big)\\
2207	&=a-b~~.
2208	\end{align*}
2209\end{proof}
2210
2211
2212 
2213\begin{thm}\label{thm:kksjdj11}
2214	The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2215	\begin{equation*}
2216	\big(\aleph_\mathcal{X}+b\big)-\big(\aleph_\mathcal{Y}+a\big)=
2217	\aleph_{(\mathcal{X}-\mathcal{Y})}-a+b~~,
2218	\end{equation*}
2219	
2220	\noindent follows from the limit definition of infinity.
2221\end{thm}
2222
2223\begin{proof}
2224	Proof follows from direct substitution of the limit definition of infinity (Definition \ref{def:RRRinf}.)  We have
2225	\begin{align*}
2226	\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)\\
2227	&=\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]\\
2228	&=\big(\mathcal{X}-\mathcal{Y}\big)\left(\lim\limits_{x\to0}\frac{1}{x}\right)-a+b\\
2229	&=\big(\mathcal{X}-\mathcal{Y}\big)\widehat\infty-a+b\\
2230	&=\aleph_{(\mathcal{X}-\mathcal{Y})}-a+b~~.
2231	\end{align*}
2232\end{proof}
2233
2234\begin{rem}
2235	Theorem \ref{thm:kksjdj11} requires clarification because we might have written
2236	\begin{align*}
2237	\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)\\
2238	&=\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]\\
2239	&= \left(\lim\limits_{x\to0}\frac{\mathcal{X}-\mathcal{Y}}{x}\right)-a+b\\
2240	&=\widehat\infty-a+b~~.
2241	\end{align*}
2242	
2243	\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
2244	\begin{equation*}
2245	\aleph_\mathcal{X}=\mathcal{X}\,\widehat\infty=\lim\limits_{x\to0}\frac{\mathcal{X}}{x}~~.
2246	\end{equation*}
2247	
2248	\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.   
2249\end{rem}
2250
2251 
2252
2253
2254\begin{thm}\label{thm:kksjdj22}
2255	The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0^+$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})\leq1$
2256	\begin{equation*}
2257	\big(\aleph_\mathcal{X}+b\big)\cdot a=\aleph_{(\mathcal{X}a)}+ba~~,
2258	\end{equation*}
2259	
2260	\noindent follows from the limit definition of infinity.
2261\end{thm}
2262
2263\begin{proof}
2264	Proof follows from direct substitution of the limit definition of infinity.  We have
2265	\begin{align*}
2266	\big(\aleph_\mathcal{X}+b\big)\cdot a&=\big(\mathcal{X}\,\widehat\infty+b\big)\cdot a\\
2267	&=\left[\mathcal{X}\left(\lim\limits_{x\to0}\frac{1}{x}\right)-b\right]\cdot a\\
2268	&=\mathcal{X}a \left(\lim\limits_{x\to0}\frac{1}{x}\right)+ba\\
2269	&=\mathcal{X}a\widehat\infty+ba\\
2270	&=\aleph_{(\mathcal{X}a)}+ba~~.
2271	\end{align*}
2272\end{proof}
2273
2274
2275\begin{thm}\label{thm:kksjdj21}
2276	The property of Axioms \ref{ax:plus} and \ref{ax:1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2277	\begin{equation*}
2278	\big(\aleph_\mathcal{X}-b\big)\cdot\big(\aleph_\mathcal{Y}-a\big)=\aleph_{(\aleph_{(\!\mathcal{XY}\!)}+a\mathcal{X}+b\mathcal{Y})}+ ba~~,
2279	\end{equation*}
2280	
2281	\noindent follows from the limit definition of infinity.
2282\end{thm}
2283
2284\begin{proof}
2285	Proof of the present theorem follows from direct substitution of the limit definition of infinity.  We have
2286	\begin{align*}
2287	\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)\\
2288	&=\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]\\
2289	&= \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
2290	\end{align*}	
2291	
2292	\noindent If we wrote here 
2293	\begin{equation*}
2294	\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~~,
2295	\end{equation*}
2296	
2297	\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
2298	\begin{equation*}
2299	\widehat\infty\cdot\widehat\infty=\widehat\infty\cdot\aleph_1=\aleph_{\widehat\infty}\neq\aleph_1=\widehat\infty~~.
2300	\end{equation*}
2301	
2302	\noindent Therefore, we finish the proof as
2303	\begin{align*}
2304	\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\\
2305	&=\aleph_{\mathcal{XY}\left(\lim\limits_{x\to0}\frac{1}{x}\right) } -\aleph_a-\aleph_b+ba\\
2306	&=\aleph_{\mathcal{XY}}\cdot\infty-\aleph_{a+b}+ba\\
2307	&=\aleph_{(\aleph_{(\!\mathcal{XY}\!)}+a\mathcal{X}+b\mathcal{Y})}+ ba~~.
2308	\end{align*}
2309\end{proof}
2310
2311
2312
2313
2314
2315\begin{thm}\label{thm:kksgjdj}
2316	The property of Axiom \ref{ax:div1g1g1g1}  giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2317	\begin{equation*}
2318	\cfrac{\aleph_\mathcal{X}+b}{\widehat\infty}=\mathcal{X}~~,
2319	\end{equation*}
2320	
2321	\noindent follows from the limit definition of infinity.
2322\end{thm}
2323
2324
2325\begin{proof}
2326	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 
2327	\begin{align*}
2328	\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}~~.
2329	\end{align*} 
2330\end{proof}
2331
2332
2333\begin{rem}
2334	The property of Axiom \ref{ax:div1g1g1g1} giving for $a,b\in\mathbb{R}_0$ and $0<\min(\mathcal{X,Y})\leq\max(\mathcal{X,Y})<1$
2335	\begin{equation*}
2336	\cfrac{a}{\aleph_\mathcal{X}+b}=0~~,
2337	\end{equation*}
2338	
2339	\noindent does not follow from the limit definition of infinity.If we wrote
2340	\begin{equation*}
2341	\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}}}~~,
2342	\end{equation*}
2343	
2344	\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.
2345\end{rem}
2346
2347\begin{thm}\label{thm:g34k3k222}
2348	The quotient of any $\mathbb{R}_0$ number divided by any number with a non-vanishing big part is identically zero.
2349\end{thm}
2350
2351\begin{proof}
2352	Suppose $x,b\in\mathbb{R}_0^+$ and $0<\mathcal{X}<1$ and that
2353	\begin{equation*}
2354	\cfrac{x}{\aleph_\mathcal{X}+b}=z~~.
2355	\end{equation*}
2356	
2357	\noindent Axiom \ref{ax:dssfgss33} allows us to take $x\in\mathbb{R}_0$ out of the quotient so we may write
2358	\begin{equation*}
2359	\cfrac{1}{\aleph_\mathcal{X}+b}=\cfrac{z}{x}~~.
2360	\end{equation*}
2361	
2362	\noindent The quotient is only well defined for $z=0$.	
2363\end{proof}
2364
2365
2366\begin{thm}
2367	Quotients of the form $\mathbb{R}^\mathcal{X}_0\div\mathbb{R}^\mathcal{Y}_0$ are always equal to $\frac{\mathcal{X}}{\mathcal{Y}}$.
2368\end{thm}
2369 
2370\begin{proof}
2371	By Theorem \ref{thm:g34k3k222}, we have
2372	\begin{equation*}
2373	\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}~~.
2374	\end{equation*}
2375	
2376	\noindent If $a=0$, then
2377	\begin{equation*}
2378	\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}}~~.
2379	\end{equation*}
2380	
2381	\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
2382	\begin{equation*}
2383	\cfrac{\aleph_\mathcal{X}}{\aleph_\mathcal{Y}+a}=c~~.
2384	\end{equation*}
2385	
2386	\noindent Further suppose that $\mathcal{X}<\mathcal{Y}$ so that we may assume $0<c<1$.  Then $c$ has a multiplicative inverse and
2387	\begin{equation*}
2388	\cfrac{\aleph_\mathcal{X}}{\aleph_{(c\mathcal{Y})}+ca}=1~~.
2389	\end{equation*} 
2390	
2391	\noindent Then
2392	\begin{equation*}
2393	\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}}~~.
2394	\end{equation*}
2395	
2396	\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.	
2397\end{proof} 
2398
2399 
2400 
2401 \begin{exa}
2402 	This example demonstrates that the associativity of multiplication and division for $\mathbb{R}_0$ numbers such as $c$.  Consider the expression
2403 	\begin{equation*}
2404 	c\cdot\cfrac{\aleph_\mathcal{X}+b}{\aleph_\mathcal{Y}+a}=c\cdot\cfrac{\mathcal{X}}{\mathcal{Y}}=\cfrac{c\mathcal{X}}{\mathcal{Y}}~~.
2405 	\end{equation*}
2406 	
2407 	\noindent If we move $c$ into the quotient and perform the multiplication before the division, then
2408 	\begin{equation*}
2409 	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}}~~,
2410 	\end{equation*}
2411 	
2412 	\noindent demonstrates that the operation remains well-defined with the special associative operations for $\mathbb{R}_0$
2413 \end{exa}
2414 
2415 \begin{exa}
2416 	This example treats the negative exponent inverse notation.  We have
2417 	\begin{equation*}
2418 	\cfrac{x}{\aleph_\mathcal{X}+b}=0 \quad\centernot\implies\quad \cfrac{\aleph_\mathcal{X}+b}{x}=\cfrac{1}{0}~~.
2419 	\end{equation*}
2420 	
2421 	\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
2422 	\begin{equation*}
2423 	\cfrac{\aleph_\mathcal{X}+b}{x}=\aleph_{\left(\frac{\mathcal{X}}{x}\right)}+\frac{b}{x}~~,\qquad\text{and}\qquad \left(\cfrac{x}{\aleph_\mathcal{X}+b}\right)^{\!-1}=\cfrac{1}{\left(\cfrac{x}{\aleph_\mathcal{X}+b}\right)}=\text{undefined}~~.
2424 	\end{equation*}
2425 \end{exa}
2426  
2427 
2428 
2429
2430
2431\subsection{Field Axioms}\label{sec:fieldAX}
2432
2433 
2434	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.
2435	
2436	If $\widehat\infty$ retains multiplicative absorption, then for $n,b\in\mathbb{N}$ we have
2437	\begin{equation*}
2438	n\big(\widehat\infty-b\big)\leq\big(\widehat\infty-b\big)~~.
2439	\end{equation*}
2440	
2441	\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
2442	\begin{equation*}
2443	\big(  \widehat\infty-b \big)-\big(  \widehat\infty-a \big)=a-b~~,
2444	\end{equation*}
2445	
2446	\noindent but the notion of the sum is not.  With multiplicative absorption in place, adding two $\mathbb{R}_0^1$ numbers yields 
2447	\begin{align}\label{eq:it96y0hho}
2448	\big(  \widehat\infty-b \big)+\big(  \widehat\infty-a \big)&=2\widehat\infty -\big( b+a \big)=\widehat\infty -\big( b+a \big)~~.	
2449	\end{align}
2450	
2451	\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 $\aleph_{0.9}+\aleph_{0.9}=\aleph_{1.8}\ggg\aleph_1$.  
2452	
2453	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
2454	\begin{align*}
2455	\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)~~.	
2456	\end{align*}
2457	
2458	\noindent Assuming the associative property of addition, we may arrange the LHS brackets to write
2459	\begin{align*}
2460	\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)\\
2461	\widehat\infty+\big[c-\big( b+a \big)\big]&=c-\big(b+a\big)~~.	
2462	\end{align*}
2463	
2464	\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. 
2465	
2466	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.
2467
2468\begin{defin}
2469	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}.
2470\end{defin}
2471
2472\begin{axio}\label{ax:fieldplus}
2473	The addition axioms for fields are
2474	\begin{itemize}
2475		\item (A1) $S$ is closed under addition: If $x,y\in S$, then $x+y\in S$.
2476		\item (A2) Addition is commutative: If $x,y\in S$, then $x+y=y+x$.
2477		\item (A3) Addition is associative: If $x,y,z\in S$, then $(x+y)+z=x+(y+z)$.
2478		\item (A4) There exists an additive identity element $0$ in $S$: If $x\in S$, then $x+0=x$. 
2479		\item (A5) Every $x\in S$ has an additive inverse: If $x\in S$, then there exists $-x\in S$ such that $x+(-x)=0$.
2480	\end{itemize}
2481\end{axio}
2482
2483
2484\begin{rem}
2485	The arithmetic axioms do not obey (A1) but they do obey (A2)-(A5).
2486\end{rem}
2487
2488
2489
2490\begin{axio}\label{ax:fieldtimes}
2491	The multiplication axioms for fields are
2492	\begin{itemize}
2493		\item (M1) $S$ is closed under multiplication: If $x,y\in S$, then $x\cdot y\in S$.
2494		\item (M2) Multiplication is commutative: If $x,y\in S$, then $x\cdot y=y\cdot x$.
2495		\item (M3) Multiplication is associative: If $x,y,z\in S$, then $(x\cdot y)\cdot z=x\cdot(y\cdot z)$.
2496		\item (M4) There exists a multiplicative identity element $1\neq0$ in $S$: If $x\in S$, then $x\cdot 1=x$. 
2497		\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 $x\cdot x^{-1}=1$.
2498	\end{itemize}
2499\end{axio}
2500
2501
2502\begin{rem}
2503	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}.
2504\end{rem}
2505 
2506
2507\begin{defin}
2508	An ordered field is a field $F$ together with a relation $<$ which satisfies the field ordering axioms: Axiom \ref{ax:fieldord}.
2509\end{defin}
2510
2511\begin{axio}\label{ax:fieldord}
2512	The field ordering axioms are
2513	\begin{itemize}
2514		\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 $x>y$.
2515		\item (O2) The $<$ relation is transitive: If $x,y,x\in F$, then $x<y$ and $y<z$ together imply $x<z$.
2516		\item (O3) If $x,y,z\in F$, then $x<y$ implies $x+z<y+z$.
2517		\item (O4) If $x,y,z \in F$ and $z>0$, then $x<y$ implies $x\cdot z< y\cdot z$.
2518	\end{itemize}
2519
2520	\noindent It is understood that $x<y$ means $y>x$.
2521\end{axio}
2522 
2523
2524\begin{thm}\label{thm:443eee}
2525	For any $\mathcal{X}>0$, $\aleph_\mathcal{X}$ is an upper bound of $\mathbb{R}_0$.
2526\end{thm} 
2527
2528\begin{proof}
2529	An upper bound of a set is greater than or equal to every element of that set.  Suppose $X,Y\in\mathbf{AB}$ and
2530	\begin{equation}
2531	x\in\mathbb{R}_0~~,~~x\in X\qquad\text{and}\qquad \aleph_\mathcal{X}\in Y~~.\nonumber
2532	\end{equation}
2533	
2534	\noindent Then it follows that
2535	\begin{equation}
2536	\mathcal{D}_{\!\mathbf{AB}}(AX)=0~~,\qquad\text{and}\qquad \mathcal{D}_{\!\mathbf{AB}}(AY)=\mathcal{X}~~.\nonumber
2537	\end{equation}
2538	
2539	\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 $\mathcal{X}>0$.
