#  Naïve Infinitesimal Analysis
Its Construction and Its Properties

Anggha Nugraha
<EMAIL_ADDRESS>Maarten McKubre-Jordens  Hannes Diener

###### Abstract

This paper aims to build a new understanding of the nonstandard mathematical
analysis. The main contribution of this paper is the construction of a new set
of numbers, $\mathbb{R^{Z_{<}}}$, which includes infinities and
infinitesimals. The construction of this new set is done naïvely in the sense
that it does not require any heavy mathematical machinery, and so it will be
much less problematic in a long term. Despite its naïvety character, the set
$\mathbb{R^{Z_{<}}}$ is still a robust and rewarding set to work in. We
further develop some analysis and topological properties of it, where not only
we recover most of the basic theories that we have classically, but we also
introduce some new enthralling notions in them. The computability issue of
this set is also explored. The works presented here can be seen as a
contribution to bridge constructive analysis and nonstandard analysis, which
has been extensively (and intensively) discussed in the past few years.

###### Acknowledgements

The first author received financial support from Indonesia Endowment Fund for
Education that enables the research of this article. We would also like to
thank Professor Elemér Rosinger and Dr. Josef Berger for their invaluable
inputs.

##  1 Background and Aim

There have been many attempts to rule out the existence of inconsistencies in
mathematical and scientific theories. Since the 1930s, we have known (from
Gödel’s results) that it is impossible to prove the consistency of any
interesting system (in our case, this is a system capable of dealing with
arithmetic and analysis). One of the famous examples of inconsistency is as
follows. Suppose we have a function $f\left(x\right)=ax^{2}+bx+c$ and want to
find its first derivative. By using Newton’s ‘definition of the derivative’:

$\displaystyle f^{\prime}\left(x\right)$
$\displaystyle=\frac{f\left(x+h\right)-f\left(x\right)}{h}$
$\displaystyle=\frac{a\left(x+h\right)^{2}+b\left(x+h\right)+c-ax^{2}-bx-c}{h}$
$\displaystyle=\frac{ax^{2}+2axh+2h^{2}+bx+bh+c-ax^{2}-bx-c}{h}$
$\displaystyle=\frac{2axh+2h^{2}+bh}{h}$ $\displaystyle=2ax+h+b$ (1)
$\displaystyle=2ax+b$ (2)

In the example above, the inconsistency is located in treating the variable
$h$ (some researchers speak of it as an infinitesimal). It is known from the
definition that $h$ is a small but non-trivial neighbourhood around $x$ and,
because it is used as a divisor, cannot be zero. However, the fact that it is
simply omitted at the end of the process (from Equation 1 to Equation 2)
indicates that it was, essentially, zero after all. Hence, we have an
inconsistency.

This problem of inconsistency has been ‘resolved’ in the
$19^{\textnormal{th}}$ century111If we look historically, the debates of the
use of infinitesimals have a long and vivid history. Their early appearance in
mathematics was from the Greek atomist philosopher Democritus (around 450
B.C.E.), only to be dispelled by Eudoxus (a mathematician around 350 B.C.E.)
in what was to become “Euclidean” mathematics. by the concept of limit, but
its (intuitive) naïve use is still common nowadays, e.g. in physics [48]. In
spite of that, interesting and correct results are still obtained. This
outlines how firmly inconsistent infinitesimal reasoning (which is a reasoning
with prima facie inconsistent infinitesimals) is entrenched in our scientific
community and it means that inconsistency is something that, if unavoidable,
should be handled appropriately. Actually, inconsistency would not have been
such a problem if the logic used was not explosive [50]. The problem is that
our mathematical theory is mostly based on classical logic, which is
explosive. Thus, one promising solution is to change the logic into a non-
explosive one and this is the main reason for the birth of paraconsistent
mathematics which uses paraconsistent logic as its base.

Recent advances in paraconsistent mathematics have been built on developments
in set theory [51], geometry [29], arithmetic [28], and also the elementary
research at calculus [28] and [9]. A first thorough study to apply
paraconsistent logic in real analysis was based on the early work, such as
[11] and [27]. While Rosinger in [39] and [38] tried to elaborate the basic
structure and use of inconsistent mathematics, McKubre-Jordens and Weber in
[26] analysed an axiomatic approach to the real line using paraconsistent
logic. They succeed to show that basic field and also compactness theorems
hold in that approach. They can also specify where the consistency requirement
is necessary. These preliminary works in [26] and [52] show how successful a
paraconsistent setting to analysis can be. On the side of the non-standard
analysis, it can be seen for example in [1] and [15] that it is still well-
studied and still used in many areas.

The underlying ideas of the research described on this paper are as follows.
We have two languages: $\mathfrak{L}$, the language of real numbers
$\mathbb{R}$, and $\mathfrak{L^{*}}$, the language of the set of hyperreal
numbers $\mathbb{{}^{*}R}$. The language $\mathfrak{L^{*}}$ is an extension of
the language $\mathfrak{L}$. It can be shown that each of those two sets forms
a model for the formulas in its respective language.

Speaking about the hyperreals, the basic idea of this system is to extend the
set $\mathbb{R}$ to include infinitesimal and infinite numbers without
changing any of the elementary axioms of algebra. Transfer principle holds an
important role in the formation of the set $\mathbb{{}^{*}R}$ in showing what
are still preserved in spite of this extension. However, there are some
problems with the transfer principle, notably its non-computability (see
Subsection 2.2). To avoid its use, one can logically think to simply
collapsing the two languages into one language $\mathfrak{\widehat{L}}$ which
corresponds to the set $\mathbb{\widehat{R}}$, a new set of numbers
constructed by combining the two set of axioms of $\mathbb{R}$ and
$\mathbb{{}^{*}R}$. This is what we do here (see Section 3). Nevertheless,
there is at least one big problem from this idea: a contradiction.

We can at present consider two possible ways of resolving this contradiction.
The first way is to change the base logic into paraconsistent logic. There are
many paraconsistent logics that are available at the moment. This could be a
good thing, or from another perspective, be an additional difficulty as we
need to choose wisely which paraconsistent logic we want to use at first, i.e.
which one is the most appropriate or the best for our purpose. But then, to be
able to do this, we need to know beforehand which criteria to use and this, in
itself, is still an open question.

The second way we could consider is to have a subsystem in our theory. This
idea arose from a specific reasoning strategy, Chunk & Permeate, which was
introduced by Brown and Priest in 2003 [9]. Using this strategy, we divide our
set $\mathbb{\widehat{R}}$ into some consistent chunks and build some
permeability relations between them. This process leads us to the creation of
the sets $\mathbb{R^{Z}}$ and $\mathbb{R^{Z_{<}}}$. In our view, this idea is
more sensible and promises to be more useful than the first. Moreover, after
further analysing this idea, we produce some new interesting and useful
notions that will be worth to explore even further (see Sections 4-6).

The aim of this paper is to build a new model of the nonstandard analysis. By
having the new set produces in this paper, we would have real numbers,
infinities, and infinitesimals in one set and would still be able to do our
“usual” analysis in, and with this set. Moreover, in terms of Gödel’s second
incompleteness theorem, if we can build a new structure for nonstandard
mathematical analysis which is resilient to contradiction, we would open the
door to having not just a sound, but a complete mathematical theory. To put it
simply, like Weber said, ‘In light of Gödel’s result, an inconsistent
foundation for mathematics is the only remaining candidate for completeness’
[50].

##  2 Some Preliminaries

To understand and carry our investigation, it is essential to have an accurate
grasp of the received view about formal language, the reals and hyperreals,
and the paraconsistent logic. In turn, to analyse these matters, it will be
useful to fix some terminologies.

###  2.1 Reals, Hyperreals, and Their Respective Languages

Formal language is built by its syntax and semantics. Here are the symbols
that are used in our language:

variables | : |  $a$ $b$ $c$ $\dots$ $x_{1}$ $x_{2}$ $\dots$
---|---|---
grammatical signs | : | ( ) ,
connectives | : |  $\land$ $\lor$ $\neg$ $\rightarrow$
quantifiers | : |  $\forall$ $\exists$
constant symbols | : |  $1$ $-2.5$ $\pi$ $\sqrt{5}$ $\dots$
function symbols | : |  $+$ $-$ $\sin$ $\tan$ $\dots$
relation symbols | : |  $=$ $<$ $>$ $\leq$ $\geq$ $\dots$

Like in natural language, a sentence is built by its term. Terms and sentences
are defined as usual. Example 2.1 uses the simple language $\mathfrak{I}$ to
build some statements about integer numbers, $\mathbb{Z}$.

