Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers

TL;DR

This paper explores the properties of non-Archimedean field extensions of the real numbers, revealing their rich self-similar structure and potential implications for physics, especially in handling infinities.

Contribution

It introduces ultrapower field extensions of the real numbers, highlighting their self-similar structure and potential relevance to physics, challenging the traditional uniqueness of real and complex fields.

Findings

Ultrapower fields contain infinitesimals and infinitely large elements.

These fields are non-Archimedean with rich self-similar structure.

Potential applications in physics for handling infinities.

Abstract

It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear order structure on the geometric line. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid's Geometry. The consequence of that belief is a similar belief in the uniqueness of the field $\mathbb{C}$ of complex numbers, and therefore, of the various spaces, manifolds, etc., be they finite or infinite dimensional, constructed upon the real or complex numbers, including the Hilbert spaces used in Quantum Mechanics. An near total lack of awareness follows about the {\it rich self-similar} structure of various linearly ordered scalar fields obtained through the {\it ultrapower} construction which extend the usual field $\mathbb{R}$ of real numbers. Such {\it ultrapower field} extensions contain as a rather small subset the usual field $\mathbb{R}$ of real numbers. The rich self-similar structure of such field extensions is due to {\it infinitesimals}, and thus also of {\it infinitely large} elements in such fields, which make such fields {\it non-Archimedean}. With the concept of {\it walkable world}, which has highly intuitive and pragmatic algebraic and geometric meaning, the mentioned rich self-similar structure is illustrated. The ultrapower fields presented can have a wide ranging relevance in Physics, among others, for a proper treatment of what are usually called the "infinities in Physics". The ultrapower construction which gives such non-Archimedean fields is rather simple and elementary, requiring only 101 Algebra.

Full text

**Surprising Properties of Non-Archimedean

Field Extensions of the

Real Numbers**

Elemér E  Rosinger

*Department of Mathematics

and Applied Mathematics*

University of Pretoria

Pretoria

0002 South Africa

eerosinger@hotmail.com

Dedicated to Marie-Louise Nykamp

Abstract

This, under the present form, is a replacement that is a two part paper in which the new second part was brought together with my recently posted arxiv paper, upon the suggestion of the arxiv moderators.

**PART I :

Real Scalars with a Rich Structure

of ”Walkable Worlds” :

Universes next to Universes,

and/or within Universes …

and so on, ad infinitum …**

Abstract

It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear order structure on the geometric line. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid’s Geometry. The consequence of that belief is a similar belief in the uniqueness of the field $\mathbb{C}$ of complex numbers, and therefore, of the various spaces, manifolds, etc., be they finite or infinite dimensional, constructed upon the real or complex numbers, including the Hilbert spaces used in Quantum Mechanics. An near total lack of awareness follows about the rich self-similar structure of various linearly ordered scalar fields obtained through the ultrapower construction which extend the usual field $\mathbb{R}$ of real numbers. Such ultrapower field extensions contain as a rather small subset the usual field $\mathbb{R}$ of real numbers. The rich self-similar structure of such field extensions is due to infinitesimals, and thus also of infinitely large elements in such fields, which make such fields non-Archimedean. With the concept of walkable world, which has highly intuitive and pragmatic algebraic and geometric meaning, the mentioned rich self-similar structure is illustrated. The ultrapower fields presented can have a wide ranging relevance in Physics, among others, for a proper treatment of what are usually called the ”infinities in Physics”. The ultrapower construction which gives such non-Archimedean fields is rather simple and elementary, requiring only 101 Algebra.

0. Preliminaries

It is a rather universal tacit and hardly ever questioned belief - and even more so among physicists - that there is no, and thus there cannot be any other geometric infinite straight line, except the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear order structure. Such a belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid’s Geometry.

On the other hand, as is well known in Mathematics, there are a variety of far larger infinite straight lines which have a far more rich structure, among the better known ones being those given in Topology by the so called long line, or in Analysis by the field ${}^{*}\mathbb{R}$ of nonstandard real numbers.

Needless to say, restricting oneself to scalars given by the usual field $\mathbb{R}$ of real numbers, as well as those built upon them, like the usual field $\mathbb{C}$ of complex numbers, can lead to dramatic effects in thinking in Physics. In particular, it can lead to inappropriate mathematical models of already known concepts, phenomena, processes, etc., or worse yet, it can simply prevent the very emergence of concepts in Physics due to the sheer impossibility of their mathematical modelling based on the dramatically limiting usual field $\mathbb{R}$ of real numbers used since ancient times and till the present day.

The so called ”infinities in Physics” are an example of inappropriate mathematical models, while regarding the concepts in Physics not yet formulated they are but anybody’s guess …

One possible such concept still missing in Physics, and with likely fundamental importance, is that of infinitely many different levels of precision, [13, subsection 3.3].

In this paper, we shall focus on an large class of extensions of the usual field $\mathbb{R}$ of real numbers and of its usual linear order structure with its the geometric line, a class other than the mentioned long line or nonstandard reals. These extensions will be constructed in a surprisingly easy and elementary way, much unlike for instance, the long line or nonstandard real numbers. The respective construction is called the ultrapower construction, and the resulting linearly ordered field extensions of the usual field $\mathbb{R}$ of real numbers are called ultrapower fields.

The natural aspect of the ultrapower construction is in the fact that, as seen in Model Theory, which is a branch of Mathematical Logic, this construction is in fact often present in a large variety of important mathematical contexts. And the general lack of familiarity with that fact is only due to the reduced familiarity with Mathematical Logic, let alone, Model Theory, among mathematicians, and even more so, among physicists.