2540\end{proof}
2541
2542
2543
2544\begin{cor}\label{cor:rtt35}
2545	$\mathbb{N}$ is bounded from above.
2546\end{cor}
2547
2548\begin{proof}
2549	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.
2550\end{proof} 
2551
2552
2553
2554\begin{rem}
2555	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.
2556\end{rem}
2557 
2558
2559\begin{pro}\label{mthm:u9999979y}
2560	$\mathbb{R}_0\subset\mathbb{R}$ 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.   
2561\end{pro}
2562
2563 
2564\begin{just}
2565	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 $s$.
2566\end{just}
2567
2568\begin{defin}
2569	The issue described in the justification of Proposition \ref{mthm:u9999979y} shall be referred to as the least upper bound problem.
2570\end{defin}
2571 
2572
2573\subsection{Compliance of Cauchy Equivalence Classes with the Arithmetic Axioms}\label{sec:consXXX} 
2574
2575
2576In 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}.
2577
2578\begin{thm}\label{thm:3f444042040}
2579	Every convergent rational sequence of terms $a_n\in\mathbb{Q}$ is a Cauchy sequence.
2580\end{thm}
2581 
2582\begin{proof}
2583	Per Definition \ref{def:CSq}, a sequence $\{a_n\}$ is a Cauchy sequence if and only if 
2584	\begin{equation*}
2585	\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~~.
2586	\end{equation*} 
2587	
2588	\noindent By the convergence of $\{a_n\}$, it is granted that there exists some $l\in\mathbb{R}$ such that
2589	\begin{equation*}
2590	\lim\limits_{n\to\infty}a_n=l~~.
2591	\end{equation*}
2592	
2593	\noindent  Convergence then guarantees that 
2594	\begin{equation*}
2595	\exists n,N\in\mathbb{N}\quad\text{s.t.}\quad n>N\quad\implies\quad \big|a_n-l\big|<\frac{\delta}{2}~~.
2596	\end{equation*}
2597	
2598	\noindent Then, whenever $n,m>N$, we have
2599	\begin{equation*}
2600	\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~~.
2601	\end{equation*}
2602	
2603	\noindent Therefore, every convergent rational sequence $\{a_n\}$ is a Cauchy sequence.	
2604\end{proof}
2605
2606\begin{defin}\label{def:86969696}
2607	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 $x\cdot y=[(x_n\cdot y_n)]$.
2608\end{defin}
2609
2610\begin{thm}\label{thm:cec200e0c0ec}
2611	The additive operation for equivalence classes given by Definition \ref{def:86969696} is well-defined.
2612\end{thm}
2613
2614\begin{proof}
2615	Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2616	\begin{equation*}
2617	[a]=[b]~~,\qquad\text{and}\qquad [c]=[d]~~,
2618	\end{equation*}
2619	
2620	\noindent so that 
2621	\begin{equation*}
2622	\lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\qquad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2623	\end{equation*}
2624	
2625	\noindent For addition to be proven well-defined, we need to prove that $[(a_n+c_n)]=[(b_n+d_n)]$.  This requires
2626	\begin{equation*}
2627	[(a_n+c_n)]-[(b_n+d_n)]=0~~.
2628	\end{equation*}
2629	
2630	\noindent The difference being equal to zero means that for sufficiently large $n$, and for any $\delta\in\mathbb{R}$, we must have
2631	\begin{equation*}
2632	[(a_n+c_n)]-[(b_n+d_n)]=[(a_n-b_n)]-[(c_n-d_n)]<\delta~~.
2633	\end{equation*}
2634	
2635	\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
2636	\begin{equation*}
2637	\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}.
2638	\end{equation*}
2639	
2640	\noindent Then, whenever $n,m>N$, we have
2641	\begin{equation*}
2642	\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~~.
2643	\end{equation*}
2644	
2645	\noindent This proves that $[a+c]=[b+d]$ and that, therefore, addition is a well-defined operation on Cauchy equivalence classes.
2646\end{proof}
2647
2648\begin{exa}
2649	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
2650	\begin{equation*}
2651	[\aleph_{[\mathcal{X}_1]}+x_1]=[\aleph_{[\mathcal{Y}_1]}+y_1] ~~,\qquad\text{and}\qquad[\aleph_{[\mathcal{X}_2]}+x_2]=[\aleph_{[\mathcal{Y}_2]}+y_2] ~~.
2652	\end{equation*}
2653	
2654	\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 
2655	\begin{equation*}
2656	[\aleph_{[\mathcal{X}_1]}+x_1]+[\aleph_{[\mathcal{X}_2]}+x_2]=[\aleph_{[\mathcal{Y}_1]}+y_1]+[\aleph_{[\mathcal{Y}_2]}+y_2]~~.
2657	\end{equation*}
2658	
2659	\noindent Evaluating the left and right sides independently yields
2660	\begin{align*}
2661	[\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]~~,
2662	\end{align*}
2663	
2664	\noindent and 
2665	\begin{align*}
2666	[\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]~~.
2667	\end{align*}
2668	
2669	\noindent Considering first the small parts, Definition \ref{def:86969696} gives $[x+y]=[x]+[y]$ so 
2670	\begin{equation*}
2671	[x_1+x_2]=[y_1+y_2]\quad\iff\quad [x_1]+[x_2]=[y_1]+[y_2]~~.
2672	\end{equation*}
2673	
2674	\noindent This condition follows from Theorem \ref{thm:cec200e0c0ec}.  Considering the big parts yields
2675	\begin{equation*}
2676	[\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]~~.
2677	\end{equation*}
2678	
2679	\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.
2680\end{exa}
2681
2682\begin{rem}
2683	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.
2684\end{rem}
2685
2686\begin{thm}\label{thm:bbbbb4b4}
2687	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.
2688\end{thm}
2689
2690\begin{proof}
2691	Since $\{a_n\}$ is Cauchy, we know there is some sufficiently large $m,n\in\mathbb{N}$ such that
2692	\begin{equation*}
2693	\big|a_n-a_m\big|<1~~.
2694	\end{equation*}
2695	
2696	\noindent If follows for such $n$ that
2697	\begin{equation*}
2698	\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)~~.
2699	\end{equation*}
2700	
2701	\noindent Define $M$ as the greatest element of a set with a natural number of elements
2702	\begin{equation*}
2703	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\}~~.
2704	\end{equation*}
2705	
2706	\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 $\{a_n\}$.
2707\end{proof}
2708
2709 
2710
2711\begin{thm}\label{thm:cec20f0e0c0ec}
2712	The multiplicative operation for equivalence classes given by Definition \ref{def:86969696} is well-defined.   
2713\end{thm}
2714
2715\begin{proof}
2716	Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2717	\begin{equation*}
2718	[a]=[b]~~,\qquad\text{and}\qquad [c]=[d]~~,
2719	\end{equation*}
2720	
2721	\noindent so that 
2722	\begin{equation*}
2723	\lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\qquad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2724	\end{equation*}
2725	
2726	\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 $n$
2727	\begin{equation*}
2728	[(a_n\cdot c_n)]-[(b_n\cdot d_n)]<\delta~~.
2729	\end{equation*}
2730	
2731	\noindent  To that end, insert the additive identity as a difference of cross terms so that
2732	\begin{align*}
2733	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)\\
2734	&=\big(a_n\cdot c_n- c_n\cdot b_n\big)+\big(c_n\cdot b_n -b_n\cdot d_n\big)\\
2735	&=c_n\cdot\big(a_n-  b_n\big)+b_n\cdot\big(c_n - d_n\big)~~.
2736	\end{align*}
2737	
2738	\noindent It follows that 
2739	\begin{equation*}
2740	\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)~~.
2741	\end{equation*}
2742	
2743	\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 
2744	\begin{equation*}
2745	\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)~~.
2746	\end{equation*}
2747	
2748	\noindent Since all four sequences are Cauchy, we have
2749	\begin{equation*}
2750	\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}.
2751	\end{equation*}
2752	
2753	\noindent We prove the theorem by writing 
2754	\begin{equation*}
2755	\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~~.
2756	\end{equation*}
2757\end{proof}
2758
2759
2760\begin{rem}
2761	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 $\mathcal{Y}>0$.
2762\end{rem}
2763
2764 
2765
2766\begin{rem}
2767	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.
2768\end{rem}
2769
2770\begin{defin}\label{def:869696ffefe96}
2771	If $x,y\in\mathbb{R}$ such that there are two Cauchy equivalence classes $x=[(x_n)]$ and $y=[(y_n)]$, then $x\div y=[(x_n\div y_n)]$.
2772\end{defin}
2773
2774
2775\begin{thm}\label{thm:3111r22424}
2776	The quotient operation for equivalence classes of rationals given by Definition \ref{def:86969696} is well-defined.  
2777\end{thm}
2778
2779\begin{proof}
2780	Define four Cauchy equivalence classes $[(a_n)],[(b_n)],[(c_n)],$ and $[(d_n)]$ having the properties
2781	\begin{equation*}
2782	[a]=[b]~~,\qquad\text{and}\qquad [c]=[d]~~,
2783	\end{equation*}
2784	
2785	\noindent so that 
2786	\begin{equation*}
2787	\lim\limits_{n\to\infty}\big(a_n-b_n\big)=0~~,\qquad\text{and}\qquad\lim\limits_{n\to\infty}\big(c_n-d_n\big)=0~~.
2788	\end{equation*}
2789	
2790	\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
2791	\begin{equation*}
2792	[(a_n\div c_n)]-[(b_n\div d_n)]<\delta~~.
2793	\end{equation*}
2794	
2795	\noindent  To that end, insert the additive identity as a difference of the cross terms so that
2796	\begin{align*}
2797	\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)\\
2798	&=\left( \frac{a_n}{c_n}-\frac{b_n}{c_n}\right)+\left(\frac{b_n}{c_n}-\frac{b_n}{d_n}\right)\\
2799	&=\frac{a_n-b_n}{c_n}+\frac{b_n\cdot\big(d_n-c_n\big)}{c_n\cdot d_n}~~.
2800	\end{align*}
2801	
2802	\noindent It follows that 
2803	\begin{equation*}
2804	\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)~~.
2805	\end{equation*}
2806	
2807	
2808	\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
2809	\begin{equation*}
2810	\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}.
2811	\end{equation*}
2812	
2813	\noindent We prove the theorem by writing 
2814	\begin{equation*}
2815	\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~~.
2816	\end{equation*}
2817	
2818	\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.
2819\end{proof}
2820
2821
2822
2823\begin{mainthm}\label{thm:3111r22424}
2824	The quotient operation given by Definition \ref{def:86969696} is well-defined for equivalence classes in $C_\mathbb{Q}^\mathbf{AB}\setminus C_\mathbb{Q}$.  
2825\end{mainthm}
2826
2827\begin{proof}
2828	Suppose there are four subsets of $C_\mathbb{Q}^{\mathbf{AB}}$ with the properties
2829	\begin{equation*}
2830	[\aleph_{[\mathcal{A}]}+a]=[\aleph_{[\mathcal{B}]}+b] ~~,\qquad\text{and}\qquad[\aleph_{[\mathcal{C}]}+C]=[\aleph_{[\mathcal{D}]}+d] ~~.
2831	\end{equation*}
2832	
2833	\noindent It follows from the equality of Cauchy sequences that
2834	\begin{align*}
2835	\lim\limits_{n\to\infty}\big(\mathcal{A}_n-\mathcal{B}_n\big)&=0\\
2836	\lim\limits_{n\to\infty}\big(\mathcal{C}_n-\mathcal{D}_n\big)&=0\\
2837	\lim\limits_{n\to\infty}\big(a_n-b_n\big)&=0\\
2838	\lim\limits_{n\to\infty}\big(c_n-d_n\big)&=0~~.
2839	\end{align*}
2840	
2841	\noindent For concision in notation, introduce the symbols
2842	\begin{align*}
2843	(A_n)&=(\aleph_{[(\mathcal{A}_n)]}+a_n)\\
2844	(B_n)&=(\aleph_{[(\mathcal{B}_n)]}+b_n)\\
2845	(C_n)&=(\aleph_{[(\mathcal{C}_n)]}+c_n)\\
2846	(D_n)&=(\aleph_{[(\mathcal{D}_n)]}+d_n)~~.
2847	\end{align*}
2848	
2849	
2850	\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
2851	\begin{equation*}
2852	[(A_n\div C_n)]-[(B_n\div D_n)]<\delta~~.
2853	\end{equation*}
2854	
2855	\noindent Following the form of Theorem \ref{thm:3111r22424}, we may insert the identity to obtain the inequality	
2856	\begin{equation*}
2857	\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|} ~~.
2858	\end{equation*}
2859	
2860	\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
2861	\begin{align*}
2862	[(A_n)]&=([\mathcal{A}],[a])\leq(A_0,a_0)\\
2863	[(B_n)]&=([\mathcal{B}],[b])\leq(B_0,b_0)\\
2864	[(C_n)]&=([\mathcal{C}],[c])\leq(C_0,c_0)\\
2865	[(D_n)]&=([\mathcal{D}],[d])\leq(D_0,d_0)~~,
2866	\end{align*}
2867	
2868	\noindent where the notation implies the ordering of each paired element respectively.  It follows that	
2869	\begin{align*}
2870	\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|}\\
2871	&\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|}\\
2872	&\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|}\\
2873	&\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|}\\
2874	&\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|}\\
2875	&\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|}~~.
2876	\end{align*}
2877	
2878	\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
2879	\begin{equation*}
2880	\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}.
2881	\end{equation*}
2882	
2883	\noindent We prove the theorem by writing 
2884	\begin{equation*}
2885	\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~~.
2886	\end{equation*}
2887\end{proof} 
2888 
2889 
2890
2891
2892\section{Arithmetic Applications}
2893
2894
2895
2896
2897\subsection{Properties of the Algebraic Fractional Distance Function Revisited}\label{sec:cont2}
2898 
2899
2900We 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 $AX=cAB$.  Formally we have 
2901\begin{equation*}
2902\mathcal{D}_{\!\mathbf{AB}}:AX\to[0,1]~~,\qquad\text{and}\qquad\mathcal{D}_{\!\mathbf{AB}}^\dagger:\{AX;x\}\to[0,1]~~,
2903\end{equation*}
2904
2905\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 $x$.