######  Example 2.1.

In addition to our usual connectives, variables, quantifiers and grammatical
symbols in $\mathbb{Z}$, $\mathfrak{I}$ also contains:

constant symbols | : | $\dots,-2,-1,0,1,2,\dots$
---|---|---
function symbols | : | $q(x)=x^{2}$
|  | $\text{add}(x,y)=x+y$
|  | $\text{mul}(x,y)=x\times y$
relation symbols | : | $P(x)\textnormal{ for ``}x\textnormal{ is positive''}$
|  | $E(x,y)\textnormal{ for ``}x\textnormal{ and }y\textnormal{ are equal''}$

In this language $\mathfrak{I}$, one can translate an English statement
‘squaring any integer number will give a positive number’ as $\forall x$
$P(s(x))$.

Now we can define a language for the set hyperreals. We define a language
$\mathfrak{L}$ whose every sentence, if true in reals, is also true in
hyperreals.

######  Definition 2.2 (Language $\mathfrak{L}$).

The language $\mathfrak{L}$ consists of the usual defined variables,
connectives, and grammatical signs in $\mathbb{R}$, and the following:

constant symbols | : one symbol for every real number
---|---
function symbols | : one symbol for every real-valued function of any finite
| number real variables
relation symbols | : one symbol for every relation on real numbers of any finite
| number real variables

Semantics in our language is described by its model. This model gives an
interpretation of the sentences of the language, such that we may know whether
they are true or false in that model.

######  Definition 2.3 (Model of a Language).

Suppose that we have a language $\mathfrak{A}$. A model for $\mathfrak{A}$
consists of:

  1. 1.

a set $A$ so that each constant symbol in $\mathfrak{A}$ corresponds to an
element of $A$,

  2. 2.

a set $F$ of functions on $A$ so that each function symbol in $\mathfrak{A}$
corresponds to a function in $F$,

  3. 3.

a set $R$ of relations on $A$ so that each relation symbol in $\mathfrak{A}$
corresponds to a relation in $R$.

######  Example 2.4.

For our language $\mathfrak{I}$ over integer numbers, its model is the set
$A=\mathbb{Z}$ with several functions and relations already well-defined in
$\mathbb{Z}$.

######  Theorem 2.5 (Reals as a Model).

The real number system $\mathbb{R}$ is a model for the language
$\mathfrak{L}$.

###### Proof.

Take $A=\mathbb{R}$ and $F$ and $R$ as set of all functions and relations,
respectively, which are already well-defined in $\mathbb{R}$. ∎

Then, by using the definition of a model, we defined what hyperreal number
system is.

######  Definition 2.6.

A hypereal number system is a model for the language $\mathfrak{L}$ that, in
addition to all real numbers, contains infinitesimal and infinite numbers.

Now suppose that $\mathbb{{}^{*}R}$ is the set of all hyperreal numbers. Our
goal now is to show that $\mathbb{{}^{*}R}$ is a model for the language
$\mathfrak{L}$. To show this, we need to extend the definition of relations
and functions on $\mathbb{R}$ into $\mathbb{{}^{*}R}$. This extension can also
be seen in [18].

######  Definition 2.7 (Extended Relation).

Let $R$ be a $k$-variable relation on $\mathbb{R}$, i.e. for every
$x_{1},x_{2},\dots,x_{k}$, $R(x_{1},x_{2},\dots,x_{k})$ is a sentence that is
either true or false. The extension of $R$ to $\mathbb{{}^{*}R}$ is denoted by
${}^{*}R$. Suppose that $\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{k}$
are any hyperreal numbers whose form is
$\\{x_{1n}\\},\\{x_{2n}\\},\dots,\\{x_{kn}\\}$, respectively. We define
${}^{*}R(\mathbf{x_{1}},\mathbf{x_{2}},\dots,\mathbf{x_{k}})$ as true iff

$\\{n|R(x_{1n},x_{2n},\dots,x_{kn})$ is true in $\mathbb{R}\\}$

is big.222A ‘big set’ is a set of natural numbers so large that it includes
all natural numbers with the possible exception of finitely many [18].
Otherwise, $R(\mathbf{x_{1}},\mathbf{x_{2}},\dots,\mathbf{x_{k}})$ is false.

######  Example 2.8.

Suppose that

$ⓩ=\\{1,2,3,4,5,...\\}$

By taking $k=1$ in Definition 2.7, we are able to have a relation $I(x)=$“$x$
is an integer”. The relation $I(ⓩ)$ is true. This is because the set of
indexes where relation $I(x)$ is tru, is a big set. Thus, we conclude that ⓩ
is actually a hyperinteger.

######  Definition 2.9 (Extended Function).

Let $f$ be an $k$-variables function on $\mathbb{R}$. The extension of $f$ to
$\mathbb{{}^{*}R}$ is denoted by ${}^{*}f$. Suppose that
$\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{k}$ are any hyperreal numbers
whose form is $\\{x_{1n}\\},\\{x_{2n}\\},\dots,\\{x_{kn}\\}$, respectively. We
define ${}^{*}f(\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{k})$ by

${}^{*}f(\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{k})=\\{f(x_{11},x_{21},\dots,x_{k1}),f(x_{12},x_{22},\dots,x_{k2}),f(x_{13},x_{23},\dots,x_{k3}),\dots\\}$

######  Example 2.10.

Suppose that

$ⓔ=\\{2,4,6,8,...\\}$

By taking $k=1$ in Definition 2.9, we might have, for example, a well-defined
hypersinus function:

$\sin(ⓔ)=\\{\sin(2),\sin(4),\sin(6),\sin(8),...\\}$

######  Theorem 2.11 (Hyperreals as a Model).

The set $\mathbb{{}^{*}R}$ is a model for the language $\mathfrak{L}$ that
contains infinitesimals and infinities.

###### Proof.

Take $A=\mathbb{{}^{*}R}$ in Definition 2.3 with all of the functions
${}^{*}f$ defined in Definition 2.9 and relations ${}^{*}R$ defined in
Definition 2.7. ∎

###  2.2 Transfer Principle and Its Problems

######  Definition 2.12 (Transfer Principle).

Let $S$ be a sentence in $\mathfrak{L}$. The transfer principle says that:

$S$ is true in the model $\mathbb{R}$ for $\mathfrak{L}$ iff $S$ is true in
the model $\mathbb{{}^{*}R}$ for $\mathfrak{L}$.

As Goldblatt said in [18], ‘The strength of nonstandard analysis lies in the
ability to transfer properties between $\mathbb{R}$ and $\mathbb{{}^{*}R}$.’
But, there are some serious problems with the transfer principle. Some of them
are: it is non-computable in the sense of there is no good computable
representation of the hyperreals to start with; it really depends
intrinsically on the mathematical model or language we use; we are prone to
get things wrong when not handled correctly (especially because of human
error).

Furthermore, there is an all too often overlooked, yet major deficiency with
the transfer principle: it performs particularly poorly upon a rather simple
“cost-return” analysis. Namely, on the one hand, the mathematical machinery
which must be set up in advance in order to use the transfer principle is of
such a considerable technical complication and strangeness with a lack of
step-by-step intuitive insight that, ever since 1966, when Abraham Robinson
published the first major book on the subject — that is, for more than half a
century by now — none of the more major mathematicians ever chose to switch to
the effective daily use of nonstandard analysis, except for very few among
those who have dealt with time continuous stochastic processes, and decided to
use the “Loeb Integral” introduced in 1975. On the other hand, relatively few
properties of importance can ever be transferred, since they are not — and
cannot be, within usual nonstandard analysis — formulated in terms of first
order logic.

One possible solution for overcoming (some of) these problems is simply by not
using the transfer principle, by throwing together the two sets $\mathbb{R}$
and ${}^{*}\mathbb{R}$, i,e. combining their languages and axioms. However,
Example 2.13 shows that if we just simply combine the two languages, it will
pose one big problem of contradictions which can lead to absurdity if we use
classical logic. That is the reason why we use paraconsistent logic as it is
resilient against local contradiction.

######  Example 2.13.

Take the well-ordering principle for our example. This principle says that:
“every non-empty set of natural numbers contains a least element”. Call a set
$S=\\{x\in N^{*}:x\textnormal{ is infinite}\\}$. Let $s$ be its least element.
Note that $s$ is infinite and so $s-1$. Thus, $s-1\in S$ and it makes $s$ is
not the least element. Therefore, there exists $l$ such that $l$ is the least
element of $S$ and there is no $l$ such that $l$ is the east element of $S$.