The importance of the existence and easy, elementary construction of such ultrapower fields is that, contrary to the general view in Physics, the real or complex scalars, as well as the variety of finite or infinite dimensional spaces or manifolds constructed upon them, including Hilbert spaces, can in fact have a surprisingly rich self-similar structure, one that is so far just about completely unknown in Physics, and one that can offer the modelling of a wide range of yet unknown phenomena, as well as the more appropriate modelling of the already known ones.

As mentioned, the consequences of the present highly limiting belief regarding the structure of the geometric straight line, and thus of all the scalars and spaces built upon it, can lead to unfortunate effects in Physics. Among them may simply be the inability to model properly important physical phenomena, and here one can immediately think about the considerable troubles the so called ”infinities in Physics” do cause, among others, in Quantum Field Theory. Yet more grave consequences could also come from the sheer impossibility to conceive of, let alone model properly important physical concepts, phenomena, processes, etc.

Such and relates issue have been addressed in certain of their more general features in [2-14].

In [11-13] it has been shown that certain most basic results in Relativity, Quantum Mechanics and Quantum Computation can in an easy and natural manner be reformulated in terms of the far more rich space-time structures built upon the general ultrapower construction of which the ultrapower fields are but a particular case.

Here, with the help of a highly intuitive and pragmatic algebraic and geometric concept, namely, that of a walkable world, the surprisingly rich self-similar structure of a large class of infinite geometric lines is presented, lines which correspond to linearly ordered ultrapower fields containing the usual field $\mathbb{R}$ of real numbers.

Since the Arithmetisation of Geometry by Descartes, the usual infinite geometric straight line is associated in the well known manner with the field $\mathbb{R}$ of the usual real numbers.

Such an association, fortunately, can easily be extended to hold between various far larger infinite geometric straight lines, lines with a rich self-similar structure, and on the other hand, linearly ordered fields which contain the field $\mathbb{R}$ of the usual real numbers as a rather small subset.

In view of that association, an association which will easily follow from the constructions in the sequel, we shall only deal with the linearly ordered ultrapower field extensions of the field $\mathbb{R}$ of the usual real numbers.

These ultrapower field extensions will be obtained as particular cases of the so called ultrapower construction.

1. Linearly Ordered Fields as Ultrapowers

There is a simple and elementary way to construct linearly ordered fields which contain as a rather small subset the usual field $\mathbb{R}$ of the real numbers. This construction, called the ultrapower construction, proceeds as follows, [2-14].

Given an infinite index set $\Lambda$, we take on it any ultrafilter ${\cal U}$, see Appendix, which satisfies the condition

(1.1)    ${\cal F}_{re}(\Lambda)\subseteq{\cal U}$

where

(1.2)    ${\cal F}_{re}(\Lambda)=\{~{}I\subseteq\Lambda~{}~{}|~{}~{}\Lambda\setminus I~{}~{}\mbox{is finite}~{}\}$

is called the Frechèt filter on $\Lambda$. Further, we define on the set of all real valued functions defined on $\Lambda$, that is, on the set

$~{}~{}~{}~{}~{}~{}\mathbb{R}^{\Lambda}=\{~{}x:\Lambda\longrightarrow\mathbb{R}~{}\}$

the equivalence relation $\approx_{\cal U}$ by

(1.3)    $x\approx_{\cal U}y~{}~{}\Longleftrightarrow~{}~{}\{~{}\lambda\in\Lambda~{}|~{}x(\lambda)=y(\lambda)~{}\}\in{\cal U}$

Finally, through the usual set-theoretic quotient construction, we obtain the ultrapower field

(1.4)    $\mathbb{F}_{\cal U}=\mathbb{R}^{\Lambda}/\approx_{\cal U}$

which proves to have the following two properties.

First, the mapping

(1.5)    $\mathbb{R}\ni r\longmapsto(u_{r})_{\cal U}\in\mathbb{F}_{\cal U}$

is an embedding of fields in which $\mathbb{R}$ is a strict subset of $\mathbb{F}_{\cal U}$, where $u_{r}\in\mathbb{R}^{\Lambda}$ is defined by $u_{r}(\lambda)=r$, for $\lambda\in\Lambda$, while $(u_{r})_{\cal U}$ is the coset of $u_{r}$ with respect to the equivalence relation $\approx_{\cal U}$. For simplicity we shall denote $(u_{r})_{\cal U}=r$, for $r\in\mathbb{R}$, and thus (1.5) takes the form

(1.6)    $\mathbb{R}\ni r\longmapsto r\in\mathbb{F}_{\cal U},~{}~{}~{}\mbox{or simply}~{}~{}~{}\mathbb{R}\subsetneqq\mathbb{F}_{\cal U}$

The field operations of addition and multiplication on $\mathbb{F}_{\cal U}$ are simply given by the following term-wise operations. If $\xi=(x)_{\cal U},~{}\eta=(y)_{\cal U}\in\mathbb{F}_{\cal U}$, where $x,y\in\mathbb{R}^{\Lambda}$, then