2906
2907In 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---that demonstrated by Theorem \ref{thm:algfracdisnotcont3}---is that it gives a requirement
2908\begin{equation}\label{eq:dfgr3r3434}
2909|x-\infty|<\delta\quad\iff\quad \delta>\infty~~.\nonumber
2910\end{equation}
2911
2912\noindent There is no such $\delta$.  What is the source of this discrepancy?  The source is the additive absorptive property of infinity giving $\infty-x=\infty$.  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 $\mathcal{D}^\dagger_{\!AB}$.
2913
2914
2915
2916\begin{mainthm}\label{thm:algfracdisnotcont4}
2917	The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!\mathbf{AB}}(AX)$ converges to a limit $l=1$ at $B\in\mathbf{AB}$.
2918\end{mainthm}
2919
2920
2921\begin{proof}
2922	According to the Cauchy definition of the limit of $f(x):D\to R$ at infinity, we say that 
2923	\begin{equation}
2924	\lim\limits_{x\to \widehat\infty} f(x) = l~~,\nonumber
2925	\end{equation}
2926	
2927	\noindent if and only if
2928	\begin{equation}
2929	\forall\varepsilon>0\quad\exists\delta>0\quad\text{s.t}\quad\forall x\in D~~,\nonumber
2930	\end{equation}
2931	
2932	\noindent we have
2933	\begin{equation}
2934	0<|x-\widehat\infty|<\delta\quad\implies\quad|f(x)-l |<\varepsilon~~.\nonumber
2935	\end{equation}
2936	
2937	\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 $x\in\mathbb{R}$, and use the arithmetic axioms to obtain, for example,
2938	\begin{equation}
2939	|( \widehat\infty-b )-\widehat\infty|= b~~,\qquad\text{or}\qquad |\aleph_\mathcal{X}-\widehat\infty|=\aleph_{(1-\mathcal{X})}\nonumber~~.
2940	\end{equation}
2941	
2942	\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}.  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
2943	\begin{equation}
2944	0<|x-\widehat\infty|<\aleph_{\left(\frac{\varepsilon}{2}\right)}\quad\text{and}\quad|\mathcal{D}'_{\!\mathbf{AB}}(AX)-\mathcal{D}'_{\!\mathbf{AB}}(AB)|<\varepsilon~~.\nonumber
2945	\end{equation}
2946	
2947	\noindent Evaluation of the $\delta$ expression yields
2948	\begin{equation}
2949	\widehat\infty-x<\aleph_{\left(\tfrac{\varepsilon}{2}\right)}\quad\iff\quad x>\aleph_{\left(1-\tfrac{\varepsilon}{2}\right)}~~.\nonumber
2950	\end{equation} 
2951	
2952	\noindent Definition \ref{def:algfracdis} gives $\mathcal{D}'_{\!AB}$ as
2953	\begin{equation}
2954	\mathcal{D}'_{\!AB}(AX)=\begin{cases}
2955	~~~1\qquad\quad\text{for}\quad X=B\\[8pt]
2956	\cfrac{\|AX\|}{ \|AB\|}\,~~\quad\text{for}\quad X\neq A \quad\text{and}\quad X\neq B\\[10pt]
2957	~~~0 \qquad\quad\text{for}\quad X=A \end{cases}~~,\nonumber
2958	\end{equation} 
2959	
2960	\noindent where	
2961	\begin{equation}
2962	\cfrac{\|AX\|}{ \|AB\|}  =   \cfrac{\text{len}[a,x]}{\text{len}[a,b]}    ~~.\nonumber
2963	\end{equation} 
2964
2965	
2966	\noindent Evaluation of the $\varepsilon$ expression, therefore, yields	
2967	\begin{equation}
2968	\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
2969	\end{equation}
2970	
2971	\noindent Therefore,
2972	\begin{equation}
2973	\lim\limits_{x\to \widehat\infty} \mathcal{D}'_{\!\mathbf{AB}}(AX) = 1~~.\nonumber
2974	\end{equation}	
2975	
2976	\noindent This limit demonstrates the continuity of $\mathcal{D}'_{\!\mathbf{AB}}$ at infinity.
2977\end{proof}
2978
2979
2980\begin{rem}
2981	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}.
2982\end{rem}
2983
2984
2985
2986\begin{lem}\label{thm:uniqueneighb}
2987	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 $x_2\in\mathbb{R}^{\mathcal{X}_0}_\aleph$.
2988\end{lem}
2989
2990
2991\begin{proof}
2992	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
2993	\begin{equation*}
2994	x_1=\aleph_{\mathcal{X}_1}+b_1~~,\qquad\text{and}\qquad x_2=\aleph_{\mathcal{X}_2}+b_2~~.
2995	\end{equation*}
2996	
2997	\noindent The algebraic FDF tells us that
2998	\begin{equation}
2999	\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
3000	\end{equation} 
3001	
3002	\noindent and
3003	\begin{equation}
3004	\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
3005	\end{equation} 
3006	
3007	\noindent It follows from the identity $\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)=\mathcal{D}_{\!\mathbf{AB}}(AX)$ that 
3008	\begin{equation}
3009	\min[\mathcal{D}_{\!\mathbf{AB}}(AX)] =\mathcal{X}_1~~,\qquad\text{and}\qquad \max[\mathcal{D}_{\!\mathbf{AB}}(AX)]=\mathcal{X}_2~~.\nonumber
3010	\end{equation} 
3011	
3012	\noindent By Definition \ref{def:gfdf}, $\mathcal{D}_{\!\mathbf{AB}}(AX)$ is one-to-one which requires	
3013	\begin{equation}
3014	\mathcal{X}_1=\mathcal{X}_2~~.\nonumber
3015	\end{equation} 
3016	
3017	\noindent This contradicts the assumed condition that $\mathcal{X}_1\neq\mathcal{X}_2$.
3018\end{proof}
3019
3020
3021\begin{thm}\label{thm:injjj2}  
3022	The algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$ is injective (one-to-one) on all real line segments.
3023\end{thm}
3024
3025\begin{proof} 
3026	(Proof of Conjecture \ref{conj:dv2324}.)  Recall that $\mathcal{D}'_{\!AB}:AB\to [0,1]$ is	
3027	\begin{equation}
3028	\mathcal{D}'_{\!AB}(AX)=\begin{cases}
3029	~~~1\qquad\quad&\text{for}\quad X=B\\[8pt]
3030	\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]
3031	~~~0 \qquad\quad&\text{for}\quad X=A \end{cases}~~.\nonumber
3032	\end{equation} 
3033	
3034	\noindent Injectivity requires that	
3035	\begin{equation}
3036	\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
3037	\end{equation}
3038	
3039	
3040	\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}:	
3041	\begin{equation}
3042	\min[\mathcal{D}'_{\!AB}(AX_k)] = \max[\mathcal{D}'_{\!AB}(AX_k)]=\mathcal{X}_k~~.\nonumber
3043	\end{equation} 
3044	
3045	\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 	
3046	\begin{equation}
3047	\mathcal{D}'_{\!AB}(AX)=\mathcal{D}_{\!AB}(AX)~~.\nonumber
3048	\end{equation}
3049\end{proof}
3050
3051
3052
3053\begin{conj}
3054	The algebraic fractional distance function $\mathcal{D}^\dagger_{\!AB}$ is an algebraic fractional distance function of the first kind $\mathcal{D}'_{\!AB}$.
3055\end{conj}
3056
3057
3058
3059\subsection{Some Theorems for Numbers in the Neighborhood of Infinity}\label{sec:gduidt7t}
3060
3061In 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
3062\begin{itemize}
3063	\item $L\in\mathbb{R}_0$
3064	\item $L\in\mathbb{R}^0_\aleph\setminus\mathbb{R}_0$
3065	\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.)
3066	\item $L=\widehat\infty$
3067\end{itemize}
3068
3069\noindent  We were able to prove in that section the cases $L\in\mathbb{R}_0$ and $L=\widehat\infty$ 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 $\mathbb{R}^0_\aleph\setminus\mathbb{R}_0=\varnothing$, but by now we have given the tools needed to prove the many-to-one relationship between real numbers and points in a line segment with $L\in\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}^1_0$ meaning $\text{Lit}(L)\in\mathbb{R}_0$.  This is the third case given above modified with a restriction to the natural neighborhoods of $\aleph_\mathcal{X}$ rather than the whole neighborhoods.  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.
3070 
3071
3072 
3073\begin{thm}\label{thm:finlinseggg3}
3074	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.
3075\end{thm}
3076
3077\begin{proof}
3078	By Definition \ref{def:XrepR}, every point in a line segment has an algebraic representation
3079	\begin{equation}
3080	X\equiv\mathscr{X}=[x_1,x_2]~~.\nonumber
3081	\end{equation}
3082	
3083	\noindent It follows that 
3084	\begin{equation}
3085	\min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]=\cfrac{\text{len}[0,x_1]}{\text{len}[0,\aleph_\mathcal{X}+b]} =\frac{x_1}{\aleph_\mathcal{X}+b}~~,\nonumber
3086	\end{equation} 
3087	
3088	\noindent Now suppose $x_0\in\mathbb{R}_0^+$, and $z=x_1+x_0$ so that $z> x_1$.  Then
3089	\begin{equation}
3090	\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
3091	\end{equation} 
3092	
3093	\noindent By Axiom \ref{ax:div1g1g1g1}, the $x_0$ term vanishes so we find
3094	\begin{equation}
3095	\cfrac{\text{len}[0,z]}{\text{len}[0,\aleph_\mathcal{X}+b]} =\frac{x_1}{\aleph_\mathcal{X}+b}=\min[\mathcal{D}^\dagger_{\!\mathbf{AB}}(AX)]~~.\nonumber
3096	\end{equation} 
3097		
3098	\noindent Invoking the single-valuedness of bijective functions, we find that
3099	\begin{equation*}
3100	\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~~.
3101	\end{equation*}
3102	
3103	\noindent Therefore $x_1\neq x_2$ and the theorem is proven.
3104\end{proof}
3105
3106
3107\begin{thm}\label{thm:ijdsvoiydt97c}
3108	The derivative of $f(x)=\aleph_x$ with respect to $x$ is infinite.
3109\end{thm}
3110
3111\begin{proof}
3112	The definition of the derivative of $f(x)$ with respect to $x$ is
3113	\begin{equation*}
3114	\dfrac{d}{dx}\,f(x)=\lim\limits_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}~~.
3115	\end{equation*}
3116	
3117	\noindent For $f(x)=\aleph_x$, we have
3118	\begin{align*}
3119	\dfrac{d}{dx}\,\aleph_x&=\lim\limits_{\Delta x\to0}\frac{\aleph_{(x+\Delta x)}-\aleph_x}{\Delta x}\\
3120	&=\lim\limits_{\Delta x\to0}\frac{\aleph_x+\aleph_{\Delta x}-\aleph_x}{\Delta x}\\
3121	&=\lim\limits_{\Delta x\to0}\frac{ 1 }{\Delta x}\aleph_{\Delta x}\\
3122	&=\aleph_1~~.
3123	\end{align*}
3124\end{proof}
3125
3126
3127\begin{defin}\label{def:y98t9tuigjaaaa}
3128	For $0<\mathcal{X}<1$, $\mathbb{N}_\mathcal{X}$ is a subset of real numbers such that
3129	\begin{equation*}
3130	\mathbb{N}_\mathcal{X}=\big\{ \aleph_\mathcal{X}+w~\big|~w\in\mathbb{W}  \big\}~~.
3131	\end{equation*}
3132	
3133	\noindent where $\mathbb{W}=\{...\!-2,-1,0,1,2,...\}$.  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 
3134	\begin{equation*}
3135	\mathbb{N}_1=\big\{ \widehat\infty-n  ~\big|~ n\in\mathbb{N}\big\}~~,
3136	\end{equation*}
3137	
3138	\noindent called natural numbers in the maximal neighborhood of infinity.  The set of all extended natural numbers is
3139	\begin{equation*}
3140	\mathbb{N}_\infty=\mathbb{N}\cup\{\mathbb{N}_\mathcal{X}\}\cup\mathbb{N}_1~~.
3141	\end{equation*}
3142\end{defin}
3143 
3144
3145\begin{defin}
3146	The function $E^x$ is defined as
3147	\begin{equation*}
3148	E^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}~~,
3149	\end{equation*}
3150	
3151	\noindent where the sum is taken to mean all $k\in\mathbb{N}_\infty\cup\{0\}$.  This function is called the big exponential function.
3152\end{defin}
3153
3154\begin{thm}
3155	For any $x\in\mathbb{R}_0$, the big exponential function is equal to the usual exponential function.  
3156\end{thm}
3157
3158\begin{proof}
3159	The usual exponential function is 
3160	\begin{equation*}
3161	e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}~~,
3162	\end{equation*}
3163	
3164	\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 
3165	\begin{align*}
3166	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!}+...\\
3167	&=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)!}+...~~.
3168	\end{align*}
3169	
3170	\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 
3171	\begin{align*}
3172	\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^{\left(\aleph_{\mathcal{X}}\pm k-1\right)}}{\big(\aleph_{\mathcal{X}}\pm k-1\big)!}=\sum_{\substack{k=0\\\vphantom{\hat{\mathbb{N}}}k\in\mathbb{N}_0}}^{\infty}0\cdot\frac{x^{\left(\aleph_{\mathcal{X}}\pm k-1\right)}}{\big(\aleph_{\mathcal{X}}\pm k-1\big)!}=0~~.
3173	\end{align*}
3174	
3175	\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.
3176\end{proof}
3177 
3178
3179\begin{exa}\label{exa:762}
3180	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 $n\in\mathbb{N}_\infty$.'' 
3181	Definition \ref{def:RRRinf} gives two definitions for the $\infty$ symbol, one of which is
3182	\begin{equation}
3183	\lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
3184	\end{equation} 
3185	
3186	\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
3187	\begin{equation*}
3188	 \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
3189	\end{equation*} 
3190	
3191	\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 $n\in\mathbb{N}_\infty$.
3192	
3193	This example has demonstrated the utility of $\mathbb{N}_\infty$ as a thinking device, and it also makes a distinction between the two formulae
3194	\begin{equation}
3195	\lim\limits_{x\to0^\pm}\dfrac{1}{x}=\pm\infty~~, \qquad\text{and}\qquad \lim\limits_{n\to\infty} \sum_{k=1}^{n}k= \infty  ~~.\nonumber
3196	\end{equation}
3197	
3198	\noindent In the partial sums definition, and under the $\mathbb{N}_\infty$ convention, the distinction between geometric infinity and algebraic infinity is suggested as
3199	\begin{equation*}
3200	\widehat\infty=\aleph_1~~,\qquad\text{and}\qquad\infty=\aleph_\infty~~.