In addition to the problems with the transfer principle, there are also some
downsides of the construction of the set $\mathbb{{}^{*}R}$ itself. Indeed,
the usual construction of the hyperreal set $\mathbb{R}$ involves an
ultrafilter on $\mathbb{N}$, the existence of which is justified by appealing
to the full Axiom of Choice whose validity is still a great deal to discuss
[49, 22]. Moreover, it also relies on some heavy and non-constructive
mathematical machineries such as Zorn’s lemma, the Hahn-Banach theorem,
Tychonoff’s theorem, the Stone-Cech compactification, or the boolean prime
ideal theorem. On the side of the nonstandard analysis itself, there are some
critiques as can be seen in [7, 13, 14, 19, 45, 47]. Most of them are related
with its non-constructivism and its difficulties to be used in class teaching.
This problem can be solved by building a naïve constructive non-standard set
and making sure that it is still a useful set by redefining some well-known
notions in there.

###  2.3 Paraconsistent Logics in Mathematics

Generally, paraconsistent logics are logics which permit inference from
inconsistent information in a non-trivial fashion [36]. Paraconsistent logics
are characterized by rejecting the universal validity of the principle ex
contradictione quodlibet (ECQ) which is defined below.

######  Definition 2.14 (ECQ Principle).

The principle of explosion, ECQ, is the law which states that any statement
can be proven from a contradiction.

By admitting the ECQ principle in one theory, if that theory contains a single
inconsistency, it becomes absurd or trivial. This is something that, in
paraconsistent logics, does not follow necessarily.

Paraconsistent logicians believe that some contradictions does not necessarily
make the theory absurd. It just means that one has to be very careful when
doing deductions so as to avoid falling from contradiction into an absurdity.
In other words, classical and paraconsistent logic treat contradiction in
different ways. The former treats contradiction as a global contradiction
(making the theory absurd), while the latter treats some contradictions as a
local contradiction. In other words, classical logic cannot recognise if there
is an interesting structure in the event of a contradiction.

######  Definition 2.15 (Paraconsistent Logic).

Suppose that $A$ is a logical statement. A logic is called paraconsistent
logic iff

$\exists A,B$ such that $A\land\lnot A\nvdash B$.

The symbol $\Gamma\vdash A$ simply means that there exists a proof of $A$ from
set of formulas $\Gamma$, in a certain logic.

There are at least two different approaches to paraconsistent logics. The
first is by adding another possible value, both true and false, to classical
truth values while the second one is called the relevant-approach. The idea of
the relevant-approach is simply to make sure that the conclusion of an
implication must be relevant to its premise(s). Those two paraconsistent
logics, respectively, are Priest’s Paraconsistent Logic LP⊃ and Relevant Logic
R.333Note that in general, relevant logic differs from paraconsistent logic.
When someone claims that they use relevant logic, it implies that they use
paraconsistent logic, but not vice versa. Using paraconsistent logic does not
necessarily mean using relevant logic, e.g. the logic $LP^{\supset}$ below is
not relevant logic. More explanations on each of them can be seen, for
example, in [34], [2], [35], [24], and [12].

When we are applying paraconsistent logic to a certain theory, there will be
at least two terms that we have to be aware of: inconsistency and incoherence.
The first term, inconsistency, is applicable if there occurs a contradiction
in a system. Meanwhile, the second term, incoherence, is intended for a system
which proves anything (desired or not). In classical logic, there is no
difference between these two terms because of the ECQ principle. Thus, if a
contradiction arises inside a theory, anything that the author would like to
say can be proved or inferred within that theory. This is something that
likewise does not have to happen in paraconsistent logic.

In mathematical theory, foundation of mathematics is the study of the basic
mathematical concepts and how they form more complex structures and concepts.
This study is especially important for learning the structures that form the
language of mathematics (formulas, theories, definitions, etc.), structures
that often called metamathematical concepts. A philosophical dimension is
hence central to this study. One of the most interesting topics in the
foundation of mathematics is the foundation of real structure, or analysis.

Generally, it is known that the construction of real numbers is categorised in
classical logic — while there is an advancement in paraconsistent logic such
as in [4], this has not yet been extensively explored. However, it seems
viable to make a further study of real structure by developing paraconsistent
foundations of analysis.

##  3 The Creation of The New Sets

As noted before, the transfer principle is useful as well as fairly
problematic at the same time. One way to avoid the unnecessary complications
of the transfer principle is by collapsing the two languages involved into one
language $\mathfrak{\widehat{L}}$. Simply collapsing the two, however, causes
additional problems. One of the problems that can be expected to appear is
contradiction, but we can use a paraconsistent logic to handle this when it
arises.

In this section, we construct a new number system $\mathbb{\widehat{R}}$
through its axiomatisation by ‘throwing’ the axioms of $\mathbb{R}$ and
$\mathbb{{}^{*}R}$ together. The set $\mathbb{\widehat{R}}$ is the number
system on which the language $\mathfrak{\widehat{L}}$ will be based and it
contains positive and negative infinities, and also infinitesimals.444We are
not necessarily expecting the resulting system to have contradictions, but we
will make sure that we maintain coherency by not allowing contradictions to
become an absurdity. It does make sense to insert infinities (and their
reciprocals, infinitesimals) into $\mathbb{\widehat{R}}$ as some of the
contradictions in mathematics come from their existence and also because they
are still used in today’s theory as can be seen in [48].

###  3.1 The New Set $\mathbb{\widehat{R}}$

For the sake of clarity, Axioms 3.1–3.7 give the axiomatisation of the number
system $\mathbb{\widehat{R}}$.

######  Axiom 3.1 (Additive Property of $\mathbb{\widehat{R}}$).

In the set $\mathbb{\widehat{R}}$, there is an operator $+$ that satisfies:
A1: For any $x,y\in\mathbb{\widehat{R}},x+y\in\mathbb{\widehat{R}}$. A2: For
any $x,y\in\mathbb{\widehat{R}},x+y=y+x$. A3: For any
$x,y,z\in\mathbb{\widehat{R}},(x+y)+z=x+(y+z)$. A4: There is
$0\in\mathbb{\widehat{R}}$ such that $x+0=x$ for all
$x\in\mathbb{\widehat{R}}$. A5: For each $x\in\mathbb{\widehat{R}}$, there is
$-x\in\mathbb{\widehat{R}}$ such that $x+(-x)=0$.

######  Axiom 3.2 (Multiplicative Property of $\mathbb{\widehat{R}}$).

In the set $\mathbb{\widehat{R}}$, there is an operator $\cdot$ that
satisfies:
M1: For any $x,y\in\mathbb{\widehat{R}},x\cdot y\in\mathbb{\widehat{R}}$. M2:
For any $x,y\in\mathbb{\widehat{R}},x\cdot y=y\cdot x$. M3: For any
$x,y,z\in\mathbb{\widehat{R}},(x\cdot y)\cdot z=x\cdot(y\cdot z)$. M4: There
is $1\in\mathbb{\widehat{R}}$ such that $(1=0)\rightarrow\bot$ and $\forall
x\in\mathbb{\widehat{R}}$ $x\cdot 1=x$. M5: For each
$x\in\mathbb{\widehat{R}}$, if $(x=0)\rightarrow\bot$, then there is
$y\in\mathbb{\widehat{R}}$ such that $x\cdot y=1$.

######  Axiom 3.3 (Distributive Property of $\mathbb{\widehat{R}}$).

For all $x,y,z\in\mathbb{\widehat{R}}$, $x\cdot(y+z)=(x\cdot y)+(x\cdot z)$.

######  Axiom 3.4 (Total Partial Order Property of $\mathbb{\widehat{R}}$).

There is a relation $\leq$ in $\mathbb{\widehat{R}}$, such that for each
$x,y,z\in\mathbb{\widehat{R}}$:
O1: Reflexivity: $x\leq x$, O2: Transitivity: $(x\leq y\land y\leq
z)\rightarrow x\leq z$, O3: Antisymmetry: $(x\leq y\land y\leq
x)\leftrightarrow x=y$, O4: Totality: $(x\leq y\rightarrow\bot)\rightarrow
y\leq x$. O5: Addition order: $x\leq y\rightarrow x+z\leq y+z$. O6:
Multiplication order: $(x\leq y\land z\geq 0)\rightarrow xz\leq yz$.

######  Axiom 3.5 (Completeness Property of $\mathbb{\widehat{R}}$).

Every non-empty bounded above subset of $\mathbb{\widehat{R}}$ has a least
upper bound (see Definition LABEL:def:ClassicalLeastUpperBound).

######  Axiom 3.6 (Infinitesimal Property of $\mathbb{\widehat{R}}$).

The set $\mathbb{\widehat{R}}$ has an infinitesimal (see Definition 3.8 for
what infinitesimal is).

######  Axiom 3.7 (Archimedean Property of $\mathbb{\widehat{R}}$).

For all $x,y>0$, $\exists n$ such that $nx>y$ (see Definition 3.9 for the
operator $>$).

Notice what Axiom 3.5 and Axioms 3.6 and 3.7 cause. The first axiom, which
states the completeness property of $\mathbb{\widehat{R}}$, causes
computability issues in our set. The last two axioms, they cause the
consistency trouble. Infinity and infinitesimals are formally defined in
Definition 3.8.