(1.7)    $\xi+\eta=(x+y)_{\cal U},~{}~{}~{}\xi\eta=(xy)_{\cal U}$

Second, on $\mathbb{F}_{\cal U}$ we have the linear order defined by

(1.8)    $(x)_{\cal U}\leq(y)_{\cal U}~{}~{}\Longleftrightarrow~{}~{}\{~{}\lambda\in\Lambda~{}|~{}x(\lambda)\leq y(\lambda)~{}\}\in{\cal U}$

where $x,y\in\mathbb{R}^{\Lambda}$. And this linear order is compatible with the field structure of $\mathbb{F}_{\cal U}$, which means that, for $\xi,\eta,\theta\in\mathbb{F}_{\cal U}$, we have

(1.9)    $\xi\leq\eta~{}~{}~{}\Longrightarrow~{}~{}~{}\xi+\theta\leq\eta+\theta$

while for for $\xi,\eta,\theta\in\mathbb{F}_{\cal U},~{}\theta\geq 0$, we have

(1.10)    $\xi\leq\eta~{}~{}~{}\Longrightarrow~{}~{}~{}\xi\theta\leq\eta\theta$

We recall that the field $\mathbb{R}$ of usual real numbers is linearly ordered, giving what in Geometry is called the real line. And with the above linear order in (1.8), $\mathbb{R}$ becomes a linearly ordered subfield of the much larger linearly ordered field $\mathbb{F}_{\cal U}$.

In other words, from geometric point of view, $\mathbb{F}_{\cal U}$ is a much more rich self-similar real line, than the usual real line given by $\mathbb{R}$.

Lastly, it should be noted that the nonstandard reals ${}^{*}\mathbb{R}$ are a particular case of the above ultrapower fields (1.4), see Remark 1.1. at the end of this section.

Here, it is worth recalling the general case of the above construction, even if for the sake of simplicity it will not be considered in the present paper. Namely, the above ultrapower construction is a particular case of what is well known in Model Theory - a branch of Mathematical Logic - under the name of reduced powers. Details on that general construction can be found in [2-14].

This general construction of reduced power algebras goes as follows, and again, it only requires 101 Algebra.

Let ${\cal F}$ be any filter on $\Lambda$ which satisfies

(1.11)    ${\cal F}_{re}(\Lambda)\subseteq{\cal F}$

We define on $\mathbb{R}^{\Lambda}$ the corresponding equivalence relation $\approx_{\cal F}$ by

(1.12)    $x\approx_{\cal U}y~{}~{}\Longleftrightarrow~{}~{}\{~{}\lambda\in\Lambda~{}|~{}x(\lambda)=y(\lambda)~{}\}\in{\cal F}$

Then, through the usual quotient construction, we obtain the reduced power algebra

(1.13)    $\mathbb{A}_{\cal F}=\mathbb{R}^{\Lambda}/\approx_{\cal F}$

which has the following two properties. The mapping

(1.14)    $\mathbb{R}\ni r\longmapsto(u_{r})_{\cal F}\in\mathbb{A}_{\cal F}$

is an embedding of algebras in which $\mathbb{R}$ is a strict subset of $\mathbb{A}_{\cal F}$, where $u_{r}\in\mathbb{R}^{\Lambda}$ is defined by $u_{r}(\lambda)=r$, for $\lambda\in\Lambda$, while $(u_{r})_{\cal F}$ is the coset of $u_{r}$ with respect to the equivalence relation $\approx_{\cal F}$. Further, on $\mathbb{A}_{\cal F}$ we have the partial order

(1.15)    $(x)_{\cal F}\leq(y)_{\cal F}~{}~{}\Longleftrightarrow~{}~{}\{~{}\lambda\in\Lambda~{}|~{}x(\lambda)\leq y(\lambda)~{}\}\in{\cal F}$

where $x,y\in\mathbb{R}^{\Lambda}$.

Remark 1.1.

It is worth recalling that Abraham Robinson’s field ${}^{*}\mathbb{R}$ of nonstandard reals can be obtained by (1.4) above. However, what complicates considerably the resulting Nonstandard Analysis is the intent to have the so called Transfer Principle operating in it. And according to that principle every property of the field $\mathbb{R}$ of the usual real numbers, property which cam be formulated in terms of First Order Predicate Logic, should also hold for the field ${}^{*}\mathbb{R}$ of nonstandard reals.

However, in Nonstandard Analysis it appears that one is obliged to pay a high price for achieving that intent. Indeed, the resulting necessary technical edifice makes Nonstandard Analysis quite hard even for a large majority of mathematicians, let alone physicists. On the other hand, the advantage obtained from the Transfer Principle turns out to be rather limited, since by far most of the properties of interest of $\mathbb{R}$, and let alone of ${}^{*}\mathbb{R}$, simply cannot be formulated in terms of First Order Predicate Logic.

On the contrary, here, as well as in [2-14], we do not aim to obtain any kind of transfer principle. Instead, we only make use of the reduced power construction. Consequently, the technical part can be limited to the simplicity and ease of 101 Algebra.

2. Walkable Worlds …

An essential property of the ultrapower fields (1.4) is that they are no longer Archimedean. In other words, unlike the usual field given by the real line $\mathbb{R}$, they do not satisfy the Archimedean Axiom

(2.1)    $\exists~{}~{}u>0~{}:~{}\forall~{}~{}v>0~{}:~{}\exists~{}~{}n\in\mathbb{N}~{}:~{}nu>v$

As it happens, and it is still seldom realized, the fact that the ultrapower fields (1.4) are non-Archimedean, and also are larger than $\mathbb{R}$, gives them an extremely rich both local and global structure. And this results in a surprising involved self-similar structure.

Here we shall illustrate that fact with the help of the concept of walkable world, a concept which is highly intuitive in its pragmatic algebraic and geometric meaning.