3201	\end{equation*}
3202	
3203	\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.
3204\end{exa}
3205 
3206
3207
3208
3209\begin{mainthm}\label{mthm:giutqx}
3210	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 
3211	\begin{equation*}
3212	\|AC\|=\sqrt{\vphantom{\hat A}   \|AB\|^2+  \|BC\|^2 }~~,\qquad\text{with}\qquad\|AC\|=\text{len}(AC)~~,
3213	\end{equation*}
3214	
3215	\noindent then
3216	\begin{equation*}
3217	\text{len}(AC)\not\in\mathbb{R}^0_\aleph\cup\{\mathbb{R}^\mathcal{X}_\aleph\}\cup\mathbb{R}^1_\aleph ~~.
3218	\end{equation*} 
3219\end{mainthm}
3220
3221
3222\begin{proof}
3223	The squared lengths of the legs are 
3224	\begin{equation*}
3225	\|AB\|^2=\big( \aleph_\mathcal{X}+x \big)^2=\aleph_{\left(\aleph_{(\mathcal{X}^2)}+2x\mathcal{X}\right)}+x^2~~,
3226	\end{equation*}
3227	
3228	\noindent and 
3229	\begin{equation*}
3230	\|BC\|^2=\big( \aleph_\mathcal{Y}+y \big)^2=\aleph_{\left(\aleph_{(\mathcal{Y}^2)}+2y\mathcal{Y}\right)}+y^2~~.
3231	\end{equation*}
3232	
3233	\noindent If we directly state the Pythagorean theorem in terms of the lengths then we find
3234	\begin{equation*}
3235	\|AC\|^2=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2~~.
3236	\end{equation*}
3237	
3238	\noindent Assuming $\|AC\|=\aleph_\mathcal{A}+a$, we find
3239	\begin{equation*}
3240	\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~~.
3241	\end{equation*}
3242	
3243	\noindent Setting the big parts equal yields
3244	\begin{equation*}
3245	\aleph_{(\mathcal{A}^2)}+2a\mathcal{A}=\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})~~,
3246	\end{equation*}
3247	
3248	\noindent which still has separable big and little parts.  Doing the maximum possible separation of all the big and little parts yields
3249	\begin{align*}
3250	\mathcal{A}^2&=\mathcal{X}^2+\mathcal{Y}^2\\
3251	a\mathcal{A}&=x\mathcal{X}+y\mathcal{Y}\\
3252	a^2&=x^2+y^2~~.
3253	\end{align*}
3254	
3255	\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.
3256\end{proof}
3257
3258
3259
3260\begin{defin}\label{def:oydc986sdv8}
3261	A number is a complex number $z\in\mathbb{C}$ if and only if
3262	\begin{equation}
3263	z=x+iy~~,\qquad\text{and}\qquad x,y\in\mathbb{R}~~.\nonumber
3264	\end{equation}
3265\end{defin}
3266
3267\begin{thm}\label{mthm:gfff222x}
3268	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 $AC^2\equiv \bar zz$.
3269\end{thm}
3270
3271\begin{proof}
3272	Given two legs, we want to find the hypotenuse through the Pythagorean theorem.  We assume that the legs are real line segments so that
3273	\begin{equation*}
3274	AB^2\equiv\|AB\|^2~~,\qquad\text{and}\qquad BC^2\equiv\|BC\|^2~~.
3275	\end{equation*}
3276
3277	
3278	\noindent The geometric identity 
3279	\begin{equation*}
3280	AC^2=AB^2+BC^2~~,
3281	\end{equation*}
3282	
3283	\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
3284	\begin{equation*}
3285	AC^2\equiv\langle {AC}|{AC}\rangle~~.
3286	\end{equation*}
3287	
3288	\noindent Since the legs are taken as real, the algebraic representation of each is its own complex conjugate.  Again we find
3289	\begin{equation*}
3290	\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~~,
3291	\end{equation*}
3292	
3293	\noindent and 
3294	\begin{equation*}
3295	\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~~.
3296	\end{equation*}
3297	
3298	\noindent The present statement of the Pythagorean theorem is 
3299	\begin{equation*}
3300	\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~~.
3301	\end{equation*}
3302	
3303	\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
3304	\begin{equation*}
3305	z=\aleph_{(\mathcal{X}\pm i\mathcal{Y})}+x\pm iy=\big(\aleph_\mathcal{X}+x\big)+i\big(\aleph_\mathcal{Y}+y\big)~~.
3306	\end{equation*}
3307	
3308	\noindent We have 
3309	\begin{align*}
3310	\bar zz=\big\langle AC\big|AC\big\rangle
3311	&=\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]\\
3312	&=\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]\\
3313	&=\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)\\
3314	&\qquad\qquad+\aleph_{\left(\mathcal{X}\pm i\mathcal{Y}\right)}(x\mp iy)+(x\mp iy)(x\pm iy)\\
3315	&=\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)}\\
3316	&\qquad\qquad+\aleph_{\left( x\mathcal{X}\mp iy\mathcal{X}\pm ix\mathcal{Y}+y\mathcal{Y} \right)}+x^2+y^2\\
3317	&=\aleph_{\left(\aleph_{(\mathcal{X}^2+\mathcal{Y}^2)}+2(x\mathcal{X}+y\mathcal{Y})\right)}+x^2+y^2 ~~.
3318	\end{align*}
3319	
3320	\noindent Therefore,
3321	\begin{equation*}
3322	\big\langle AC\big|AC\big\rangle=\big\langle AB\big|AB\big\rangle+\big\langle BC\big|BC\big\rangle~~.
3323	\end{equation*}  
3324\end{proof}
3325
3326\begin{cor}\label{mthm:gffff222x}
3327	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 $AC^2\equiv \vec x \cdot \vec x$.
3328\end{cor}
3329
3330\begin{proof}
3331	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 $\mathbb{R}^2$:
3332	\begin{equation*}
3333	\vec{AB}=(\aleph_\mathcal{X}+x,0) ~~,~~	\vec{BC}=(0,\aleph_\mathcal{Y}+y)~~,\qquad\text{and}\qquad 	\vec{AC}=(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)~~.
3334	\end{equation*}
3335	
3336	\noindent The Pythagorean theorem yields	
3337	\begin{align*}
3338	\vec {AC}\cdot\vec{AC}&=\vec {AB}\cdot\vec{AB}+\vec {BC}\cdot\vec{BC}~~.
3339	\end{align*}
3340	
3341	\noindent Again we find
3342	\begin{equation*}
3343	\vec{AB}\cdot\vec{AB}=\big( \aleph_\mathcal{X}+x \big)^2=\aleph_{\left(\aleph_{(\mathcal{X}^2)}+2x\mathcal{X}\right)}+x^2~~,
3344	\end{equation*}
3345	
3346	\noindent and 
3347	\begin{equation*}
3348	\vec{BC}\cdot\vec{BC}=\big( \aleph_\mathcal{Y}+y \big)^2=\aleph_{\left(\aleph_{(\mathcal{Y}^2)}+2y\mathcal{Y}\right)}+y^2~~.
3349	\end{equation*}
3350	
3351	\noindent Checking the given form of $\vec{AC}\in\mathbb{R}^2$, we find
3352	 \begin{align*}
3353	 \vec{AC}\cdot\vec{AC}&=(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)\cdot(\aleph_\mathcal{X}+x,\aleph_\mathcal{Y}+y)\\
3354	 &=\vec {AB}\cdot\vec{AB}+\vec {BC}\cdot\vec{BC}
3355	 \end{align*}
3356	 
3357	 \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.
3358\end{proof}
3359
3360\begin{exa}
3361	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.
3362	
3363	TAKE $\sqrt{\bar zz}$
3364	
3365\end{exa}
3366
3367
3368=======================
3369
3370=======================
3371
3372=======================
3373
3374
3375Is the Pythagorean Theorem a geometric statement or an algebraic one?
3376
3377
3378=======================
3379
3380=======================
3381
3382=======================
3383 
3384
3385\begin{exa}\label{exa:f3ede1xxxx3}
3386	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
3387	\begin{equation}\label{eq:jjj3j3j3}
3388	\|AC\|\sin(\alpha)=\aleph_\mathcal{Y}+y~~,\qquad\text{and}\qquad \|AC\|\cos(\alpha)=\aleph_\mathcal{X}+x~~.
3389	\end{equation}
3390	
3391	\noindent It follows that 
3392	\begin{equation*}
3393	\|AC\|=\aleph_{\left(\frac{\mathcal{Y}}{\sin(\alpha)}\right)}+\frac{y}{\sin(\alpha)}~~,\qquad\text{and}\qquad \|AC\|=\aleph_{\left(\frac{\mathcal{X}}{\cos(\alpha)}\right)}+\frac{x}{\cos(\alpha)}
3394	\end{equation*}
3395	
3396	\noindent Equating the big and little parts yields
3397	\begin{equation*}
3398	\tan(\alpha)=\frac{\mathcal{Y}}{\mathcal{X}}~~,\qquad\text{and}\qquad \tan(\alpha)=\frac{y}{x}~~.
3399	\end{equation*}
3400	
3401	\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 $\|AC\|$.  
3402\end{exa}
3403
3404
3405
3406\begin{rem}
3407	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.
3408\end{rem}
3409
3410
3411
3412
3413
3414\begin{thm}\label{thm:t24t24t24fffft}
3415	A real number with non-vanishing big part is not the product of any real number with itself, \textit{i.e.}:
3416	\begin{equation*}
3417	\centernot\exists x\in\mathbb{R}\quad\text{s.t.}\quad x^2\in\{\mathbb{R}^\mathcal{X}_\aleph\}\cup\mathbb{R}^1_\aleph
3418	\end{equation*}
3419\end{thm}
3420
3421\begin{proof}
3422	Assume there exists a square root $z=\aleph_\mathcal{Z}+a$ of $x=\aleph_\mathcal{X}+b$ so that
3423	\begin{equation}\label{eq:87587f55757}
3424	\big( \aleph_\mathcal{Z}+a \big)^2=\aleph_\mathcal{X}+b~~.
3425	\end{equation}
3426	
3427	\noindent We have
3428	\begin{equation*}
3429	\big( \aleph_\mathcal{Z}+a \big)^2=\aleph_{\left(\aleph_{(\mathcal{Z}^2)}+2a\mathcal{Z}\right)}+z^2~~,
3430	\end{equation*}
3431	
3432	\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
3433	\begin{equation*}
3434	\aleph_{\left(\mathcal{Z}^2\right)}+2a\mathcal{Z}=\mathcal{X}~~,\qquad\text{and}\qquad z^2=b~~.
3435	\end{equation*}
3436	
3437	\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 $x$.
3438\end{proof}
3439 
3440
3441
3442\begin{exa}\label{exa:f42tl2444}
3443	Consider the limit 
3444	\begin{equation*}
3445	\lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=l~~.
3446	\end{equation*}
3447	
3448	\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
3449	\begin{equation*}
3450	\lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=-\infty~~,\qquad\text{and}\qquad\lim\limits_{ b\to\widehat\infty} \aleph_\mathcal{X}-b=\aleph_\mathcal{X}-\widehat\infty=-\aleph_{(1-\mathcal{X})}~~,
3451	\end{equation*}
3452	
3453	\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.
3454	\end{exa}
3455
3456
3457 
3458
3459\subsection{The Archimedes Property of Real Numbers}\label{sec:archim}
3460
3461While there are many ways to state the Archimedes property of real numbers with symbolic logic, the modern establishment has adopted
3462\begin{equation}
3463\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
3464\end{equation}
3465
3466\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.
3467
3468
3469
3470\begin{defin}\label{def:eudox}
3471	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}.
3472	\begin{quote}
3473		``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3474	\end{quote}
3475\end{defin}
3476
3477\begin{rem}
3478	\label{rem:32322222v}
3479	As it appears in Euclid's \textit{Elements}, the straightforward mathematical statement of the property would be 
3480	\begin{equation}
3481	\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
3482	\end{equation}
3483	
3484	\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.
3485\end{rem}
3486
3487\begin{defin}
3488	In Reference \cite{EE}, Fitzpatrick translates Book 5, Definitions 1 through 5 as follows.
3489	\begin{enumerate}
3490		\item A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.
3491		\item And the greater is a multiple of the lesser whenever it is measured by the lesser.
3492		\item A ratio is a certain type of condition with respect to size of two magnitudes of the same kind.
3493		\item (Those) magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.
3494		\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.
3495	\end{enumerate}
3496\end{defin}
3497
3498\begin{rem}
3499	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.
3500	\begin{enumerate}
3501		\item In other words, $\alpha$ is said to be a part of $\beta$ if $\beta=m\alpha$.
3502		\item (\textit{No footnote given.})
3503		\item In modern notation, the ratio of two magnitudes, $\alpha$ and $\beta$, is denoted $\alpha\,:\,\beta$.
3504		\item In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.
3505		\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.
3506	\end{enumerate}
3507
3508	\noindent Footnote 5 makes it exceedingly obvious that the multipliers are ``all $m$ and $n$'' in $\mathbb{R}$.
3509\end{rem}
3510
3511\begin{thm}\label{thm:euclid}
3512	The statement 
3513	\begin{equation}
3514	\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
3515	\end{equation}
3516	
3517	\noindent is not a proper statement of the Archimedes property of real numbers as given in antiquity.	
3518\end{thm}
3519
3520\begin{proof}
3521	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:
3522	\begin{quote}
3523		``And the greater is a multiple of the lesser whenever it is measured by the lesser.''
3524	\end{quote}
3525
3526	\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:
3527	\begin{quote}
3528		``A magnitude is a part of a(nother) magnitude, the lesser of the greater, when it measures the greater.''
3529	\end{quote}
3530
3531	\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.
3532\end{proof}
3533
3534\begin{rem}
3535	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.
3536\end{rem}
3537
3538\begin{exa}
3539	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 
3540	\begin{equation}
3541	\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
3542	\end{equation}
3543	
3544	\noindent does adequately encapsulate the Archimedes property of antiquity.  Proof of this statement is given by Rudin in Reference \cite{BRUDIN} as follows.
3545	
3546		\begin{quote}
3547			``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 $A$.''
3548		\end{quote}
3549	
3550	\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}.