######  Definition 3.8.

An element $x\in\mathbb{\widehat{R}}$ is

  * •

infinitesimal iff $\forall n\in\mathbb{N}$ $|x|<\frac{1}{n}$;

  * •

finite iff $\exists r\in\mathbb{R}$ $|x|<r$;

  * •

infinite iff $\forall r\in\mathbb{R}$ $|x|>r$;

  * •

appreciable iff $x$ is finite but not an infinitesimal;

By using notation $\epsilon$ as an infinitesimal, an infinity $\omega$ is
defined as a reciprocal of $\epsilon$, i.e. $\frac{1}{\epsilon}$.

######  Definition 3.9.

For any numbers $x,y\in\mathbb{\widehat{R}}$,

  1. 1.

$x\geq y:=x<y\rightarrow\bot$

  2. 2.

$x<y:=x\leq y\land(x=y\rightarrow\bot)$

  3. 3.

$x>y:=x\geq y\land(x=y\rightarrow\bot)$

If we look further, the set $\mathbb{\widehat{R}}$ is actually an inconsistent
set. Example 3.10 gives one of these contradictions.

######  Example 3.10.

Suppose that we have a set $S=\\{x\in\mathbb{\widehat{R}}:\left\lvert
x\right\rvert<\frac{1}{n}\textnormal{ for all }n\in\mathbb{N}\\}$. In other
words, the set $S$ consists of all infinitesimals in $\mathbb{\widehat{R}}$.
It is easily proven that $S$ is not empty and bounded above. So by
Completeness Axiom, $S$ has a least upper bound. Suppose that $z$ is its least
upper bound (which also means that $z$ must be an infinitesimal). Because $z$
is an infinitesimal, $2z$ is also an infinitesimal and this means $2z$ is also
in $\mathbb{\widehat{R}}$. By using Definition 3.9, $z<2z$ and so, $z$ is not
the least upper bound of $S$. Now suppose that sup $S=2z$. The same argument
can be used to show that $2z$ is not supremum of $S$ but $3z$. We can build
this same argument infinitely to show that there does not exist $s$ such that
sup $S=s$. Thus, we have $\exists s:\textnormal{sup }S=s$ and $\nexists
s:\textnormal{sup }S=s$.

This kind of contradiction forces us to use a non-explosive logic such as
paraconsistent logic instead of classical logic to do our reasoning in
$\mathbb{\widehat{R}}$. Furthermore, we choose a particular paraconsistent
reasoning strategy, Chunk and Permeate (C&P), to resolve our dilemma. The
detail explanations of this strategy can be seen in [9].

###  3.2 Chunks in $\mathfrak{\widehat{L}}$ and The Creation of The Set
$\mathbb{R^{Z_{<}}}$

Using the C&P strategy, we divided $\mathfrak{\widehat{L}}$ into some
consistent chunks — naturally, there might be several ways to do it (e.g. one
can have an idea to divide the original set into two, three, or even more
chunks). Nevertheless, we found out that one particular way to have just two
different chunks, as provided in here, is the most interesting one as it leads
to the creation of a new model. One chunk is a set which contains Axioms
(3.1–3.4,3.6), while the other chunk is a set which contains Axioms
(3.1–3.5,3.7). The consistency of each chunk was proved by providing a model
for each of them.

####  3.2.1 Model for The First Chunk

One of the possible — and interesting — chunks is a set consists of Axioms
3.1-3.4 and Axiom 3.6. Here we proved the consistency of this chunk and also
some corollaries that we have.

It is well-established that the set of hyperreals, $\mathbb{{}^{*}R}$, clearly
satisfied those axioms [6]. Nevertheless, the construction of hyperreals
$\mathbb{{}^{*}R}$ depends on highly non-constructive arguments. In
particular, it requires an axiom of set theory, the well-ordering principle,
which assumes into existence something that cannot be constructed [21]. Here
we proposed to take a look at a simpler set. Remember that our set has to
contain not just $\mathbb{R}$, but also infinitesimals and infinities (and the
combinations of the two). We take $\mathbb{R^{Z}}$, functions from integers to
real numbers, as our base set. The member of $\mathbb{R^{Z}}$ consists of
standard and non-standard parts. The standard part of a certain number simply
shows its finite element (the real part), while the non-standard part shows
its infinite or infinitesimal part (see Definition 3.11).

######  Definition 3.11 (Member of $\mathbb{R^{Z}}$).

A typical member of $\mathbb{R^{Z}}$ has the form
$\mathbf{x}=\langle\epsilon_{-i},\widehat{x},{\epsilon_{j}}\rangle$ where
$\widehat{x}\in\mathbb{R}$ and $\epsilon_{n}$ denotes the sequence of the
constant part of infininitesimals if $n>0$, and infinities if $n<0$.

Notice that the symbol $\widehat{x}$ in Definition 3.11 signs the standard
part of a number in $\mathbb{R^{Z}}$. Thus, the member of $\mathbb{R^{Z}}$ can
be seen as a sequence of infinite numbers. Example 3.12 gives an overview of
how to write a number as a member of $\mathbb{R^{Z}}$.

######  Example 3.12 (Numbers in $\mathbb{R^{Z}}$).

  1. 1.

The number 1 is written as $\textbf{1}=\langle\dots
0,0,\widehat{1},0,0,\dots\rangle$.

  2. 2.

The number $\epsilon$ is written as $\mathbf{\epsilon}=\langle\dots
0,\widehat{0},1,0,\dots\rangle$.

  3. 3.

The number $\omega$ (one of the infinities) is written as
$\mathbf{\omega}=\langle\dots 0,1,\widehat{0},0,\dots\rangle$.

  4. 4.

The number $2+2\epsilon-\omega^{2}$ is written as
$\mathbf{2+2\epsilon-\omega^{2}}=\langle\dots,0,-1,0,\widehat{2},2,0,\dots\rangle$.

By using this form, all of the possible numbers can be written in
$\mathbb{R^{Z}}$. However, this infinite form is problematic in a number of
ways. For example, multiplication cannot be easily defined and there might
exist multiple inverses if the set $\mathbb{R^{Z}}$ was going to be used (see
Example 3.16 in [32, p. 47]). Because of this, the semi-infinite form is
motivated and the modified set is denoted by $\mathbb{R^{Z_{<}}}$. The only
difference between $\mathbb{R^{Z_{<}}}$ and $\mathbb{R}$ is that, for any
number $\mathbf{x}$, we will not have an infinite sequence on the left side of
its standard part. See example 3.13 and compare to Example 3.12.

######  Example 3.13 (Numbers in $\mathbb{R^{Z_{<}}}$).

  1. 1.

A number 1 is written as $\textbf{1}=\langle\widehat{1},0,0,\dots\rangle$.

  2. 2.

A number $\epsilon$ is written as
$\mathbf{\epsilon}=\langle\widehat{0},1,0,\dots\rangle$.

  3. 3.

A number $\omega$ (one of the infinities) is written as
$\mathbf{\omega}=\langle 1,\widehat{0},0,\dots\rangle$.

  4. 4.

A number $2+2\epsilon-\omega^{2}$ is written as
$\mathbf{2+2\epsilon-\omega^{2}}=\langle-1,0,\widehat{2},2,0,\cdots\rangle$.

Some of the properties of the set $\mathbb{R^{Z_{<}}}$ are as follows, while
their proof (when needed) and some examples of them can be seen in [32, p.
51–57].

######  Definition 3.14 (Addition and Multiplication in $\mathbb{R^{Z_{<}}}$).

For any number $\mathbf{x}=\langle x_{z}\rangle$ and $\mathbf{y}=\langle
y_{z}\rangle$ in $\mathbb{R^{Z_{<}}}$, define:

$\displaystyle\mathbf{x}+\mathbf{y}=\langle x_{z}+y_{z}:z\in\mathbb{Z}\rangle$

and $\mathbf{x}\times\mathbf{y}$ is calculated by:

$\displaystyle\mathbf{x}\times\mathbf{y}=\left(\sum_{i=-m}a_{i}\epsilon^{i}\right)\widehat{\times}\left(\sum_{j=-n}b_{j}\epsilon^{j}\right)=\left(\sum_{k\in\mathbb{Z}}c_{k}\epsilon^{k}\right),$

where $c_{k}=\sum_{i+j=k}a_{i}b_{j}$.

######  Definition 3.15 (Order in $\mathbb{R^{Z_{<}}}$).

The set $\mathbb{R^{Z_{<}}}$ is endowed with $\widehat{\leq}$, the
lexicographical ordering.

######  Proposition 3.16.

The set $\mathbb{R^{Z_{<}}}$ satisfies the additive property in Axiom 3.1.