Let us briefly recall some of the basic features of the ultrapower fields (1.4), features which give their rich self-similar structure.

One that follows immediately from the fact that they are non-Archimedean is that their elements $t\in\mathbb{F}_{\cal U}$ are of three kind, namely, infinitesimal, finite, and infinitely large, as defined by the following respective conditions

(2.2)    $\forall~{}~{}r\in\mathbb{R},~{}r>0~{}:~{}t\in(-r,r)$

(2.3)    $\exists~{}~{}r\in\mathbb{R},~{}r>0~{}:~{}t\in(-r,r)$

(2.4)    $\forall~{}~{}r\in\mathbb{R},~{}r>0~{}:~{}t\notin(-r,r)$

where for $a,b\in\mathbb{F}_{\cal U}$, we denote as usual $(a,b)=\{s\in\mathbb{F}_{\cal U}~{}|~{}a<s<b\}$. Now, following Leibniz, one denotes

(2.5)    $monad(0)=\{~{}t\in\mathbb{F}_{\cal U}~{}|~{}~{}t~{}~{}\mbox{is~{}infinitesimal}~{}\}$

and calls it the monad of $0\in\mathbb{F}_{\cal U}$, while following Keisler, [1], one denotes

(2.6)    $Gal(0)=\{~{}t\in\mathbb{F}_{\cal U}~{}|~{}~{}t~{}~{}\mbox{is~{}finite}~{}\}$

and calls it the Galaxy of $0\in\mathbb{F}_{\cal U}$.

It is easy to see that

(2.7)    $Gal(0)=\bigcup_{r\in\mathbb{R}}monad(r)$

where for $t\in\mathbb{F}_{\cal U}$, we denote

(2.8)    $monad(t)=t+monad(0)$

Finally

(2.9)    $\mathbb{F}_{\cal U}\setminus Gal(0)$

is the set of infinitely large elements in the ultrapower field $\mathbb{F}_{\cal U}$.

In this way, all the elements of $\mathbb{F}_{\cal U}$, be they infinitesimal, finite, or infinitely large, have been expressed respectively in (2.5) by the monad of $0\in\mathbb{F}_{\cal U}$, in (2.7) by the Galaxy of $0\in\mathbb{F}_{\cal U}$, and in (2.9). And as one notes, all these sets can in fact be expressed in terms of the monad of $0\in\mathbb{F}_{\cal U}$ alone.

And now, to the walkable worlds …

Let $t,u\in\mathbb{F}_{\cal U}$, with $u>0$. Then we denote

(2.10)    \begin{array}[]{l}WW(t,u)=\{~{}s\in\mathbb{F}_{\cal U}~{}|~{}\exists~{}n\in\mathbb{N}~{}:~{}s\in(t-nu,t+nu)~{}\}=\\ \\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\bigcup_{\,n\in\mathbb{N}}~{}(t-nu,t+nu)\end{array}

which is the set of elements $s\in\mathbb{F}_{\cal U}$ that can be reached in either direction starting at $t$, by a finite number of steps of length $u$. Thus $WW(t,u)$ is the walkable world around $t$, by steps of length $u$.

Obviously

(2.11)    $Gal(0)=WW(0,1)$

and

(2.12)    $s\in WW(t,u)~{}~{}\Longrightarrow~{}~{}WW(s,u)=WW(t,u)$

while

(2.13)    $WW(t,v)=WW(t,u)$

for all $v\in\mathbb{F}_{\cal U},~{}v>0$, such that

(2.14)    $mu\leq v\leq(m+1)u$

for some $m\in\mathbb{N}$.

Furthermore, for every $t,u\in\mathbb{F}_{\cal U}$, with $u>0$, we have the order isomorphism

(2.15)    $WW(t,u)\ni s\longmapsto(s-t)/u\in WW(0,1)$

thus any two walkable worlds are order isomorphic.

The non-Archimedean nature of $\mathbb{F}_{\cal U}$ results, among others, in

(2.16)    $monad(0)\neq WW(t,u),~{}~{}~{}\mathbb{F}_{\cal U}\neq WW(t,u)$

and in fact

(2.17)    $WW(t,u)$ is an infinitely small part of $\mathbb{F}_{\cal U}$

for all $t,u\in\mathbb{F}_{\cal U}$, with $u>0$.

Remark 2.1.

In view of (2.11), in any given ultrapower field $\mathbb{F}_{\cal U}$, the whole of $Gal(0)$ is only one single walkable world, while as seen in the next section, there are infinitely many walkable worlds, either disjoint from one another, or nested within one another.

Here is already one of the essential differences with ultrapower fields, when compared with the usual field of the real line $\mathbb{R}$. Indeed, in the latter case, what corresponds to $Gal(0)=WW(0,1)$ is the whole of $\mathbb{R}$, namely, for every $t,u\in\mathbb{R},~{}u>0$, we have

(2.18)    $\{~{}s\in\mathbb{R}~{}|~{}\exists~{}n\in\mathbb{N}~{}:~{}s\in(t-nu,t+nu)~{}\}=\mathbb{R}$

In other words, the usual field of the real line $\mathbb{R}$ is but only one single walkable world, while in the case of ultrapower fields, each walkable world is merely an infinitely small part of such a field, as seen in (2.17).

This difference clearly illustrates how much limitation is imposed upon the structure of $\mathbb{R}$ by the acceptance of the Archimedean Axiom.