3551	\end{exa}
3552
3553
3554
3555\begin{rem}
3556	If we adopt
3557	\begin{equation}
3558	\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
3559	\end{equation}
3560	
3561	\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 
3562	\begin{equation*}
3563	\widehat\infty-a<\widehat\infty-b~~,
3564	\end{equation*}
3565	
3566	\noindent then we cannot multiply the LHS by a number greater than one and have a real-valued product due to the identity $\aleph_\mathcal{X}=\mathcal{X}
3567	\cdot\widehat\infty$.  If we multiply by a positive number less than one, call it $\delta$, then 
3568	\begin{equation*}
3569	\aleph_\delta-\delta a<\widehat\infty-a<\widehat\infty-b~~,
3570	\end{equation*}
3571	
3572	\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
3573	\begin{equation}\label{eq:699869869880}
3574	\mathbb{R}^+=(0,\infty)\setminus\mathbb{R}^1_\aleph~~.
3575	\end{equation}
3576	
3577	\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 $(a,b)\subset(0,\infty)$.
3578	
3579	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:
3580		\begin{quote}
3581		``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3582	\end{quote}
3583
3584	\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. 
3585\end{rem}
3586
3587\begin{defin}\label{def:lllllllllll3}
3588	The most general statement of the ancient Archimedes property of real numbers is 
3589	\begin{equation}
3590	\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
3591	\end{equation} 
3592\end{defin}
3593
3594
3595\begin{mainthm}\label{mthm:tc87dc5}
3596	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.
3597\end{mainthm}
3598
3599\begin{proof}
3600	By Definition \ref{def:lllllllllll3}, it suffices to demonstrate
3601	\begin{equation}
3602	\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
3603	\end{equation} 
3604	
3605	\noindent We will consider the general forms
3606	\begin{equation*}
3607	x=\aleph_\mathcal{X}+b~~,\qquad\text{and}\qquad y=\aleph_\mathcal{Y}+a~~,
3608	\end{equation*}
3609	
3610	\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
3611	\begin{equation*}
3612	\aleph_\mathcal{X}+b<\aleph_\mathcal{Y}+a~~.
3613	\end{equation*}
3614	
3615	\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.
3616	
3617	$~$
3618	
3619	\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
3620	\begin{equation*}
3621	\big(\aleph_\mathcal{Z}+z\big)b=\aleph_{(\mathcal{Z}b)}+zb\quad>\quad a~~.
3622	\end{equation*}   
3623	
3624	$~$
3625	
3626	\noindent $\bullet$ ($x\in\mathbb{R}^0_\aleph$ and $y\in
3627	\{\mathbb{R}^\mathcal{Y}_\aleph\}$) 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
3628	\begin{equation*}
3629	\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~~.
3630	\end{equation*}  
3631	
3632	\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 $(\mathcal{Y},1)$.  
3633	
3634	$~$
3635	
3636	\noindent $\bullet$ ($x\in\mathbb{R}^0_\aleph$ and $y\in
3637	\mathbb{R}^1_\aleph$) 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
3638	\begin{equation*}
3639	\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~~.
3640	\end{equation*}   
3641	
3642	$~$
3643	
3644	\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
3645	\begin{equation*}
3646	\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~~.
3647	\end{equation*}   
3648	
3649	$~$
3650	
3651	\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
3652	\begin{equation*}
3653	z\left(\aleph_{\mathcal{X}}+b\right)=\aleph_{(z\mathcal{X})}+zb \quad>\quad \aleph_{\mathcal{X}}+a~~.
3654	\end{equation*}   
3655	
3656	$~$
3657	
3658	\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
3659	\begin{equation*}
3660	\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 }~~.
3661	\end{equation*}   
3662	
3663	$~$
3664	
3665	\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
3666	\begin{equation*}
3667	\aleph_{1}+b \quad>\quad \frac{1}{2}\big(\aleph_1-a\big)=\aleph_{\left(\frac{1}{2}\right)}-\frac{a}{2 }~~.
3668	\end{equation*}   
3669	
3670	 \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.	 
3671\end{proof}
3672
3673\begin{rem}
3674	 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 
3675	 \begin{equation}
3676	 \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
3677	 \end{equation} 
3678	 
3679	 \noindent This form is nice because it uses only a single multiplication operation and exactly reflects Fitzpatrick's footnote:
3680	 \begin{quote}
3681	 	``In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.''
3682	 \end{quote}
3683	 
3684	 \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:
3685	  \begin{quote}
3686	 	``Magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another.''
3687	 \end{quote}
3688 
3689 	\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.
3690 	 \begin{quote}
3691 		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 $n_2$.
3692 	\end{quote}
3693 
3694 	\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.
3695 	
3696 	In general we have made a rather large statement of Euclid's few original words in the form 
3697 	\begin{equation}\label{eq:689985986f}
3698 	\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~~. 
3699 	\end{equation} 
3700 	
3701 	\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.
3702\end{rem}
3703 
3704
3705\begin{axio}\label{ax:nncncncn}
3706	The Archimedes property of 1D transfinitely continued real numbers is
3707	\begin{equation*}
3708	\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~~. 
3709	\end{equation*}
3710	
3711	\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.
3712\end{axio}
3713
3714\begin{rem}
3715	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.
3716\end{rem} 
3717 
3718 
3719
3720\section{The Topology of the Real Number Line}\label{sec:topoR}
3721
3722
3723
3724\subsection{Basic Set Properties}
3725
3726In this section, we give some elementary set properties of the natural neighborhoods and begin to approach the connection to the whole neighborhoods.  Recall that natural neighborhoods $\mathbb{R}_0^\mathcal{X}$ are defined with little part $|b|\in\mathbb{R}_0^0$ and whole neighborhoods are defined with $|b|\in\mathbb{R}^0_\aleph$.
3727
3728\begin{thm}\label{thm:opee}
3729	Every natural neighborhood in $\{\mathbb{R}_0^\mathcal{X}\}$ is an open set.
3730\end{thm}
3731 
3732
3733\begin{proof}
3734	By Definition \ref{def:ffmdmmdd3md}, the set of all intermediate natural neighborhoods of infinity is
3735	\begin{equation*}
3736	\big\{\mathbb{R}^\mathcal{X}_0\big\}=\{ \aleph_\mathcal{X}+b~| ~ b\in  \mathbb{R}_0 ,~0<\mathcal{X}<1   \} ~~. 
3737	\end{equation*}
3738	
3739	\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	
3740	\begin{equation}
3741	\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
3742	\end{equation}
3743	
3744	\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 	
3745	\begin{equation}
3746	(\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
3747	\end{equation}
3748	
3749	Alternatively, no set in $\{\mathbb{R}_0^\mathcal{X}\}$ contains its boundary points so each such set is open. 
3750\end{proof}
3751
3752
3753
3754
3755\begin{thm}\label{thm:f3444xx}
3756	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 $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$.
3757\end{thm}
3758
3759\begin{proof} 
3760	Consider the interval
3761	\begin{equation}
3762	(\aleph_{\mathcal{X}},\aleph_{\mathcal{Y}})\subset\mathbb{R}~~.\nonumber
3763	\end{equation}
3764	
3765	\noindent By Definition \ref{def:3533553}, the number at the center of this interval is 
3766	\begin{equation}
3767	\cfrac{\aleph_{\mathcal{Y}}+\aleph_{\mathcal{X}}}{2}=\aleph_{\left(\!\frac{\mathcal{Y}+\mathcal{X}}{2}\!\right)}~~.\nonumber
3768	\end{equation}
3769	
3770	\noindent We have
3771	\begin{equation}
3772	\mathcal{X}<\frac{\mathcal{Y}+\mathcal{X}}{2}<\mathcal{Y}~~,\nonumber
3773	\end{equation}
3774	
3775	\noindent so let $\mathcal{Z}=\frac{\mathcal{Y}+\mathcal{X}}{2}$.  Any number $z\in Z\in\mathbf{AB}$ of the form
3776	\begin{equation}
3777	z=\aleph_{\mathcal{Z}}+z_0~~,\qquad\text{for}\qquad |z_0|\in\mathbb{R}_0~~,\nonumber
3778	\end{equation}
3779	
3780	\noindent will be such that 
3781	\begin{equation}
3782	\mathcal{D}_{\!\mathbf{AB}}(AZ)=\mathcal{Z}~~.\nonumber
3783	\end{equation}
3784	
3785	\noindent Since $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$, the theorem is proven.
3786\end{proof}
3787
3788
3789\begin{cor}
3790	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 $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$.
3791\end{cor}
3792
3793\begin{proof}
3794	Following the proof of Theorem \ref{thm:f3444xx}, we arrive at a number $z\in Z\in\mathbf{AB}$ of the form 
3795	\begin{equation}
3796	z=\aleph_{\mathcal{Z}}+z_0~~,\qquad\text{for}\qquad z_0\in\mathbb{R}^0_\aleph~~,\nonumber
3797	\end{equation}
3798	
3799	\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
3800	\begin{equation}
3801	\text{Big}(z)=\aleph_\mathcal{Z}\quad\iff\quad \mathcal{D}_{\!\mathbf{AB}}(AZ)=\mathcal{Z}~~.\nonumber
3802	\end{equation}
3803	
3804	\noindent Proof follows from $\mathcal{X}<\mathcal{Z}<\mathcal{Y}$, as in Theorem \ref{thm:f3444xx}.
3805\end{proof}
3806
3807
3808
3809\begin{defin}
3810	An open set $S$ is disconnected if and only if there exist two open, non-empty sets $U$ and $V$ such that
3811	\begin{equation*}
3812	S=U\cup V~~,\qquad\text{and}\qquad U\cap V=\varnothing~~.
3813	\end{equation*}
3814	
3815	\noindent If a set is not disconnected, then it is connected.
3816\end{defin}
3817 
3818
3819\begin{cor}\label{cor:et3443454}
3820	$\mathbb{R}^{\mathcal{X}}_0\cup\mathbb{R}^{\mathcal{Y}}_0$ is a disconnected set for any $0\leq\mathcal{X}<\mathcal{Y}\leq1$.
3821\end{cor}
3822
3823\begin{proof}
3824	An open set is disconnected if it is the union of two disjoint, non-empty open sets.  By Theorem \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.}:
3825	\begin{equation}
3826	\mathbb{R}^{\mathcal{X}}_0\cap\mathbb{R}^{\mathcal{Y}}_0=\varnothing~~.\nonumber
3827	\end{equation}
3828	
3829	\noindent The union  $\mathbb{R}^{\mathcal{X}}_0\cup\mathbb{R}^{\mathcal{Y}}_0$ satisfies the definition of a disconnected set.  
3830\end{proof}
3831 
3832
3833
3834
3835\begin{rem}
3836	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 $\mathcal{X}=1$.
3837\end{rem}
3838
3839
3840\begin{defin}\label{def:77777yy}
3841	To streamline notation, define
3842	\begin{align*}
3843	\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\\
3844	\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~~.
3845	\end{align*}
3846\end{defin}
3847 
3848\begin{defin}\label{def:ugu2g2g2g222}
3849	The complement of $\mathbb{R}_0^\mathcal{X}$ in $\mathbb{R}^\mathcal{X}_\aleph$ is $\mathbb{R}^\mathcal{X}_C$:
3850	\begin{equation}
3851	\mathbb{R}^\mathcal{X}_C=\mathbb{R}^\mathcal{X}_\aleph\setminus \mathbb{R}_0^\mathcal{X}~~.\nonumber
3852	\end{equation}
3853\end{defin}
3854
3855
3856\begin{thm}\label{thm:nosdx}
3857	There exist more positive real numbers than are in $\mathbb{R}^\cup_0$.  In other words,
3858	\begin{equation*}
3859	\mathbb{R}^+\setminus\mathbb{R}^\cup_0\neq\varnothing~~.
3860	\end{equation*}
3861\end{thm}
3862
3863
3864
3865\begin{proof} 
3866	By the definition of an interval, an 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}.
3867\end{proof} 
3868
3869\begin{rem}
3870	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.
3871\end{rem}
3872
3873\begin{mainthm}\label{mthm:y7958t787555}
3874	There exist more positive real numbers than are in $\mathbb{R}^\cup_\aleph$.  In other words,
3875	\begin{equation*}
3876	\mathbb{R}^+\setminus\mathbb{R}^\cup_\aleph\neq\varnothing~~.
3877	\end{equation*}
3878\end{mainthm}
3879
3880
3881
3882\begin{proof} 
3883	$\mathbb{R}^+$ 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.
3884\end{proof}
3885
3886
3887
3888
3889
3890
3891\subsection{Cantor-like Sets of Real Numbers}\label{sec:bbbhhv}
3892
3893In 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.   
3894
3895
3896\begin{defin}\label{def:cantordddd}
3897	Munkres constructs a Cantor set $C$ as follows \cite{MUNK}.  
3898	
3899	\begin{quote}
3900		``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
3901		\begin{equation*}
3902		A_n=A_{n-1}-\bigcup\limits_{k=0}^\infty\left(\frac{1+3k}{3^n},\frac{2+3k}{3^n}\right)~~.
3903		\end{equation*}
3904		
3905		\noindent The intersection 
3906		\begin{equation*}
3907		C=\bigcap\limits_{n\in\mathbb{Z}^+}A_n~~,
3908		\end{equation*}
3909		
3910		\noindent is called the [\textit{ternary}] Cantor set; it is a subspace of $[0,1]$.''
3911	\end{quote}
3912\end{defin}
3913 
3914
3915\begin{rem}
3916	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.
3917\end{rem}
3918
3919
3920
3921\begin{defin}\label{def:v8v87wtugiddd}
3922	Define two Cantor-like sets
3923	\begin{equation*}
3924	\mathbb{F}_0=[0,\infty]\setminus\mathbb{R}^\cup_0~~,\qquad\text{and}\qquad \mathbb{F}_\aleph=[0,\infty]\setminus\mathbb{R}^\cup_\aleph~~.
3925	\end{equation*}
3926\end{defin}
3927
3928\begin{cor}
3929	Neither $\mathbb{F}_0$ nor $\mathbb{F}_\aleph$ is the empty set.
3930\end{cor}
3931
3932\begin{proof}
3933	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 theorem is proven.
3934\end{proof}
3935
3936\begin{rem}
3937	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.
3938\end{rem}
3939
3940
3941\begin{lem}\label{thm:utt2u2g211} 
3942	$\mathbb{F}_\aleph$ is a subset of $\mathbb{F}_0$ or it is exactly equal to $\mathbb{F}_0$, \textit{i.e.}: $\mathbb{F}_\aleph\subseteq\mathbb{F}_0$.
3943\end{lem}
3944
3945\begin{proof}
3946	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}.)
3947\end{proof}
3948
3949  
3950
3951\begin{thm}\label{thm:i8758755757}
3952	$\mathbb{F}_0$ and $\mathbb{F}_\aleph$ are closed subsets of $[0,\infty]$.
3953\end{thm}
3954
3955\begin{proof}
3956	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 $[0,\infty]$.
3957\end{proof}
3958
3959\begin{rem}
3960	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.