######  Proposition 3.17.

The set $\mathbb{R^{Z_{<}}}$ satisfies the multiplicative property in Axiom
3.2.

######  Proposition 3.18.

The set $\mathbb{R^{Z_{<}}}$ satisfies the distributive property in Axiom 3.3.

######  Proposition 3.19.

The set $\mathbb{R^{Z_{<}}}$ satisfies the total order property in Axiom 3.4.

The following results show how to find an inverse of any members of
$\mathbb{R^{Z_{<}}}$ and its uniqueness property.

######  Proposition 3.20.

The number $\mathbf{1+\omega}$ has a unique inverse.

######  Proposition 3.21.

The number $\mathbf{\omega\widehat{+}\epsilon}$ has a unique inverse.

######  Lemma 3.22.

For any $r,s\in\mathbb{R}$, a number
$r\mathbf{\epsilon}\widehat{+}s\mathbf{\omega}$ has a unique inverse.

######  Lemma 3.23.

For any $r\in\mathbb{R}$, a number $r\widehat{+}\mathbf{\omega}$ (or
$r\widehat{+}\mathbf{\epsilon})$ has a unique inverse.

######  Theorem 3.24.

For any number $\mathbf{x}\in\mathbb{R^{Z_{<}}}$, $\mathbf{x}$ has a unique
inverse.

####  3.2.2 Model for The Second Chunk

The most evident model for this second chunk is the set of real numbers,
$\mathbb{R}$. Thus, so far, we have already had two chunks in
$\mathfrak{\widehat{L}}$ and we proved their consistencies by providing a
model for each of them.

###  3.3 Grossone Theory and The Set $\mathbb{R^{Z_{<}}}$

Theories that contain infinities have always been an issue and have attracted
much research, for example [10, 17, 20, 25, 37]. Note that the arithmetic
developed for infinite numbers was quite different with respect to the finite
arithmetic that we are used to dealing with. For example, Sergeyev in [44]
created the Grossone theory. The basic idea of this theory is to treat
infinity as an ‘normal’ number, so that our usual arithmetic rules apply. He
named this infinite number Grossone and denoted it with ①. The four axioms
that form this Grossone theory and its details can be seen [44].

It is very important to emphasis that ① is a number, and so it works as a
usual number. For example, there exist numbers such as $\text{\char
172}-100,\text{\char 172}^{3}+16,\ln\text{\char 172}$, and etc. Also for
instance, $\text{\char 172}-1<\text{\char 172}$.555This is unlike the way the
usual infinity, $\infty$, behaves where for example, $\infty-1=\infty$. It
also differs from how Cantor’s cardinal numbers behave. The introduction for
this new number ① makes us able to rewrite the set of natural numbers
$\mathbb{N}$ as:

$\mathbb{N}=\\{1,2,3,4,5,6,\dots,\text{\char 172}-2,\text{\char
172}-1,\text{\char 172}\\}$.

Furthermore, adding the Infinite Unit Axiom (IUA) to the axioms of natural
numbers will define the set of extended natural numbers $\mathbb{{}^{*}N}$:

$\mathbb{{}^{*}N}=\\{1,2,3,4,5,6,\dots,\text{\char 172}-1,\text{\char
172}\,\text{\char 172}+1,\dots,\text{\char 172}^{2}-1,\text{\char
172}^{2},\dots\\}$,

and the set $\mathbb{{}^{*}Z}$, extended integer numbers, can be defined from
there.

Here we argued that our new set $\mathbb{R^{Z_{<}}}$ provides the model of
Grossone theory and therefore proves its consistency rather in a deftly way.

######  Definition 3.25.

In our system $\mathbb{R^{Z_{<}}}$, the number ① is written as:

$\text{\char 172}=\langle 1,\widehat{0},0,\dots\rangle$.

######  Proposition 3.26.

For every finite number $\mathbf{r}\in\mathbb{R^{Z_{<}}}$,
$\mathbf{r}<\text{\char 172}$.

###### Proof.

The order in set $\mathbb{R^{Z_{<}}}$ is defined lexicographically. Now
suppose that $\mathbf{r}=\langle 0,\widehat{r},0,0,\dots\rangle$ where
$\widehat{r}\in\mathbb{R}$. Then it is clear that $\mathbf{r}<\text{\char
172}$. ∎

######  Proposition 3.27.

All of the equations in the Identity Axiom of Grossone theory are also hold in
$\mathbb{R^{Z_{<}}}$.

The fractional form of ① can also be defined in $\mathbb{R^{Z_{<}}}$ as:

for any $n\in\mathbb{N}$, $\frac{\text{\char
172}}{n}=\langle\frac{1}{n},\widehat{0},0,\dots\rangle$.

Speaking about the inverse, one of the advantages of having the set
$\mathbb{R^{Z_{<}}}$ is to be able to see what the inverse of a number looks
like, not like in the Grossone theory. See Example 3.28 for more details.

######  Example 3.28.

In Grossone theory, the inverse of $\frac{1}{\text{\char 172}}+\text{\char
172}$ is just $\frac{1}{\frac{1}{\text{\char 172}}+\text{\char 172}}$. While
in our set $\mathbb{R^{Z_{<}}}$, $\frac{1}{\text{\char 172}}+\text{\char 172}$
is written as $\epsilon+\omega$ and its inverse is

$\langle\widehat{0},1,0,-1,0,1,0,-1,0,\dots\rangle=\epsilon-\epsilon^{3}+\epsilon^{5}-\epsilon^{7}+\dots$.

In other words, the more explicit form of $\frac{1}{\frac{1}{\text{\char
172}}+\text{\char 172}}$ is a series $(-1)^{n+1}\text{\char 172}^{-(2n-1)}$
for $n=1,2,3,\dots\in\mathbb{N}$.

In [23], Gabriele Lolli analysed and built a formal foundation of the Grossone
theory based on Peano’s second order arithmetic. He also gave a slightly
different notion of some axioms that Sergeyev used. One of the important
theorems in Lolli’s paper is the proof that Grossone theory – or at least his
version of it – is consistent. However, as he said also in [23], ‘The
statement of the theorem is of course conditional, as apparent from the proof,
upon the consistency of $\text{PA}_{\mu}^{2}$’ while ‘its model theoretic
proof is technically rather demanding’.

Thus, through what presented in this subsection, we have proposed a new way to
prove the consistency of Grossone theory by providing a straightforward model
of it. There is no need for complicated model-theoretic proofs. The set
$\mathbb{R^{Z_{<}}}$ is enough to establish the consistency of Grossone theory
in general. Moreover, the development in the next sessions can also be seen,
at least in part, as a contribution to the development of Grossone theory.

##  4 Topology on The Set $\mathbb{R^{Z_{<}}}$

Some topological properties of the set $\mathbb{R^{Z_{<}}}$ are discussed in
this section. However, there are a number of definitions and issues that
should be addressed first in order to understand how those properties will be
applied to our set properly.

###  4.1 Metrics in $\mathbb{R^{Z_{<}}}$

We defined what is meant by a distance (metric) between each pair of elements
of $\mathbb{R^{Z_{<}}}$.

######  Definition 4.1.

A metric $\rho$ in a set $X$ is a function

$\displaystyle\rho:X\times X\rightarrow\left[0,\infty\right)$

where for all $x,y,z\in X$, these four conditions are satisfied:

  1. 1.

$\rho(x,y)\geq 0$,

  2. 2.

$\rho(x,y)=0$ if and only if $x=y$,

  3. 3.

$\rho(x,y)=d(y,x)$,

  4. 4.

$\rho(x,z)\leq\rho(x,y)+\rho(y,z)$.

When that function $\rho$ satisfies all of the four conditions above except
the second one, $\rho$ is called a pseudo-metric666It is not without reason
that we introduced the concept of the pseudo-metric here. This kind of metric
will make sense when we are in $\mathbb{R}$. For example, the distance between
$0$ and $\epsilon$ is $0$ as our lens is not strong enough to distinguish
those two numbers in $\mathbb{R}$. on $X$.

Now we define two functions $\operatorname{\texttt{d}}$ and
$\operatorname{\texttt{d}_{\texttt{$\psi$}}}$ in $\mathbb{R^{Z_{<}}}$ as
follows:

######  Definition 4.2.

For all $\mathbf{x},\mathbf{y}\in\mathbb{R^{Z_{<}}}$,

$\displaystyle\operatorname{\texttt{d}}:\mathbb{R^{Z_{<}}}\times\mathbb{R^{Z_{<}}}\rightarrow\mathbb{R^{Z_{<}}}\textnormal{
and
}\operatorname{\texttt{d}_{\texttt{$\psi$}}}:\mathbb{R^{Z_{<}}}\times\mathbb{R^{Z_{<}}}\rightarrow\mathbb{R}$

where
$\operatorname{\texttt{d}}(\mathbf{x},\mathbf{y})=\left\lvert\mathbf{y}-\mathbf{x}\right\rvert$
and
$\operatorname{\texttt{d}_{\texttt{$\psi$}}}(\mathbf{x},\mathbf{y})=\texttt{St}(\left\lvert\mathbf{y}-\mathbf{x}\right\rvert)$.