**3. Universes Next To and/or Within Universes, and so on

     Ad Infinitum …**

Let us see now the way two arbitrary walkable worlds $WW(t,u)$ and $WW(s,v)$, with $t,u,s,v\in\mathbb{F}_{\cal U},~{}u,v>0$, can relate to one another. As it turns out, we can distinguish the following three situations regarding their relative position to one another :

Case I  :  The two walkable worlds are the same, namely

(3.1)    $WW(t,u)=WW(s,v)$

Case II  :  The two walkable worlds are disjoint, thus in particular, they are different, namely

(3.2)    $WW(t,u)\cap WW(s,v)=\phi$

Case III  :  The two walkable worlds are not disjoint, yet they are nevertheless different, namely

(3.3)    $WW(t,u)\cap WW(s,v)\neq\phi,~{}~{}~{}WW(t,u)\neq WW(s,v)$

Now, a specific feature with walkable worlds is that in the last case, we must necessarily have either the nesting

(3.3.1)    $WW(t,u)$ is an infinitesimal part of $WW(s,v)$

or the nesting

(3.3.2)    $WW(s,v)$ is an infinitesimal part of $WW(t,u)$

In other words, if two different walkable worlds intersect, that is, in the case of (3.3), then one of them must strictly contain the other, and in fact, it must contain it as an infinitesimal part.

Furthermore, concerning (3.2), there are infinitely many walkable worlds which are pair-wise disjoint. As for (3.3), the respective nestings in (3.3.1) and (3.3.2) always have an infinite depth, that is, there are always infinitely many infinitely smaller and smaller walkable worlds nested in one another.

Let us go more into detail on the possible relation between two walkable worlds in the case (3.3). We note that, given $u,v\in\mathbb{F}_{\cal U},~{}u,v>0$, we can in view of the linear order on $\mathbb{F}_{\cal U}$ always assume that

(3.4)    $v\leq u$

Thus the following two alternatives result

(3.5)    either $u/v$ is finite, or $u/v$ is infinitely large

and correspondingly, either

(3.6)    $\exists~{}~{}n\in\mathbb{N},~{}n\geq 1:~{}u\leq nv$

or

(3.7)    $\forall~{}~{}n\in\mathbb{N},~{}n\geq 1:~{}nv\leq u$

Now in view of (2.13), (2.14), the alternative (3.5) is not compatible with (3.3), since it leads to (3.1). Thus we remain with (3.7). And then (3.3.2) follows.

What is important to note here is that (3.3.2), with the assumption (3.4), will always have $u$ and $v$ in one and only one of the following six situations :

1. both $u$ and $v$ are infinitesimal

2. $u$ is finite, $v$ is infinitesimal

3. both $u$ and $v$ are finite

4. $u$ is infinitely large, $v$ is infinitesimal

5. $u$ is infinitely large, $v$ is finite

6. both $u$ and $v$ are infinitely large

We can conclude with

Theorem 3.1.

The structure of the field $\mathbb{R}$ of usual real numbers in terms of walkable worlds is given by the simple relation

(3.8)    $\mathbb{R}=WW(0,1)$

On the other hand, the structure of the reduced power field $\mathbb{F}_{\cal U}$ in terms of walkable worlds is given by the relation

(3.9)     $\mathbb{F}_{\cal U}=\bigcup_{s,u\in\,\mathbb{F}_{\cal U},\,u>\,\,0}\,WW(s,u)$

and the walkable worlds in(3.9) form a nesting on $\mathbb{F}_{\cal U}$ in the above richly structured manner.

$\Box$

Here, related with Theorem 3.1. above, we recall

Definition 3.1.

A family of nonvoid subsets $A_{i}\subseteq X$, with $i\in I$, is called a nesting on $X$, if and only if

(3.10)    $X=\bigcup_{i\in I}\,A_{i}$

and, for $i,j\in I$, we have

(3.11)    $A_{i}\cap A_{j}\neq\phi~{}~{}\Longrightarrow~{}~{}A_{i}\subseteq A_{j}~{}\mbox{or}~{}A_{j}\subseteq A_{i}$

4. The Rich Self-Similar Structure of Ultrapower Fields

For the sake of easier comparison, let us recall the self-similarity properties of the usual field $\mathbb{R}$ of real numbers. In this regard, we have the self-similarity property given by the following bijective, order reversing mapping

(4.1)    $\mathbb{R}\setminus(-1,1)\ni r\longmapsto 1/r\in[-1,1]\setminus\{0\}$

thus the unbounded set

$~{}~{}~{}~{}~{}~{}\mathbb{R}\setminus(-1,1)=(-\infty,-1]\cup[1,\infty)$

has through the mapping (4.1) the inverse linear order structure of the bounded set

$~{}~{}~{}~{}~{}~{}[-1,1]\setminus\{0\}=[-1,0)\cup(0,1]$

Now by translation and scaling, we obtain the family of self-similarities of the usual field $\mathbb{R}$ of real numbers, given by the bijective, order reversing mappings

(4.2)    $\mathbb{R}\setminus(-a,a)\ni r\longmapsto(1/r)+r_{0}\in[r_{0}-\frac{1}{a},r_{0}+\frac{1}{a}]\setminus\{r_{0}\}$

where $r_{0},a\in\mathbb{R},a>0$.