3961\end{rem}
3962
3963\begin{defin}\label{def:7t8787xx}
3964	The connected elements of $\mathbb{F}_0$ are 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
3965	\begin{equation*}
3966	\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~~.
3967	\end{equation*}
3968	
3969	\noindent Each $\mathbb{F}(n)$ is connected and every two $\mathbb{F}(n),\mathbb{F}(m)$ are disconnected whenever $n\neq m$.
3970\end{defin}
3971
3972\begin{rem}
3973	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 $\mathbb{F}(n)$.  
3974\end{rem}
3975 
3976\begin{pro}\label{thm:2323353ddd}
3977	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. 
3978\end{pro} 
3979
3980
3981\begin{just}
3982	This proposition regards the extrema of a set of sets so those extrema will be sets themselves.  We will neglect the subscripts $0$ and $\aleph$ in this proof.  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
3983	\begin{equation*}
3984	\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)\,...~~.
3985	\end{equation*}
3986	
3987	\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. 
3988	
3989	\begin{defin}\label{def:jhu777}
3990		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
3991	\begin{equation*}
3992		\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~~.
3993	\end{equation*}
3994	
3995	\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, $\aleph(1)=\aleph_0=0$.
3996	\end{defin}
3997
3998	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
3999	\begin{equation*}
4000	[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) \,...~~.
4001	\end{equation*}
4002	
4003	\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}.
4004\end{just}
4005
4006
4007
4008\begin{rem}
4009	In Example \ref{exa:f42tl2444}, we considered the limits 
4010	\begin{equation*}
4011	\lim\limits_{ b\to\infty} \aleph_\mathcal{X}-b=-\infty~~,\qquad\text{and}\qquad\lim\limits_{ b\to\widehat\infty} \aleph_\mathcal{X}-b=\aleph_\mathcal{X}-\widehat\infty=-\aleph_{(1-\mathcal{X})}~~,
4012	\end{equation*}
4013	
4014	\noindent as two desirable modes of limit behavior.  Now the $\mathbb{F}(n)$ notation suggests a third desirable behavior such that 
4015	\begin{equation*}
4016	\lim\limits_{ b\to\aleph(2)} \aleph(n)+b=\aleph(n+1)~~.
4017	\end{equation*}  
4018	
4019	\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 $\frac{d}{dx}\aleph_x=\infty$ (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.  
4020	
4021	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.   
4022\end{rem}
4023
4024\begin{defin}
4025	To avoid the problematic $(n)$ notation, label each connected element fo $\mathbb{F}$ as $\mathbb{F}^\mathcal{X}$.  For every $\mathbb{R}^\mathcal{X}$, there exists a unique $\mathbb{F}^\mathcal{X}$ such that
4026	\begin{equation*}
4027	x\in\mathbb{R}^\mathcal{X}~~,~~ z\in\mathbb{F}^\mathcal{X}\quad\implies\quad x<z~~, 
4028	\end{equation*}
4029	
4030	\noindent and 	
4031	\begin{equation*}
4032	 y\in\mathbb{R}^\mathcal{Y}~~,~~\mathcal{Y}>\mathcal{X}\quad\implies\quad y>z~~.
4033	\end{equation*}
4034	
4035	\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}.
4036\end{defin}
4037
4038
4039 
4040
4041
4042
4043\subsection{Paradoxes Related to Infinitesimals}\label{sec:pdoxes}
4044
4045In 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 and the other paradoxes are resolved in Section \ref{sec:r3r23r23r3r}.
4046
4047
4048\begin{defin}
4049	$\mathcal{F}(n)\in\mathbb{R}$ is the unique real number in the center of $\mathbb{F}_0(n)$ and $\mathbb{F}_\aleph(n)\subset\mathbb{F}_0(n)$ 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 $\mathbb{F}_\aleph^\mathcal{X}$.
4050\end{defin}
4051
4052
4053\begin{thm}
4054	The number $\mathcal{F}(1)=\mathcal{F}^0$ has infinitesimal fractional magnitude with respect to $\mathbf{AB}$.
4055\end{thm}
4056 
4057\begin{proof}
4058	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
4059	\begin{equation*}
4060	\forall x\in\mathbb{R}^+ \quad\exists \varepsilon\not\in\mathbb{R}\quad\text{s.t.}\quad 0<\varepsilon<x~~.
4061	\end{equation*}
4062	
4063	\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.
4064\end{proof}
4065
4066\begin{defin}
4067	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.
4068\end{defin}
4069
4070
4071\begin{thm}
4072	Every $x\in\mathbb{F}$ is an immeasurable number.
4073\end{thm}
4074
4075\begin{proof}
4076	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 $X\in \mathbf{AB}$.
4077\end{proof}
4078
4079 
4080\begin{rem}\label{rem:87585zzzz}
4081	  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.
4082\end{rem}
4083
4084
4085
4086
4087
4088
4089\begin{pdox}\label{pdox:332323}
4090	Every $\mathcal{F}(n)$ has the property
4091	\begin{equation}\label{pdox:t875874rsss}
4092	\mathcal{F}(n)=\cfrac{\aleph(n)+\aleph(n+1)}{2}~~.
4093	\end{equation}
4094	
4095	\noindent Every $\mathbb{R}_0^\mathcal{X}$ can be obtained by a translation operation on another element of $\{\mathbb{R}^\mathcal{X}_0\}$.  Doing setwise arithmetic, we may write, for instance
4096	\begin{equation*}
4097	\hat{T}(\aleph_\delta)\mathbb{R}_0^{(\mathcal{X}-\delta)}=\aleph_\delta+\mathbb{R}_0^{(\mathcal{X}-\delta)}=\mathbb{R}_0^\mathcal{X}~~.
4098	\end{equation*}
4099	
4100	\noindent Letting $AB\equiv [\aleph(n),\aleph(n+1)]$ for some $n\geq 2$, the translational invariance requires 
4101	\begin{equation*}
4102	\text{len} \big( \mathbb{R}_0(n)\cap AB \big)=\text{len} \big( \mathbb{R}_0(n+1)\cap AB\big)~~.
4103	\end{equation*}
4104	
4105	\noindent Since 
4106	\begin{equation*}
4107	AB=\big\{\mathbb{R}_0(n)\cap AB\big\}\cup\mathbb{F}_0(n)\cup\big\{\mathbb{R}_0(n+1)\cap AB\big\}~~,
4108	\end{equation*}
4109	
4110	\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 $\mathbb{R}_0(n)$ 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 
4111	\begin{equation*}
4112	\mathcal{F}(n)=\cfrac{\aleph_\mathcal{X}+\aleph_\mathcal{Y}}{2}=\aleph_{\left(\frac{\mathcal{X}+\mathcal{Y}}{2}\right)}~~.
4113	\end{equation*}
4114	
4115	\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 $\aleph_\mathcal{X}$.
4116\end{pdox} 
4117
4118\begin{pres}
4119	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.}:
4120	\begin{equation*}
4121	\mathcal{F}^\mathcal{X}\neq\cfrac{\aleph^\mathcal{X}+\aleph^\mathcal{Y}}{2}~~.
4122	\end{equation*}
4123\end{pres}
4124
4125\begin{pdox}\label{pdox:3323v3}
4126	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 
4127	\begin{equation*}
4128	\text{len}\big(\mathbb{R}_0(1)\big)=\frac{1}{2}\,\text{len}\big(\mathbb{R}_0(2)\big)~~,\qquad\text{and}\qquad \mathcal{F}(1)=\frac{1}{3}\,\mathcal{F}(2)~~.
4129	\end{equation*}
4130	
4131	\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.  This contradicts the ordering 
4132	\begin{equation*}
4133	x\in\mathbb{R}_0(n)\quad\implies\quad x<\mathcal{F}(n)~~.
4134	\end{equation*}
4135\end{pdox}
4136
4137
4138\begin{pres}
4139	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 $\mathcal{F}(2)$.
4140\end{pres}
4141
4142
4143\begin{pdox}\label{pdox:33232vv3}
4144	 Each $\mathbb{F}_0(n)\subset \mathbb{F}_0$ is a closed, connected interval.  It is required, then, that
4145	\begin{equation*}
4146	\mathbb{F}_0(1)=[a,b]~~,\qquad\text{and}\qquad \mathcal{F}(1)=\frac{b-a}{2}~~.
4147	\end{equation*}
4148	
4149	\noindent It immediately follows that 
4150	\begin{equation*}
4151	\sup\mathbb{R}_0=b-2\mathcal{F}(1)=a~~.
4152	\end{equation*}
4153	
4154	\noindent This is paradoxical for the reasons presented in Proposition \ref{mthm:u9999979y}: $\mathbb{R}_0$ ought not have a supremum.
4155\end{pdox}
4156
4157
4158\begin{pdox}\label{pdox:33232vv4}
4159	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
4160	\begin{equation*}
4161	\mathcal{F}(1)=\frac{\aleph_{\mathcal{X}_\text{min}}}{2}~~,
4162	\end{equation*}
4163	
4164	\noindent is inadmissibly paradoxical.
4165\end{pdox}
4166
4167 
4168 
4169\subsection{Complements of Natural Neighborhoods }\label{sec:bbbhh} 
4170 
4171In 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:
4172\begin{itemize}
4173	\item  We have defined $\mathbb{R}_0^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}=\mathbb{R}_\aleph^\mathcal{X}$.
4174	\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.   
4175	\item We have not yet defined any arithmetic operations for $x\in\mathbb{F}^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}$
4176	\item We have not yet given an equivalence class construction for $x\in\mathbb{F}^\mathcal{X}\cup\mathbb{R}_C^\mathcal{X}$.
4177\end{itemize}
4178
4179\begin{rem}
4180	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.
4181\end{rem}
4182
4183\begin{thm}\label{thm:85yfhhfeee}
4184	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 $\text{len}[0,\mathcal{F}^0]\lll\text{len}\,\mathbf{AB}$.
4185\end{thm}
4186 
4187
4188
4189\begin{figure}[t]
4190	\begin{center}
4191		\includegraphics[scale=0.22]{numline.png}
4192		\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. }\label{fig:f9t99}
4193	\end{center}
4194\end{figure}
4195
4196
4197\begin{proof}
4198	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 $m\in\mathbb{N}$
4199	\begin{equation*}
4200	\cfrac{AX}{AB}-\cfrac{AA}{AB}>\frac{1}{m}~~.
4201	\end{equation*}
4202	
4203	\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 $AB\equiv[0,\mathcal{F}^0]$.
4204\end{proof}
4205
4206
4207\begin{cor}\label{thm:rgrrggr3rg}
4208	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
4209	\begin{equation}
4210	\mathcal{D}_{\!AB}(AX)=0~~.\nonumber
4211	\end{equation}
4212\end{cor}
4213
4214
4215\begin{proof}
4216	By the property $\mathbb{R}_0^0\subseteq\mathbb{R}_\aleph^0$, proof follows from Theorem \ref{thm:85yfhhfeee}.
4217	
4218	$~$
4219	
4220	Alternatively, $\mathcal{D}_{\!AB}$ is such that	
4221	\begin{equation}
4222	\mathcal{D}_{\!AB}(AX)=\mathcal{D}^\dagger_{\!AB}(AX)=\cfrac{x}{\mathcal{F}_1}~~.\nonumber
4223	\end{equation}
4224	
4225	\noindent The case of $x=0$ is trivial.  To prove the other cases by contradiction, suppose $z>0$ and that 	
4226	\begin{equation}
4227	\cfrac{x}{\mathcal{F}^0}=z~~.\nonumber
4228	\end{equation}
4229	
4230	\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	
4231	\begin{equation}
4232	\cfrac{x}{z\mathcal{F}^0}=1\quad\iff\quad z^{-1}x=\mathcal{F}^0~~.\nonumber
4233	\end{equation}
4234	
4235	\noindent This is a contradiction because $z^{-1}x\in\mathbb{R}_0$ but $\mathcal{F}^0$ is greater than any $\mathbb{R}_0$. (If $x$ was allowed to have a non-zero big part $\aleph_\mathcal{X}$, then we could not write the ``if and only if'' relation here because division is many-valued in such cases.  However, the condition $x\in\mathbb{R}_0$ requires single-valuedness for non-zero quotients.)
4236\end{proof}
4237
4238\begin{rem}
4239	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 $AB=[0,\mathcal{F}^0]$, we could define a set of whole neighborhoods along $AB$
4240	\begin{equation*}
4241	\mathbb{R}^\mathcal{X}_\mathcal{F}=\big\{ \digamma_{\!\mathcal{X}}+b~\big|~ b\in\mathbb{R}^0_\aleph  \big\}~~,
4242	\end{equation*}
4243	
4244	\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 $[0,\infty]\to[0,\mathcal{F}^0]$.  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 \textit{ad infinitem} and $\mathbb{R}^0_\aleph$ would never leave the left endpoint.  
4245	
4246	To accommodate the interpretation of the positive branch of $\mathbb{R}$ as a Euclidean line segment, 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
4247	\begin{equation*}
4248	\mathbb{F}(n)\stackrel{?}{=}  \big\{  a\mathcal{F}(n)+b  ~\big|~ a,|b|\in\mathbb{R}_\aleph^0  \big\}~~,
4249	\end{equation*}  
4250	
4251	\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, $\mathbb{R}_C(n)$ being the first, we ought to combine them into a single mystery set.  We have not proven that $\mathbb{R}_\aleph(n)\setminus\mathbb{R}_0(n)=\varnothing$ but neither have we proven the existence of such numbers as we have with $\aleph_\mathcal{X}$ and, hopefully, $\mathcal{F}(n)$.  
4252	
4253	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 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 $\mathbb{F}_0=\mathbb{F}_\aleph$.  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 $\mathbb{F}$.
4254\end{rem}
4255
4256
4257\begin{conj}\label{conj:ZZjjj3j3333}
4258	Every number having zero fractional magnitude with respect to $\mathbf{AB}$ is an element of $\mathbb{R}_0$.  In other words,
4259	\begin{equation*}
4260	\mathbb{R}^\mathcal{X}_\aleph=\mathbb{R}^\mathcal{X}_0~~,\qquad\text{and}\qquad \mathbb{R}_C^\mathcal{X}=\varnothing~~.
4261	\end{equation*}
4262\end{conj}
4263
4264\begin{rem}
4265	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.
4266\end{rem}
4267 
4268 
4269 
4270  
4271 
4272 \subsection{Dedekind Cuts and The Least Upper Bound Property}\label{sec:r3r23r23r3r}
4273 
4274In this section, we will give an algebraic definition for $\mathcal{F}^\mathcal{X}$ as a partition of $C^\mathbf{AB}_\mathbb{Q}$.  We will resolve the paradoxes of Section \ref{sec:pdoxes} by making a distinction between arithmatic and non-arithmatic numbers.