It can be easily verified that $\operatorname{\texttt{d}}$ is a metric in
$\mathbb{R^{Z_{<}}}$ (and so
$\left(\mathbb{R^{Z_{<}}},\operatorname{\texttt{d}}\right)$ forms a metric
space) and $\operatorname{\texttt{d}_{\texttt{$\psi$}}}$ is a pseudo-metric in
$\mathbb{R^{Z_{<}}}$ (and so
$\left(\mathbb{R^{Z_{<}}},\operatorname{\texttt{d}_{\texttt{$\psi$}}}\right)$
forms a pseudo-metric space).

###  4.2 Balls and Open Sets in $\mathbb{R^{Z_{<}}}$

Now that we have the notion of distance in $\mathbb{R^{Z_{<}}}$, we can define
what it means to be an open set in $\mathbb{R^{Z_{<}}}$ by first defining what
a ball is in $\mathbb{R^{Z_{<}}}$.

######  Definition 4.3.

A ball of radius $\mathbf{y}$ around the point
$\mathbf{x}\in\mathbb{R^{Z_{<}}}$ is

$\displaystyle
B_{\mathbf{x}}(\mathbf{y})=\\{\mathbf{z}\in\mathbb{R^{Z_{<}}}\mid
d_{1}(\mathbf{x},\mathbf{z})<\mathbf{y}\\},$

where $d_{1}(\mathbf{x},\mathbf{y})$ is either
$\operatorname{\texttt{d}}(\mathbf{x},\mathbf{y})$ or
$\operatorname{\texttt{d}_{\texttt{$\psi$}}}(\mathbf{x},\mathbf{y})$.

We require this additional definition in order to set forth our explanation
about balls properly:

######  Definition 4.4.

The sets $\Delta^{m}$ and $\Delta^{\underset{\downarrow}{m}}$ are defined as
follows:

$\Delta^{m}=\\{\mathbf{x}:\mathbf{x}=a_{m}\mathbf{\epsilon}^{m}\\}$ and
$\Delta^{\underset{\downarrow}{m}}=\bigcup\limits_{n\geq m}\Delta^{n}$,

where $m\in\mathbb{N}\cup\\{0\\}$, $a_{m}\in\mathbb{R}$ and $a_{m}\neq 0$
whenever $m\geq 1$.

We have to be careful here as unlike in classical topology, there are
different notions of balls that can be described as follows. The first
possible notion of balls is when we use $\operatorname{\texttt{d}}$ as our
metric and having $\mathbf{y}>0$ as our radius. In this case, in
$\mathbb{R^{Z_{<}}}$, the ball around a point $\mathbf{x}$ with $\mathbf{y}$
radius is an interval $(\mathbf{x}-\mathbf{y},\mathbf{x}+\mathbf{y})$. Note
that by using $\mathbf{y}$ as the radius, beside having the usual balls with
“real” radius (St-balls) (that is when $\mathbf{y}=r\in\mathbb{R}$), we also
have some infinitesimally small balls (e-balls) when
$\mathbf{y}=\mathfrak{e}\in\Delta^{\underset{\downarrow}{m}}$ for any given
$m$. The second possible notion is while we use the same metric
$\operatorname{\texttt{d}}$, we have $\nicefrac{{1}}{{n}}$ for some
$n\in\mathbb{N}$ as its radius. This produces balls (rat-balls) in the form of
$\left(\mathbf{x}-\nicefrac{{1}}{{n}},\mathbf{x}+\nicefrac{{1}}{{n}}\right)$.
The third possibility is by using
$\operatorname{\texttt{d}_{\texttt{$\psi$}}}$ as our metric. In this case,
interestingly, the balls around a point $\mathbf{x}$ with
$\nicefrac{{1}}{{n}}$ radius will be in the form of the following set:

$\\{\mathbf{y}\mid\texttt{St}(\mathbf{y})\in(\texttt{St}(\mathbf{x})-\nicefrac{{1}}{{n}},\texttt{St}(\mathbf{x})+\nicefrac{{1}}{{n}})\\}$.

We call this kind of balls as psi-balls. See Table 1 for the summary of these
possibilities of balls in our sets.

Table 1: Some types of balls both in $\mathbb{R}$ and $\mathbb{R^{Z_{<}}}$
Balls in $\mathbb{R}$ and $\mathbb{R^{Z_{<}}}$
---
The Set | The Metric | The Form of The Balls
$\mathbb{R}$ |  $\rho(x,y)$ $=\left\lvert x-y\right\rvert$ |  $B_{x}(r)$ $=(x-r,x+r)$
$\mathbb{R^{Z_{<}}}$ |  $\operatorname{\texttt{d}}(\mathbf{x},\mathbf{y})$ $=\left\lvert\mathbf{x}-\mathbf{y}\right\rvert$ |  $B_{\mathbf{x}}(\mathbf{r})$ $=(\mathbf{x}-\mathbf{r},\mathbf{x}+\mathbf{r})$
$\mathbb{R^{Z_{<}}}$ |  $\operatorname{\texttt{d}}(\mathbf{x},\mathbf{y})$ $=\left\lvert\mathbf{x}-\mathbf{y}\right\rvert$ | $B_{\mathbf{x}}(\nicefrac{{1}}{{n}})=(\mathbf{x}-\nicefrac{{1}}{{n}},\mathbf{x}+\nicefrac{{1}}{{n}})$
$\mathbb{R^{Z_{<}}}$ | $\operatorname{\texttt{d}_{\texttt{$\psi$}}}(\mathbf{x},\mathbf{y})=\texttt{St}(\left\lvert\mathbf{x}-\mathbf{y}\right\rvert)$ | $B_{\mathbf{x}}(\nicefrac{{1}}{{n}})=B_{\texttt{St}(\mathbf{x})}(\nicefrac{{1}}{{n}})=\\{\mathbf{y}\mid\texttt{St}(\mathbf{y})\in$
|  |  $\quad(\texttt{St}(\mathbf{x})-\nicefrac{{1}}{{n}},\texttt{St}(\mathbf{x})+\nicefrac{{1}}{{n}})\\}$

######  Remark 4.5.

In $\mathbb{R}$, the $\mathfrak{e}$-ball does not exist, whereas in
$\mathbb{R^{Z_{<}}}$ there are infinitely many $\mathfrak{e}$-balls around
every point there.

Finally, we defined what it means to be an open set in $\mathbb{R^{Z_{<}}}$.
Notice that because we had two notions of ball in our set, i.e. St-balls and
$\mathfrak{e}$-ball, it led us to two different notions of openness as
follows.

######  Definition 4.6 (St-open).

A subset $O\subseteq\mathbb{R^{Z_{<}}}$ is St-open iff

$\forall x\in O$ $\exists n\in\mathbb{N}\textnormal{ s.t.
}B_{x}\left(\frac{1}{n}\right)\subseteq O$.

######  Definition 4.7 ($\mathfrak{e}$-open).

A subset $O\subseteq\mathbb{R^{Z_{<}}}$ is e-open iff

$\forall x\in O$
$\exists\mathfrak{e}\in\Delta^{\underset{\downarrow}{m}}\textnormal{ s.t.
}B_{x}(\mathfrak{e})\subseteq O$.

Remember that the set $\Delta^{\underset{\downarrow}{m}}$ is defined in
Definition 4.4.

######  Example 4.8.

The interval $(\textbf{2},\textbf{3})$ in $\mathbb{R^{Z_{<}}}$ is St-open and
also $\mathfrak{e}$-open.

######  Example 4.9.

The interval $(\textbf{0},\mathbf{\mathfrak{e}})$ in $\mathbb{R^{Z_{<}}}$ is
$\mathfrak{e}$-open, but not St-open.

Example 4.9 gives us the theorem below:

######  Theorem 4.10.

For any set $U\subseteq\mathbb{R^{Z_{<}}}$,

$\displaystyle\mathbb{R^{Z_{<}}}\not\models\textit{If }U\textit{ is {St}-open,
then }U\textit{ is }\mathfrak{e}\textit{-open}.$

Using the two definition of openness given in Definitions 4.6 and 4.7, we
defined what it means by two points are topologically distinguishable. There
are also two different notions of distinguishable points as can be seen in
Definitions 4.11 and 4.12.

######  Definition 4.11 (St-distinguishable).

Any two points in $\mathbb{R^{Z_{<}}}$ are St-distinguishable if and only if
there is a St-open set containing precisely one of the two points.