Here we can note that none of the self-similarities (4.2) refers to the structure at any given point $r_{0}\in\mathbb{R}$, but only to the structure of the sets

(4.3)    $[r_{0}-a,r_{0}+a]\setminus\{r_{0}\}=[r_{0}-a,r_{0})\cup(r_{0},r_{0}+a],~{}~{}~{}~{}a>0$

around points $r_{0}\in\mathbb{R}$, sets which are whole neighbourhoods of $r_{0}$ from which, however, the point $r_{0}$ itself has been taken out. This is obviously inevitable, since each point $r_{0}\in\mathbb{R}$ is at a finite strictly positive - thus not infinitesimal - distance from any other point in $\mathbb{R}$.

In addition, we also have the self-similarities

(4.4)    $\mathbb{R}\stackrel{{\scriptstyle f}}{{\longrightarrow}}(a,b)$

where $-\infty\leq a<b\leq\infty$, while $f$ can be any bijective order preserving continuous mapping.

On the other hand, with the ultrapower fields $\mathbb{F}_{\cal U}$, their self-similar structures are far more rich, due to the presence of their infinitesimals, and thus as well, of their infinitely large elements. Indeed, this time, the self-similarities can also refer to the whole monad of each point, except for the point itself.

Let us start with a self-similarity of any ultrapower field $\mathbb{F}_{\cal U}$ which does not exist in the case of the usual real line $\mathbb{R}$. Namely, it is easy to see that we have the order reversing bijective mapping

(4.5)    $(\mathbb{F}_{\cal U}\setminus Gal(0))\ni t\longmapsto 1/t\in(monad(0)\setminus\{0\})$

which means that the set of all infinitely large elements in $\mathbb{F}_{\cal U}$ has the inverse order structure of the set of infinitesimal elements from which one excludes 0.

This shows the important fact that the infinitesimally local structure, and on the other hand, the global structure of $\mathbb{F}_{\cal U}$ do in fact mirror one another, a property which has no correspondence in the case of the usual field $\mathbb{R}$ of real numbers.

Also, through translation and scaling, we have, for each $t_{0},u\in\mathbb{F}_{\cal U},u>0$, the order reversing bijective mapping

(4.6)    $\mathbb{F}_{\cal U}\setminus(-u,u)\ni t\longmapsto(1/t)+t_{0}\in[t_{0}-\frac{1}{u},t_{0}+\frac{1}{u}]\setminus\{t_{0}\}$

where $\mathbb{F}_{\cal U}\setminus(-u,u)$ will always contain infinitely large elements.

These again are self-similarities not present in the case of the usual real line $\mathbb{R}$.

Furthermore, in (4.6) we have a far more rich possibility for translations and scalings than in the usual case of the real line $\mathbb{R}$. Indeed, in addition to translations and scalings with non-zero finite elements $r_{0},a\in\mathbb{R},a>0$, as in (4.3), we can now also translate and scale with all $t_{0},u\in\mathbb{F}_{\cal U},u>0$, thus with all infinitely large elements, as well as with all infinitesimal elements, except for scaling with $0\in\mathbb{F}_{\cal U}$.

Let us consider the above in some detail by listing the different possibilities for the sets

(4.7)    $[t_{0}-\frac{1}{u},t_{0}+\frac{1}{u}]\setminus\{t_{0}\}$

in (4.6).

First of all, these sets are no longer mere subsets in $\mathbb{R}$, but instead, they are subsets in $\mathbb{F}_{\cal U}$, and will always contain infinitesimals, since they contain nonvoid intervals. Furthermore, as seen below, they may also contain infinitely large elements.

Also, $t_{0},u\in\mathbb{F}_{\cal U},u>0$ in (4.7) can independently be finite, infinitesimal, or infinitely large, thus resulting in 9 possible combinations and 6 distinct outcomes regarding the set (4.7), which we list below. This is in sharp contradistinction with the case in (4.3) which applies to the real line $\mathbb{R}$. Indeed :

1. Let us start the listing of these 9 different cases and 6 distinct outcomes with both $t_{0}$ and $u$ being finite. Then obviously (4.7) is a subset of $Gal(0)$, and it has the finite, non-infinitesimal length $2u$.

2. When $t_{0}$ is finite and $u$ is infinitesimal, then the set (4.7) is infinitely large, and is no longer contained in $Gal(0)$, however, it contains $Gal(0)\setminus\{t_{0}\}$.

3. If $t_{0}$ is finite, but $u$ is infinitely large, then (4.7) is again a subset of $Gal(0)$, and in fact, it has the infinitesimal length $2u$, which means that it is a subset of $monad(t_{0})$.

4. Let us now assume that $t_{0}$ is infinitesimal and $u$ finite. Then regarding the set (4.1), we are back to case 1) above.

5. If both $t_{0}$ and $u$ are infinitesimal then the set (4.7) is as in 2) above.

6. When $t_{0}$ is infinitesimal and $u$ is infinitely large, the set (4.7) is as in 3) above.

7. Let us now take $t_{0}$ infinitely large and $u$ finite. Then the set (4.7) is disjoint from $Gal(0)$, and it has the finite, non-infinitesimal length $2u$.

8. When $t_{0}$ infinitely large and $u$ infinitesimal, then the set (4.7) is again not contained in $Gal(0)$, and it has the infinitely large length $2u$. Furthermore, depending on the relationship between $|t_{0}|$ and $1/u$, it may, or it may not intersect $Gal(0)$.

9. Finally, when both $t_{0}$ and $u$ are infinitely large, then the set (4.7) is disjoint from $Gal(0)$, and it has the infinitesimal length $2u$.