4275
4276
4277
4278\begin{defin}
4279	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.
4280	\begin{itemize}
4281		\item $L$ is non-empty.
4282		\item $L\neq\mathbb{Q}$
4283		\item If $x,y\in\mathbb{Q}$, if $x<y$, and if $y\in L$, then $x\in L$.
4284		\item If $x\in L$, then there exists $y\in L$ such that $y>x$.
4285	\end{itemize}
4286\end{defin}
4287 
4288 \begin{rem}
4289 	Now we will extend the Dedekind cut as a partition of $C_\mathbb{Q}^{\mathbf{AB}}$.
4290 \end{rem}
4291 
4292 
4293 
4294 \begin{defin}
4295 	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.
4296 	\begin{itemize}
4297 		\item $L$ is non-empty.
4298 		\item $L\neq C_\mathbb{Q}^\mathbf{AB}$
4299 		\item If $x,y\in C_\mathbb{Q}^\mathbf{AB}$, if $x<y$, and if $y\in L$, then $x\in L$.
4300 		\item If $x\in L$, then there exists $y\in L$ such that $y>x$.
4301 	\end{itemize}
4302 \end{defin}
4303
4304
4305\begin{thm}\label{thm:sdwd1cdwc}
4306	The number $\mathcal{F}^0$ is an extended Dedekind partition of $C_\mathbb{Q}^\mathbf{AB}$.
4307\end{thm}
4308
4309\begin{proof}
4310	Let
4311	\begin{equation*}
4312	L=\big\{  [x]\subset C_\mathbb{Q}^\mathbf{AB}   ~\big|~  \text{Big}(x)=0 \big\}~~,\qquad\text{and}\qquad R=\big\{  [x]\subset C_\mathbb{Q}^\mathbf{AB}   ~\big|~  \text{Big}(x)>0 \big\}~~,
4313	\end{equation*}
4314	
4315	\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.
4316\end{proof}
4317 
4318\begin{axio}
4319	$\mathcal{F}^\mathcal{X}$ is the only number in $\mathbb{F}^\mathcal{X}\subset \mathbb{R}$.
4320\end{axio}
4321 
4322\begin{rem}
4323	Now we have 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 $C_\mathbb{Q}^\mathbf{AB}$.  The arithmetic of such numbers is given in Section \ref{sec:5}.  Following the form of Theorem \ref{thm:sdwd1cdwc}, every immeasurable $\mathcal{F}^\mathcal{X}$ is now formally constructed as a partition of $C_\mathbb{Q}^\mathbf{AB}$.  Since the arithmetic axioms are proven well-defined for $[x]\subset C^\mathbf{AB}_\mathbb{Q}$ in Section \ref{sec:consXXX}, we cannot simply throw $\mathcal{F}^\mathcal{X}$ in there.  There is not an equivalence class $[\mathcal{F}^\mathcal{X}]\subset C_\mathbb{Q}^\mathbf{AB}$ for every (or any) $[\mathcal{X}]\subset C_\mathbb{Q}$.  Even if we forced arithmetic onto $\mathcal{F}^\mathcal{X}$ with more axioms, we would immediately hit the least upper bound problem: the issue raised in Proposition \ref{mthm:u9999979y}.  Theorem \ref{thm:sdwd1cdwc} clearly defines $\mathcal{F}^0$ as the supremum of $\mathbb{R}_0$.  If arithmetic is defined for $x\in\mathbb{F}$ in the usual way, then $\mathcal{F}^0-1\in\mathbb{R}_0$.  Call that number $x$.  By the closure of $\mathbb{R}_0$, $x+2$ is also an element of $\mathbb{R}_0$.  This contradicts the identity $\mathcal{F}^0=\sup(\mathbb{R}_0)$.  
4324\end{rem}
4325
4326
4327
4328\begin{defin}
4329	If a real number is in $[x]\subset C^\mathbf{AB}_\mathbb{Q}$, then it is called an arithmatic number (pronounced arith-matic.)  All measurable numbers are arithmatic.  If a a real number is a partition of $C^\mathbf{AB}_\mathbb{Q}$ not given by any element of that $C^\mathbf{AB}_\mathbb{Q}$, then it is called a non-arithmatic number.  All immeasurable numbers are non-arithmatic.
4330\end{defin}
4331
4332\begin{axio}\label{ax:f9tw9ef5}
4333	Arithmetic operations are not defined for non-arithmetic numbers.
4334\end{axio}
4335
4336
4337
4338
4339
4340\begin{axio}\label{ax:69957d5c}
4341	The ordering of $\mathbb{R}$ is such that $\mathcal{F}^\mathcal{X}=\sup(\mathbb{R}_0^\mathcal{X})$
4342\end{axio}
4343
4344
4345\begin{defin}
4346	If every $S\subset F$ has a least upper bound $\sup(S)\in F$, then $F$ is said to have the least upper bound property.  All 1D connected intervals have the least upper bound property.
4347\end{defin}
4348
4349\begin{pro}
4350	(The least upper bound problem: Restatement of Proposition \ref{mthm:u9999979y}.)  $\mathbb{R}$ does not have the least upper bound property because $\mathbb{R}_0$ cannot have a supremum.
4351\end{pro}
4352
4353\begin{refut}
4354	Axioms \ref{ax:f9tw9ef5} and \ref{ax:69957d5c} are such that
4355	\begin{equation*}
4356	\mathcal{F}=\sup(\mathbb{R}_0)\quad\implies\quad\sup(\mathbb{R}_0)\pm x=\text{undefined}~~.
4357	\end{equation*}
4358	
4359	\noindent The arithmetic can not be used to demonstrate the condition in the justification of this proposition and $\mathbb{R}_0$ most certainly can have a supremum.  The supremum of each open set of numbers that are $100\times\mathcal{X}\%$ of the way down the real number line is $\mathcal{F^X}$, a non-arithmatic number.
4360\end{refut}
4361
4362\begin{rem}
4363	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 all numbers were supposed to be algebraic until the connectedness of the interval demanded non-algebraic numbers to fill in the gaps.  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 polynomials with rational coefficients, and non-algebraic numbers which are not.  Non-algebraic numbers are supposed to exist because they are needed to fill in the gaps 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
4364	
4365	
4366\end{rem}
4367
4368
4369
4370
4371
4372
4373
4374\begin{pres}\label{pdox:33232vv3aaa}
4375	(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 $\mathcal{F^X}$.
4376\end{pres}
4377
4378
4379\begin{pres}\label{pdox:33232vv4qeve}
4380	(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 $\mathcal{F^X}$.
4381\end{pres}
4382
4383\begin{rem}
4384	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 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.  
4385	
4386	Measurable and immeasurable numbers are so markedly different in their qualia that it is certainly reasonable.  Keeping in mind that $\mathbb{R}^\cup$ numbers with zero little part are squarely in the middle of the natural neighborhoods, if we specified $\mathcal{F^X}$ as a Dedekind partition with a little part, then $\mathcal{F^X}$ would be forced into the arithmetic axioms through the proofs in Section \ref{sec:consXXX}.  The immeasurable numbers are the only possible exception for an ordered element of $(-\widehat\infty,\widehat\infty)$ to be a cut in the real number line and somehow not automatically thrown into the arithmetic axioms.  
4387	
4388	DID I WANDER OFF HERE FROM WHERE I WAS GOING?
4389	
4390	GO BACK TO IT!	
4391\end{rem}
4392
4393arithmetic is inherently geometric
4394
4395MORE BLURB
4396
4397===================
4398
4399
4400
4401
4402
4403
4404============
4405
4406SHOW $\mathcal{F}$ SATISFIES ARCHIMEDES
4407
4408
4409let the $\mathcal{F^X}$ be the natural numbers on a higher level of $\aleph$
4410
4411The number $\mathcal{F}^\mathcal{X}$ sews together the connected intervals by the ordering relation
4412
4413metric becomes undefined -- breaks the metric space!!!
4414
4415=========
4416
4417REALS ARE ``TOTALLLY ORDERED''
4418
4419PROVE THAT THAT ORDERING RELATION DEFINED AS THE SUPREMUM WORKS WELL IN THE ABSENCE OF GEOMETRIC ARITHMETIC
4420
4421
4422 
4423 \subsection{The Topology of the Real Number Line}\label{sec:topo}
4424 
4425 
4426The 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     
4427\begin{equation}
4428\mathbb{R}=(-\infty,\infty)~~.\nonumber
4429\end{equation}
4430
4431\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?
4432
4433We 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.
4434
4435\begin{axio}
4436	The fundamental axiom of algebraic construction.  The basis of the topology of the real number line is the usual one.  
4437\end{axio}
4438
4439
4440
4441\begin{defin}\label{def:762222}
4442	The basis $B_0$ for the usual topology on $\mathbb{R}$ is the collection of all 1D open intervals such that
4443	\begin{equation*}
4444	B_0=\big\{  (a,b)  ~\big|~  [a],[b] \subset C_\mathbb{Q} ,~ a<b\big\}~~.
4445	\end{equation*} 
4446	
4447	The conectedness of $\mathbb{R}$ is implicit in the interval notation.	
4448\end{defin}
4449
4450\begin{rem}
4451	Definition \ref{def:762222} is such that ``having the usual topology'' is exactly equal to Axiom \ref{ax:mainR} granting that $\mathbb{R}=(-\infty,\infty)$.
4452\end{rem}
4453
4454
4455\begin{axio} \label{ax:mainR2}
4456	The fundamental axiom of geometric construction.  Non-negative real numbers are algebraic representations of points in an infinitely long line segment, \textit{i.e.}: 
4457	\begin{equation} 
4458	\mathbb{R}^+=\big\{  x ~\big|~ x\in X\in\mathbf{AB} ,~x>0,~x<\widehat\infty \big\}~~.\nonumber
4459	\end{equation}
4460\end{axio}
4461
4462
4463
4464
4465\begin{defin}
4466	The basis for the fractional distance topology on $\mathbb{R}$ is $B=B_\mathcal{X}\cup B_\infty$ such that 
4467	\begin{align*}
4468	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\}\\
4469	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\}~.
4470	\end{align*}   
4471\end{defin}
4472
4473
4474\begin{thm}
4475	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.
4476\end{thm}
4477
4478\begin{proof}
4479	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 
4480	\begin{equation*}
4481	B_{C_\mathbb{{Q}}}=\big\{    (a,b)    ~\big|~  [a],[b]\subset C_\mathbb{Q},~[a]<[b]  \big\}~~,
4482	\end{equation*}
4483	
4484	\noindent one for each of an uncountably infinite number of $[\mathcal{X}]$.
4485\end{proof}
4486
4487
4488\begin{thm}
4489	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 distane topology is more coarse than the usual topology.
4490\end{thm}
4491
4492\begin{proof}
4493	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 $C_\mathbb{Q}^\mathbf{AB}$.
4494\end{proof}
4495
4496\begin{rem}
4497	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.
4498\end{rem}
4499
4500\begin{axio}\label{ax:chvhvhvh1}
4501	A number is a cut in a real number line and there exist an infinite number of such lines, well-ordered.
4502\end{axio}
4503
4504\begin{axio}\label{ax:chvhvhvh}
4505	The real numbers as topological space are called $X$.  The topological space is
4506	\begin{equation*}
4507	X=\big\{(-\aleph_1,\aleph_1);\mathbb{T}\big\}~~,
4508	\end{equation*}
4509	
4510	\noindent where 
4511	\begin{equation*}
4512	(-\aleph_1,\aleph_1)=(-\widehat\infty,\widehat\infty)\qquad\text{and}\qquad \mathbb{T}=(-\aleph_\infty,\aleph_\infty)=(-\infty,\infty)~~.
4513	\end{equation*}
4514\end{axio}
4515
4516\begin{rem}
4517	Is a geometric point $X$ also a topological space $X$?
4518	
4519	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}.)  
4520	 
4521	
4522	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 $\mathbb{F}$.
4523
4524		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.
4525	\end{rem}
4526	
4527	\begin{axio}\label{ax:vaxcjdco}
4528		Every $x\in\mathbb{R}$ may be constructed algebraically as a Cartesian product of Cauchy equivalence classes of rational numbers.
4529	\end{axio}
4530	
4531	\begin{rem}
4532		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.
4533	\end{rem}
4534	
4535	
4536	============
4537	
4538	TWO WAYS OF DEFINING THE TOPOLOGICAL SPACE:
4539	 metric space
4540	 
4541	 
4542	 \begin{rem}
4543	 	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?
4544	 \end{rem}
4545
4546 
4547
4548\section{The Riemann Hypothesis}\label{sec:g54584688}
4549
4550 
4551The 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}.  
4552	
4553When 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
4554\begin{equation*}
4555\mathbb{R}\setminus x=(-\infty,x)\cup(x,\infty)~~,
4556\end{equation*}
4557	
4558	
4559\noindent as an alternative identical formulation of the Euclidean statement 
4560\begin{equation*}
4561x\in\mathbb{R}^+\quad\iff\quad(0,\infty)=(0,x]\cup(x,\infty)~~.
4562\end{equation*}
4563	
4564\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 $z$.
4565 
4566
4567
4568\subsection{Non-trivial Zeros in the Critical Strip}
4569
4570In this section, we will prove the negation of the Riemann hypothesis.
4571 
4572
4573
4574\begin{defin}
4575	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
4576	\begin{align}
4577	\zeta(z)&=\sum_{n=1}\frac{1}{n^z}~~.\nonumber
4578	\end{align}	
4579	 
4580	\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}
4581	\begin{equation}
4582	\zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~.\nonumber
4583	\end{equation}
4584\end{defin}
4585
4586\begin{rem}
4587	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.
4588\end{rem}
4589
4590
4591\begin{pro}\label{prop:68wdf9t}
4592	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.
4593\end{pro}
4594
4595\begin{refut}
4596	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
4597	\begin{equation*}
4598	\big|z-z_0\big|>\big(\aleph_\mathcal{X}+a\big)~~,
4599	\end{equation*}
4600	
4601	\noindent to show that the Taylor series does not converge (when $\mathcal{X}>0$.)  We have
4602	\begin{align*}
4603	f(z)&=f(z_0)+f'(z_0)\big(z-z_0\big)+\dfrac{f''(z_0)}{2!}\big(z-z_0\big)^2+...~~.
4604	\end{align*}
4605	
4606	\noindent The first term in the series vanishes by definition and so, therefore, we have by assumption
4607	\begin{align*}
4608	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 ~~.
4609	\end{align*}
4610	
4611	\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}.
4612\end{refut}
4613
4614
4615
4616\begin{axio}\label{ax:yguib}
4617	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.