######  Definition 4.12 ($\mathfrak{e}$-distinguishable).

Any two points in $\mathbb{R^{Z_{<}}}$ are e-distinguishable if and only if
there is an $\mathfrak{e}$-open set containing precisely one of the two
points.

###  4.3 Topological Spaces

######  Definition 4.13.

Let $X$ be a non-empty set and $\uptau$ a collection of subsets of $X$ such
that:

  1. 1.

$X\in\uptau$,

  2. 2.

$\emptyset\in\uptau$,

  3. 3.

If $O_{1},O_{2},\dots,O_{n}\in\uptau$, then $\bigcap_{k=1}^{n}O_{k}\in\uptau$,

  4. 4.

If $O_{\alpha}\in\uptau$ for all $\alpha\in A$, then $\bigcup_{\alpha\in
A}O_{\alpha}\in\uptau$.

The pair of objects $(X,\uptau)$ is called a topological space where $X$ is
called the underlying set, the collection $\uptau$ is called the topology in
$X$, and the members of $\uptau$ are called open sets.

Note that if $\uptau$ is the collection of open sets of a metric space
$(\mathcal{X},\rho)$, then $(\mathcal{X},\uptau)$ is a topological metric
space, i.e. a topological space associated with the metric space $(X,\rho)$.
There are at least three interesting topologies in $\mathbb{R^{Z_{<}}}$ as can
be seen in Definition 4.14 below.

######  Definition 4.14.

The standard topology $\uptau_{\textnormal{St}}$ on the set
$\mathbb{R^{Z_{<}}}$ is the topology generated by all unions of St-balls. The
$\mathfrak{e}$-topology in $\mathbb{R^{Z_{<}}}$, $\uptau_{\mathfrak{e}}$, is
the topology generated by all unions of $\mathfrak{e}$-balls and the third
topology in $\mathbb{R^{Z_{<}}}$ is pseudo-topology, $\uptau_{\psi}$, when it
is induced by $\operatorname{\texttt{d}_{\texttt{$\psi$}}}$.

######  Axiom 4.15.

$(\mathbb{R^{Z_{<}}},\uptau_{n}),$
$(\mathbb{R^{Z_{<}}},\uptau_{\mathfrak{e}}),$ and
$(\mathbb{R^{Z_{<}}},\uptau_{\psi})$ form topological metric space with
$\operatorname{\texttt{d}}$ as their metrics (for the first two) and
$\operatorname{\texttt{d}_{\texttt{$\psi$}}}$ for the third one.

######  Theorem 4.16.

$(\mathbb{R^{Z_{<}}},\uptau_{n})$ is not a Hausdorff space but it is a
preregular space.777Hausdorff space and preregular space are defined as usual.

###### Proof.

$\mathbb{R^{Z_{<}}}$ does not form a Hausdorff space because under the
topology $\uptau_{\textnormal{St}}$, there are two distinct points,
$\mathbf{\epsilon}=\langle\widehat{0},1\rangle$ and
$\mathbf{\epsilon}+\textbf{1}=\langle\widehat{1},1\rangle$ for example, which
are not neighbourhood-separable. It is impossible to separate those two points
with St-ballss as $\nicefrac{{1}}{{n}}>\mathfrak{e}$ for every
$n\in\mathbb{N}$ and $\mathfrak{e}\in\Delta^{\underset{\downarrow}{m}}$.
However, it is a preregular space as every pair of two St-distinguishable
points in $\mathbb{R^{Z_{<}}}$ can be separated by two disjoint
neighbourhoods. This follows directly from Definition 4.11. ∎

######  Theorem 4.17.

$(\mathbb{R^{Z_{<}}},\uptau_{\mathfrak{e}})$ is a non-connected space and it
forms a Hausdorff space.

###### Proof.

We observe that for all $\mathbf{x_{0}}\in\mathbb{R^{Z_{<}}}$ and
$\mathfrak{e}\in\Delta^{\underset{\downarrow}{m}}$, the balls
$B_{\mathfrak{e}}(\mathbf{x_{0}})$ are $\mathfrak{e}$-open and so is the whole
space. To show that $\mathbb{R^{Z_{<}}}$ is not connected, let

$\displaystyle S_{1}$
$\displaystyle=\\{\mathbf{x}\in\mathbb{R^{Z_{<}}}\mid(\mathbf{x}\leq\textbf{0})\textnormal{
or }(\mathbf{x}>\textbf{0}\textnormal{ and
}\mathbf{x}\in\Delta^{\underset{\downarrow}{m}})\\}\textnormal{ and }$
$\displaystyle S_{2}$
$\displaystyle=\\{\mathbf{x}\in\mathbb{R^{Z_{<}}}\mid(\mathbf{x}>\textbf{0})\textnormal{
and }\mathbf{x}\notin\Delta^{\underset{\downarrow}{m}}\\}.$

The sets $S_{1}$ and $S_{2}$ are $\mathfrak{e}$-open, disjoint and moreover,
we have that $\mathbb{R^{Z_{<}}}=S_{1}\cup S_{2}$ (and so $\mathbb{R^{Z_{<}}}$
is not connected). For any $\mathbf{x},\mathbf{y}\in\mathbb{R^{Z_{<}}}$,
$B_{x}(\nicefrac{{\operatorname{\texttt{d}}(x,y)}}{{2}})$ and
$B_{y}(\nicefrac{{\operatorname{\texttt{d}}(x,y)}}{{2}})$ are open and
disjoint. Thus, $\mathbb{R^{Z_{<}}}$ forms a Hausdorff space. ∎

We will now state the usual definition of the basis of a topology $\uptau$.

######  Definition 4.18.

Let $(X,\uptau)$ be a topological space. A basis for the topology $\uptau$ is
a collection $\mathcal{B}$ of subsets from $\uptau$ such that every
$U\in\uptau$ is the union of some collections of sets in $\mathcal{B}$, i.e.

$\forall U\in\uptau$, $\exists\mathcal{B^{*}}\subseteq\mathcal{B}$ s.t.
$U=\bigcup\limits_{B\in\mathcal{B^{*}}}B$

######  Example 4.19.

On $\mathbb{R}$ with its usual topology, the set $\mathcal{B}=\\{(a,b):a<b\\}$
is a topological basis.

######  Definition 4.20.

Let $(X,\uptau)$ be a topological space and let $x\in X$. A local basis of $x$
is a collection of open neighbourhoods of $x$, $\mathcal{B}_{x}$, such that
for all $U\in\uptau$ with $x\in U$, $\exists B\in\mathcal{B}_{x}$ such that
$x\in B\subset U$.

######  Definition 4.21.

Let $(X,\uptau)$ be a topological space. Then $(X,\uptau)$ is first-countable
if every point $x\in X$ has a countable local basis.

######  Definition 4.22.

Let $(X,\uptau)$ be a topological space. Then $(X,\uptau)$ is second-countable
if there exists a basis $\mathcal{B}$ of $\uptau$ that is countable.

######  Theorem 4.23.

$(\mathbb{R^{Z_{<}}},\tau_{\mathfrak{e}})$ is first countable but not second-
countable.888Note that the space $(\mathbb{R^{Z_{<}}},\tau_{n})$ is still
second-countable.

###### Proof.

From Axiom 4.15 and because every metric space is first-countable, it follows
that $(\mathbb{R^{Z_{<}}},\tau_{\mathfrak{e}})$ is first-countable. However,
there cannot be any countable bases in $\tau_{\mathfrak{e}}$ as the
uncountably many open sets $O_{x}=(x-\mathfrak{e},x+\mathfrak{e})$ are
disjoint. ∎

##  5 Calculus on $\mathbb{R^{Z_{<}}}$

It has been proved previously that the set $\mathbb{R^{Z_{<}}}$ forms a field.
Remember that for any $\mathbf{x}\in\mathbb{R^{Z_{<}}}$,

$\mathbf{x}=\langle
x_{-n},x_{-(n-1)},\dots,x_{-2},x_{-1},\widehat{x},x_{1},x_{2},x_{3},\dots\rangle$

where

St$(\mathbf{x})=\widehat{x}$,

Nst${}_{\epsilon}(\mathbf{x})=\\{x_{1},x_{2},x_{3},\dots\\}$, and

Nst${}_{\omega}(\mathbf{x})=\\{x_{-n},x_{-(n-1)},\dots x_{-1}\\}$.

In other words, for every $\mathbf{x}\in\mathbb{R^{Z_{<}}}$,

$\mathbf{x}=$ Nstω$(\mathbf{x})$ $+$ St$(\mathbf{x})$ $+$ Nstϵ$(\mathbf{x})$.