We conclude that he local structure of $\mathbb{F}_{\cal U}$ is given by

(4.8)    $Gal(0)=\bigcup_{r\in\mathbb{R}}\,(r+monad(0))$

while the global structure of $\mathbb{F}_{\cal U}$ is given by

(4.9)    $\mathbb{F}_{\cal U}=(~{}\bigcup_{\lambda\in\Lambda}\,Gal(-s_{\lambda})~{})~{}\bigcup~{}Gal(0)~{}\bigcup~{}(~{}\bigcup_{\lambda\in\Lambda}\,Gal(s_{\lambda})~{})$

where $\Lambda$ is an uncountable set of indices, while $s_{\lambda}\in\mathbb{F}_{\cal U}$ are positive infinite, and such that $s_{\mu}-s_{\lambda}$ is infinite, for $\lambda,\mu\in\Lambda,\lambda\neq\mu$.

Here we can point to a self-similar aspect of the interrelation between the local and global structure of $\mathbb{F}_{\cal U}$ which may remind us of a typical feature of fractals. Indeed, similar with (4.8), the relation (4.9) can also be expressed in terms monads, namely

(4.10)    $\mathbb{F}_{\cal U}=(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r-s_{\lambda}+monad(0))~{})~{}\bigcup~{}$

                           $(~{}\bigcup_{r\in\mathbb{R}}\,(r+monad(0))~{})$

                           $~{}\bigcup~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r+s_{\lambda}+monad(0))~{})$

In this way, in view of (4.5), we obtain the self-similar order reversing bijection, which is now expressed solely in terms of $mon(0)$, namely

(4.11)    $[~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r-s_{\lambda}+monad(0))~{})$

                  $~{}\bigcup~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r+s_{\lambda}+monad(0))~{})~{}]\ni t\longmapsto$

                                                             $\longmapsto 1/t\in[~{}monad(0)\setminus\{0\}~{}]$

and conversely

(4.12)    $[~{}monad(0)\setminus\{0\}~{}]\ni t\longmapsto$

                                            $\longmapsto 1/t\in[~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r-s_{\lambda}+monad(0))~{})$

                                                             $~{}\bigcup~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r+s_{\lambda}+monad(0))~{})~{}]$

As we can note, the above bijections in (4.11), (4.12) are given by the very simple algebraic, explicit, and order reversing mapping $s\longmapsto 1/s$, which involves what is essentially a field operation, namely, division. And these two bijections take the place of the much simpler order reversing bijections in the case of the usual real line $\mathbb{R}$, namely

(4.13)    $(\mathbb{R}\setminus(-1,1))\ni r\longmapsto 1/r\in([-1,1]\setminus\{0\})$

(4.14)    $([-1,1]\setminus\{0\})\ni r\longmapsto 1/r\in(\mathbb{R}\setminus(-1,1))$

The considerable difference between (4.11), (4.12), and on the other hand, (4.13), (4.14) is obvious. Indeed, in the former two, which describe the self-similar structure of $\mathbb{F}_{\cal U}$, the order reversing bijections represent the set

$~{}~{}~{}~{}~{}~{}mon(0)\setminus\{0\}$

through the set

$~{}~{}~{}~{}~{}~{}[~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r-s_{\lambda}+monad(0))~{})$

                           $~{}\bigcup~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r+s_{\lambda}+monad(0))~{})~{}]$

which contains uncountably many translates of the set $mon(0)$. And it is precisely this manifestly rich self-similarity of the set $mon(0)$ of monads which is the novelty in the non-Archimedean structure of $\mathbb{F}_{\cal U}$, when compared with the much simpler Archimedean structure of $\mathbb{R}$. This novelty is remarkable since it makes $mon(0)$ have the very same complexity with the whole of

$~{}~{}~{}~{}~{}~{}\mathbb{F}_{\cal U}\setminus Gal(0)=[~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r-s_{\lambda}+monad(0))~{})$

                                            $~{}\bigcup~{}(~{}\bigcup_{r\in\mathbb{R},\,\,\lambda\in\Lambda}\,(r+s_{\lambda}+monad(0))~{})~{}]$

In this way $mon(0)$, which is but the set of infinitesimals, thus it cannot be represented in terms of the usual field $\mathbb{R}$ of real numbers, turns out to have the very same complexity as the set $\mathbb{F}_{\cal U}\setminus Gal(0)$ of all infinitely large numbers, which again cannot be represented in terms of the usual field $\mathbb{R}$ of real numbers.

5. What Possible Relevance in Physics ?

In sharp contrast with the rich self-similar structure of the ultrapower fields $\mathbb{F}_{\cal U}$, which by necessity are not Archimedean, and thus exhibit the mentioned wealth of walkable worlds, all the known theories of Physics are - and have so far been stuck - into only one single such walkable world, see (3.8), namely, the one given by the usual field $\mathbb{R}$ of real numbers, and the scalars, manifolds, spaces, etc., built upon $\mathbb{R}$. No wonder that so called ”infinities in Physics” trouble various theories of Physics, and furthermore, lead to highly questionable ad-hock attempts at solution, such as various so called renormalization methods.

On the other hand, whatever quantities prove to be ”infinity” in Physics becomes just another usual and regular element in the ultrapower fields $\mathbb{F}_{\cal U}$, thus all algebraic operations can be effectuated upon such quantities, without absolutely any concern or restrictions.

And for all that, all it takes is simply to discard the Archimedean Axiom.

But beyond the issue of ”infinities” in Physics lies the far deeper and more important issue of possible different levels of precision which may be relevant in modelling natural processes.