4618\end{axio}
4619
4620\begin{rem}
4621	Various proofs of Axiom \ref{ax:yguib} are well-known.  They are taken for granted.
4622\end{rem}
4623
4624\begin{mainthm}\label{thm:11qqgg4gq}
4625	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
4626	\begin{equation*}
4627	\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\big|\gamma_{n+1}-\gamma_n\big|=0~~.
4628	\end{equation*}
4629\end{mainthm}
4630
4631 
4632
4633\begin{proof}
4634	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.
4635	
4636	\begin{quote}
4637		``For every large $T$, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying
4638		\begin{equation*}
4639		|\gamma-T|<\dfrac{A}{\log\log\log T}~~.\text{''}
4640		\end{equation*}
4641	\end{quote}
4642	
4643	\noindent Note that $A$ is some constant $A\in\mathbb{R}_0$.  For proof by contradiction, assume
4644	\begin{equation*}
4645	\lim\limits_{n\to(\aleph_\mathcal{X}+b)}\big|\gamma_{n+1}-\gamma_n\big|\neq0~~.
4646	\end{equation*}
4647	
4648	\noindent Then there exists some $m(n)$ and some $a \in \mathbb{R}_0^+$ such that
4649	\begin{equation*}
4650	\lim\limits_{m(n)\to(\aleph_\mathcal{X}+b)}\big|\gamma_{m(n)+1}-\gamma_{m(n)}\big|>2a~~.
4651	\end{equation*}
4652	
4653	\noindent Let $T_n$ be the average of $\gamma_{m(n)+1}$ and $\gamma_{m(n)}$ so
4654	\begin{equation*}
4655	T_n=\dfrac{\gamma_{m(n)+1}+\gamma_{m(n)}}{2}~~.
4656	\end{equation*}
4657	
4658	\noindent Now we have
4659	\begin{equation*}
4660	\lim\limits_{T_n\to(\aleph_\mathcal{X}+b)}\big| \gamma-T_n  \big|>a~~,
4661	\end{equation*}
4662	
4663	\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
4664	\begin{equation*}
4665	|\gamma-T_n|<\dfrac{A}{\log\log\log T_n}~~,\qquad\text{whenever}\qquad\dfrac{A}{\log\log\log T_n}<a~~.
4666	\end{equation*}
4667	
4668	\noindent The limit $T_n \to \aleph_\mathcal{X}+b$ is exactly such a case because
4669	\begin{equation*}
4670	\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)~~.
4671	\end{equation*}
4672	
4673	\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.
4674\end{proof}
4675
4676
4677
4678\begin{rem}
4679	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}.
4680\end{rem}
4681
4682
4683
4684\begin{cor}\label{mthm:llool}
4685	The Riemann $\zeta$ function has zeros within the critical strip yet off the critical line.  
4686\end{cor}
4687
4688\begin{proof}
4689	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.  
4690\end{proof}
4691
4692
4693\subsection{Non-trivial Zeros in the Neighborhood of Minus Infinity}
4694
4695
4696The 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. 
4697
4698
4699
4700
4701\begin{thm}\label{thm:5y74757}
4702	The Riemann $\zeta$ function is equal to one for any $\text{Re}(z_0)\in\mathbb{R}^\mathcal{X}_\aleph$ such that $0<\mathcal{X}\leq1$.
4703\end{thm}
4704
4705\begin{proof}
4706	Observe that the Dirichlet sum form of $\zeta$
4707	\begin{equation}
4708	\zeta(z)=\sum_{n\in\mathbb{N}}\cfrac{1}{n^z}~~,\nonumber
4709	\end{equation}
4710	
4711	\noindent  takes $z_0=\big(\aleph_\mathcal{X}+b\big)+iy_0$ as	
4712	\begin{align}
4713	\zeta(z_0)&=\sum_{n=1}\cfrac{1}{n^{(\aleph_\mathcal{X}+b)+iy_0}}\nonumber\\
4714	&=\sum_{n=1}\cfrac{n^{-b}n^{-iy_0}}{n^{\aleph_\mathcal{X}}}\nonumber\\
4715	&=\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\\
4716	&=1+\sum_{n=2}\cfrac{n^{-b}}{\widehat\infty}\bigg(\cos(y_0\ln n)-i\sin(y_0\ln n)\bigg)=1~~.\nonumber
4717	\end{align}	
4718\end{proof}
4719
4720\begin{mainthm}\label{thm:5rg3r3rf757}
4721	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.
4722\end{mainthm} 
4723
4724\begin{proof}
4725	Riemann's functional form of $\zeta$ \cite{RIEMANN} is
4726	\begin{equation}
4727	\zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~.\nonumber
4728	\end{equation}
4729	
4730	\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
4731	\begin{align}
4732	\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\\
4733	&=\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
4734	\end{align}
4735	
4736	\noindent For the limit involving $\Gamma$, we will compute the limit as a product of two limits.  We separate terms as
4737	\begin{align}
4738	\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
4739	\end{align}
4740	
4741	\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 
4742	\begin{equation}
4743	1=\cfrac{z-(\aleph_\mathcal{X}+b)}{z-(\aleph_\mathcal{X}+b)}~~.\nonumber
4744	\end{equation}
4745	
4746	
4747	
4748	\noindent Inserting the identity yields
4749	\begin{align}
4750	\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
4751	\end{align}
4752	
4753	\noindent Let
4754	\begin{equation}
4755	A=\Gamma(z)\bigg(z-(\aleph_\mathcal{X}+b)\bigg)~~,\qquad\text{and}\qquad B=\cfrac{(2\pi)^{-z}}{z-(\aleph_\mathcal{X}+b)}~~.\nonumber
4756	\end{equation}
4757	
4758	\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
4759	\begin{align}
4760	\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
4761	\end{align}
4762	
4763	\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
4764	\begin{align*}
4765	\Gamma\big(z\big)&=\Gamma\big(z+1\big)z^{-1}~~.
4766	\end{align*}
4767	
4768	\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
4769	\begin{align}
4770	\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
4771	\end{align}
4772	
4773	\noindent Moving the infinite product to the other side yields
4774	\begin{align}
4775	\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
4776	\end{align}
4777	
4778	\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  	
4779	
4780	\begin{equation}
4781	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
4782	\end{equation}
4783	
4784	\noindent To evaluate the limit of AB, we will take the limits of $A$ and $B$ separately.  The limit of $A$ is
4785	\begin{align*}
4786	\lim\limits_{z\to(\aleph_\mathcal{X}+b)}A&=\Gamma\big[(\aleph_\mathcal{X}+b)-(\aleph_\mathcal{X}+b)+1\big]\times\\
4787	&\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)~~.
4788	\end{align*}
4789	
4790	
4791	\noindent Axiom \ref{ax:plus} gives $(\aleph_\mathcal{X}+b)-(\aleph_\mathcal{X}+b)=0$ so 
4792	\begin{align}
4793	\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
4794	\end{align}
4795	
4796	\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
4797	\begin{align}
4798	\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\\
4799	&=\lim\limits_{z\to(\aleph_\mathcal{X}+b)}\dfrac{d}{dz} e^{-z\ln (2\pi)}\nonumber\\
4800	&=-\ln (2\pi)\ e^{-(\aleph_\mathcal{X}+b)\ln (2\pi)}\nonumber\\
4801	&=-\ln (2\pi)\ \cfrac{e^{-b\ln (2\pi)}}{\big(e^{\mathcal{X}}\big)^{\!\widehat\infty}}\nonumber\\
4802	&= -\ln(2\pi)\cfrac{e^{-b\ln (2\pi)}}{\widehat\infty}\nonumber\\
4803	&=0~~.\nonumber
4804	\end{align}
4805	
4806	\noindent Therefore, we find that the limit of $AB$ is $0$.  It follows that
4807	\begin{align}
4808	\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
4809	\end{align}
4810\end{proof} 
4811
4812
4813
4814 
4815
4816\begin{rem}
4817	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		
4818	\begin{align}
4819	\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
4820	\end{align}
4821	
4822	\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 $\zeta(1-z)$
4823	\begin{equation}
4824	\zeta(z)=\dfrac{\,(2\pi)^z}{\pi}\sin\left(\dfrac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z)~~,\nonumber
4825	\end{equation}
4826	
4827	
4828	\noindent we will compute $\zeta(\aleph_\mathcal{X})$.  Evaluation yields
4829	\begin{align}
4830	\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
4831	\end{align}
4832	
4833	\noindent This equation is undefined and we cannot obtain a contradiction.	
4834\end{rem}
4835
4836
4837
4838\begin{rem}
4839	Patterson writes the following in reference \cite{ZZZ4}.  
4840	
4841	\begin{quote}
4842		``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 
4843		\begin{equation*}
4844		\zeta(s)=\prod_{p \in\text{primes}}\big(1-p^{-s}\big)^{-1}~~.
4845		\end{equation*}
4846		
4847		\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.''
4848	\end{quote}
4849	
4850	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.  
4851\end{rem}
4852
4853
4854\begin{cor}\label{thm:5y74f757}
4855	The Euler product from of $\zeta$ has non-trivial zeros with negative real parts in $\mathbb{R}_\infty$. 
4856\end{cor}
4857
4858
4859
4860
4861\begin{proof}
4862	Consider a number $z_0\in\mathbb{C}$ such that	
4863	\begin{equation}
4864	z_0=-(\aleph_\mathcal{X}+b)+iy_0~~,\qquad\text{where}\qquad b,y_0\in\mathbb{R}_0~~.\nonumber
4865	\end{equation}
4866	
4867	\noindent Observe that the Euler product form of $\zeta$ \cite{EULER2} takes $z_0$ as	
4868	\begin{align}
4869	\zeta(z_0)&=\prod_{p}\cfrac{1}{1-p^{(\aleph_\mathcal{X}+b)-iy_0}}\nonumber\\
4870	&=  \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\\
4871	&=\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~~.
4872	\end{align}
4873	
4874	\noindent Let $y_0\ln P=2n\pi$ for some prime $P$ and $n\in\mathbb{N}$ or $n=0$.  Then
4875	\begin{align}
4876	\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
4877	\end{align} 
4878\end{proof}
4879
4880
4881\begin{conj}
4882	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.
4883\end{conj}
4884
4885
4886
4887
4888\newpage
4889\appendix
4890
4891
4892\numberwithin{thm}{section}
4893
4894\section{Developing Mathematical Systems Historically}
4895
4896
4897Because 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}.
4898
4899\begin{quote}
4900	``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}.''  
4901\end{quote}
4902
4903This 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.
4904
4905\begin{quote}
4906	``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.''
4907\end{quote}
4908
4909\noindent O'Connor and Robertson cite Wallis as writing the following.
4910
4911\begin{quote}
4912	``[S]uch proportion is not to be expressed in the commonly received ways of notation.''
4913\end{quote}
4914
4915Wallis 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.
4916
4917\begin{quote}
4918	``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.''
4919\end{quote}
4920
4921
4922\noindent Still yet further emphasizing the rightful place of new notation in mathematics, O'Connor and Robertson write the following.
4923
4924
4925\begin{quote}
4926	``Strictly speaking, only that which is logically impossible (i.e.: which contradicts itself) counts as impossible for the mathematician.''
4927\end{quote}
4928
4929\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 $\widehat\infty$.  
4930
4931We 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.
4932
4933
4934\begin{quote}
4935	``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.''
4936\end{quote}
4937
4938\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.
4939
4940\begin{quote}
4941	``[\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.''
4942\end{quote} 
4943
4944In 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.
4945
4946In 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.
4947
4948\begin{quote}
4949	``[\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 $b$.''
4950\end{quote}
4951
4952\noindent What Hilbert wrote specifically was the following.
4953
4954\begin{quote}
4955	``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 $B$.''
4956\end{quote}
4957
4958\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.
4959
4960Regarding the very ancient history, O'Connor and Robertson write the following.
4961
4962\begin{quote}
4963	``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.''
4964\end{quote}
4965
4966\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.
4967
4968In 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.
4969
4970\begin{quote}
4971	``[\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.''
4972\end{quote}
4973
4974
4975
4976O'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).
4977
4978$~$
4979
4980\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.
4981
4982\begin{quote}
4983	(RF) In other words, $\alpha$ is said to be a part of $\beta$ if $\beta=m\alpha$.
4984	
4985	(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.
4986	
4987	(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 $n>1$.
4988\end{quote}
4989
4990When 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.
4991
4992Many 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.
4993
4994
4995\begin{axio}\label{ax:fhf924792491}
4996	There exists a non-empty set $\{\mathbb{R}^\mathcal{X}\}$ of real numbers greater than any natural number.
4997\end{axio}
4998
4999\begin{axio}\label{ax:fhf924792492}
5000	There does not exist any real number greater than every natural number so, therefore, $\mathbb{R}\setminus\mathbb{R}_0=\varnothing$.
5001\end{axio}
5002
5003
5004An 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.
5005
5006$~$
5007
5008
5009
5010
5011\noindent \textbf{Book V, Definition 2} And the greater is a multiple of the lesser whenever it is measured by the lesser.
5012
5013\begin{quote}
5014	(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 $10\!:\!9$.
5015	
5016	(OR) Then comes the definition of ratio.
5017\end{quote}
5018
5019
5020$~$
5021
5022\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.
5023
5024\begin{quote}
5025	(RF) In modern notation, the ratio of two magnitudes, $\alpha$ and $\beta$, is denoted $\alpha\,:\,\beta$.
5026	
5027	(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.
5028	
5029	(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.
5030\end{quote}
5031
5032$~$
5033
5034
5035\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.
5036
5037\begin{quote}
5038	(RF) In other words, $\alpha$ has a ratio with respect to $\beta$ if $m\alpha>\beta$ and $n\beta>\alpha$, for some $m$ and $n$.
5039	
5040	(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 
5041	\begin{equation*}
5042	(-1,1)=(-1,0]\cup(0,1)~~,
5043	\end{equation*}
5044	
5045	\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 positve real number would be the greatest real number.
5046	
5047	(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 $b$.
5048\end{quote}
5049
5050$~$
5051
5052
5053\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.
5054
5055\begin{quote}
5056	(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.
5057	
5058	(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.
5059	
5060	(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
5061	\begin{align*}
5062	na > mb \quad&\text{if and only if}\quad nc > md\\
5063	na = mb \quad&\text{if and only if}\quad nc = md\\
5064	na < mb  \quad&\text{if and only if}\quad nc < md~~.
5065	\end{align*}
5066
5067	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.
5068\end{quote}
5069
5070The 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.  
5071
5072\newpage
5073
5074\bibliographystyle{unsrt}
5075\bibliography{TOIC}{}
5076%%%END LARGE 
5077\end{document}