Here, with the acceptance of infinitesimals and infinitely large large scalars, and thus with the necessary setting aside of the Archimedean Axiom, we may obviously face two rather different alternatives, namely

• •

the new properties and meanings in reduced power algebras do not correspond to any possible physical meaning, or on the contrary

• •

such new properties and meanings which appear in ultrapower fields may possibly correspond to not yet explored physical realities.

One such possible such new physical reality is that of

• •

Increased, or alternatively, Decreased Precision in Physical Measurements.

As a general issue, relating not only to Relativity or the Quanta, but possibly to Classical Physics as well, the presence of infinitesimal and infinitely large scalars in ultrapower fields may correspond to a new possibility of having no less than two radically different kind of measurements when it comes to their relative precision.

Namely, one has an increased precision in measurement, when measurement is done in terms of usual finite scalars, and one obtains as result some infinitesimal scalar in such ultrapower fields.

Alternatively, the presence of infinitely large scalars in such ultrapower fields may simply indicate that they were obtained in terms of finite scalars, and thus are but the result of a measurement with decreased precision.

In this regard, we can therefore have the following relative situations

• •

infinitesimal scalars are the result of increased precision measurements done in terms of finite or infinite scalars,

• •

finite scalars are the result of increased precision measurements done in terms of infinite scalars,

• •

finite or infinitely large scalars are the result of decreased precision measurements done in terms of infinitesimal scalars,

• •

infinitely large scalars are the result of decreased precision measurements done in terms of infinitesimal or finite scalars.

And surprisingly, one can also have the following relative situations

• •

infinitesimal scalars are the result of increased precision measurements done in terms of some less infinitesimal scalars,

• •

infinitesimal scalars are the result of decreased precision measurements done in terms of some more infinitesimal scalars,

• •

infinitely large scalars are the result of increased precision measurements done in terms of some more infinitely large scalars,

• •

infinitely large scalars are the result of decreased precision measurements done in terms of some less infinitely large scalars.

Indeed, one of the basic features of ultrapower fields is precisely their rich self-similar structure which distinguishes not only between infinitesimal, finite and infinitely large scalars, but also within the infinitely small scalars themselves, as much as within the infinitely large scalars taken all alone. Specifically, infinitesimal scalars can be infinitely smaller, or on the contrary, infinitely larger than other infinitesimals. And similarly, infinitely large scalars can be infinitely smaller, or on the contrary, infinitely larger than other infinitely large scalars.

Here, however, we can note that such a possible interpretation of increased, or decreased precision which is relative, is in fact not new. Indeed, in terms of usual scalars, be they real or complex, there is a marked dichotomy between finite scalars, and on the other hand, the so called ”infinities” which may on occasion arise from operations with finite scalars. And such simple ”formulas” like $\infty+1=\infty$, are in fact expressing that fact. Namely, on one hand, from the point of view of ”infinity”, the finite number 1 has such an increased precision as to be irrelevant with respect to addition, while on the other hand, from the point of view of the finite number 1, the ”infinity” has such an decreased precision as to be similarly irrelevant when involved in addition.

As for the issue of certain universal constants in Physics, given the above possibility in interpretation leading to relative precision measurement, be it increased or decreased, one can reconsider the status of certain universal physical constants, such as for instance, the Planck constant h and the constant c giving the velocity of light in vacuum.

Indeed, when considered from our everyday macroscopic experience, h is supposed to be unusually small, while on the contrary, c is very large. Consequently, one may see h as a sort of ”infinitesimal”, while c then looks like ”infinitely large”.

The fact is that, within ultrapower fields, such an alternative view of h and c is possible. Therefore, one may find it appropriate to explore the possible physical meaning, or otherwise, that may possibly be associated with such an interpretation.

Appendix

Given a nonvoid set $\Lambda$, by a filter ${\cal F}$ on $\Lambda$ one understands a set of subsets $F\subseteq\Lambda$ with the following four properties

(A1)    ${\cal F}\neq\phi$

(A2)    $\phi\notin{\cal F}$

(A3)    $F,G\in{\cal F}~{}~{}\Longrightarrow~{}~{}F\cap G\in{\cal F}$

(A4)    $F\in{\cal F},~{}F\subseteq G\subseteq\Lambda~{}~{}\Longrightarrow~{}~{}G\in{\cal F}$

As an example, let $\Lambda=\mathbb{N}$ and let

$~{}~{}~{}~{}~{}~{}{\cal F}_{fre}$

be the set of all subsets $F\subseteq\mathbb{N}$ such that $\mathbb{N}\setminus F$ is a finite subset of $\mathbb{N}$. Then ${\cal F}_{fre}$ is a filter on $\Lambda=\mathbb{N}$, and it is called the Fréchet filter.

An ultrafilter on a nonvoid set $\Lambda$ is any filter ${\cal U}$ which has the additional property

(A5)    $F\subseteq\Lambda~{}~{}~{}\Longrightarrow~{}~{}~{}F\in{\cal U}$   or   $\Lambda\setminus F\in{\cal U}$

Clearly, ${\cal F}_{fre}$ is not an ultrafilter on $\Lambda=\mathbb{N}$, since if we take $F$ as the set of all even numbers, then $\Lambda\setminus F$ is the set of all odd numbers, and none of them belongs to ${\cal F}_{fre}$.

The existence of ultrafilters on every nonvoid set $\Lambda$ is guaranteed by the Axiom of Choice in Set Theory.