Note that we can think of St(), Nstϵ(), and Nstω() as linear functions – that
is for any $\mathbf{x},\mathbf{y}\in\mathbb{R^{Z_{<}}}$ and a constant
$c\in\mathbb{R}$,

St$(\mathbf{x}+\mathbf{y})=$St$(\mathbf{x})+$St$(\mathbf{y})$,
St$(c\mathbf{x})=c$St$(\mathbf{x})$,

Nstϵ$(\mathbf{x}+\mathbf{y})=$Nstϵ$(\mathbf{x})+$Nstϵ$(\mathbf{y})$,
Nstϵ$(c\mathbf{x})=c$NStϵ$(\mathbf{x})$,

Nstω$(\mathbf{x}+\mathbf{y})=$Nstω$(\mathbf{x})+$Nstω$(\mathbf{y})$, and
Nstω$(c\mathbf{x})=c$Nstω$(\mathbf{x})$.

######  Definition 5.1.

Suppose that $\texttt{ni}_{\epsilon}(\mathbf{x})$ denotes the non-
infinitesimal part of $\mathbf{x}\in\mathbb{R^{Z_{<}}}$, i.e.
$\texttt{ni}_{\epsilon}(\mathbf{x})=\texttt{Nst}_{\omega}(\mathbf{x})+\texttt{St}(\mathbf{x})$
and function in $\mathbb{R^{Z_{<}}}$ be defined in the usual way. Then a
function $f$ in $\mathbb{R^{Z_{<}}}$ is microstable if and only if

$\texttt{ni}_{\epsilon}(f(x+\epsilon))=\texttt{ni}_{\epsilon}(f(x))$,

######  Example 5.2.

Suppose that a function $f$ in $\mathbb{R^{Z_{<}}}$ is defined as follows:

$f(\mathbf{x})=\begin{cases}1,&\text{if }\texttt{St}(\mathbf{x})>0\\\
0,&\text{else.}\end{cases}$

Then $f(\mathbf{x})$ is a microstable function.

######  Theorem 5.3.

Microstability is closed under addition, multiplication, and
composition.999The proof of this theorem can be seen in [32, p. 73].

Now for every function $f$ defined in $\mathbb{R^{Z_{<}}}$, we are going to
have the operator $\texttt{Der}_{f}$ which takes a 2-tuple in
$\left(\mathbb{R}\times\mathbb{R^{Z_{<}}}\right)$ as its input and returns a
member of $\mathbb{R^{Z_{<}}}$ as the output, i.e.:

$\texttt{Der}_{f}:\mathbb{R}\times\mathbb{R^{Z_{<}}}\rightarrow\mathbb{R^{Z_{<}}}$.

Eventually, this operator will be called a derivative of $f$.

Using Newton’s original definition (and a slight change of notation), if a
function $f(x)$ is differentiable, then its derivative is given by:

$\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})=\cfrac{f(\mathbf{x}+\mathbf{\epsilon})-f(\mathbf{x})}{\mathbf{\epsilon}}.$
(3)

Now suppose that we want to find a derivative of $f$ where $f$ is a function
defined in $\mathbb{R^{Z_{<}}}$. We can certainly use Equation 3 to calculate
it as that equation holds for any function $f$. But how is this calculation
related to the calculus practised in classical mathematics? Note that using
Newton’s definition to calculate the derivative will necessarily involve an
inconsistent step. This inconsistency is located in the treatment given to the
infinitesimal number. Thus it makes sense that in order to explore the problem
posed above, we will use a paraconsistent reasoning strategy which is called
Chunk and Permeate.

###  5.1 Chunk and Permeate for Derivative in $\mathbb{R^{Z_{<}}}$

Details on the Chunk & Permeate reasoning strategy can be seen in [9]. Before
applying this strategy for the derivative in $\mathbb{R^{Z_{<}}}$, define a
set $E$ which consists of any algebraic terms such that they satisfy:

$\texttt{St}(\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon}))=f\textnormal{
}^{\prime}(x)$,

where $f\textnormal{ }^{\prime}(x)$ denotes the usual derivative of $f$ in
$\mathbb{R}$. We will need this set $E$ when we try to define the permeability
relation between chunks.

######  Proposition 5.4.

The set $E$ as defined above is inhabited.

###### Proof.

We want to show that the set $E$ has at least one element in it. It is clear
that the identity function $\texttt{id}(x)=x$ is in $E$ because for all
$\epsilon$:

$\displaystyle\texttt{St}(\texttt{Der}_{x}(\mathbf{x},\mathbf{\epsilon}))$
$\displaystyle=\texttt{St}\left(\cfrac{\mathbf{x}+\mathbf{\epsilon}-\mathbf{x}}{\mathbf{\epsilon}}\right)$
$\displaystyle=\texttt{St}\left(\cfrac{\mathbf{\epsilon}}{\mathbf{\epsilon}}\right)$
$\displaystyle=\texttt{St}(\textbf{1})=1=f\textnormal{ }^{\prime}(x).$

∎

######  Theorem 5.5.

If $f$ and $g$ are microstable functions in $E$ and $c$ is any real constant,
then

  1. 1.

$f\pm g$ are in $E$,

  2. 2.

$cf$ is in $E$,

  3. 3.

$fg$ is in $E$,

  4. 4.

$\cfrac{f}{g}$ is in $E$, and

  5. 5.

$f\circ g$ is in $E$.

The proof of the above theorem is rather long and so can be seen in [32, p.
76].

Now we are ready to construct the chunk and permeate structure, called
$\mathfrak{\widehat{R}}$, which is formally written as
$\mathfrak{\widehat{R}}=\langle\\{\Sigma_{S},\Sigma_{T}\\},\rho,T\rangle$
where the source chunk $\Sigma_{S}$ is the language of $\mathbb{R^{Z_{<}}}$,
the target chunk $\Sigma_{T}$ is the language of $\mathbb{R}$, and $\rho$ is
the permeability relation between $S$ and $T$.

##### The source chunk $\Sigma_{S}$

As stated before, this chunk is actually the language of the set
$\mathbb{{R^{Z_{<}}}}$ and therefore, it consists of all six of its axioms.
The source chunk requires one additional axiom to define what it means by
derivative. This additional axiom can be stated as:

S1: $Df=\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})$

where $\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})$ is defined in Equation
3.

##### The target chunk $\Sigma_{T}$

Again, the target chunk contains the usual axiom for the set or real numbers,
$\mathbb{R}$. There is only one additional axiom needed for this chunk:

T1: $\forall\mathbf{x}$ $\mathbf{x}=\texttt{St}(\mathbf{x})$.

Note that the axiom T1 above is actually equivalent to saying that
$\forall\mathbf{x}$ $\texttt{Nst}(\mathbf{x})=0$.

##### The permeability relation

The permeability relation $\rho(S,T)$ is the set of equations of the form

$Df=g$

where $f\in E$. The function $g$ which is permeated by this permeability
relation will be the first derivative of $f$ in $\mathbb{R}$. This
permeability relation shows that the derivative notion is permeable to the set
$\mathbb{R}$.

######  Example 5.6.

Suppose that $f(\mathbf{x})=\textbf{3}\mathbf{x}$ for all $\mathbf{x}$. First,
working within $\Sigma_{S}$, the operator $D$ is applied to $f$ such that:

$\displaystyle Df$
$\displaystyle=\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})$
$\displaystyle=\cfrac{\textbf{3}(\mathbf{x}+\mathbf{\epsilon})-\textbf{3}\mathbf{x}}{\mathbf{\epsilon}}$
$\displaystyle=\cfrac{\textbf{3}\mathbf{\epsilon}}{\mathbf{\epsilon}}=\textbf{3}.$

Note that
$\texttt{St}(\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon}))=\texttt{St}(\textbf{3})=3=f\textnormal{
}^{\prime}(x)$, and so $f(x)\in E$. Permeating the last equation of $Df$ above
to $\Sigma_{T}$ gives us:

$\displaystyle Df=3$

and so the derivative of $f(x)=3x$ is $3$.

######  Example 5.7.

Suppose that
$f(\mathbf{x})=\mathbf{x}^{\textbf{2}}+\textbf{2}\mathbf{x}+\textbf{3}$ for
all $\mathbf{x}$. First, working within $\Sigma_{S}$, the operator $D$ is
applied to $f$ such that:

$\displaystyle Df$
$\displaystyle=\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})$
$\displaystyle=\cfrac{(\mathbf{x}+\mathbf{\epsilon})^{\textbf{2}}+\textbf{2}(\mathbf{x}+\mathbf{\epsilon})+\textbf{3}-\mathbf{x}^{\textbf{2}}-\textbf{2}\mathbf{x}-\textbf{3}}{\mathbf{\epsilon}}$
$\displaystyle=\cfrac{\textbf{2}\mathbf{x}\mathbf{\epsilon}+\mathbf{\epsilon}^{\textbf{2}}+\textbf{2}\mathbf{\epsilon}}{\mathbf{\epsilon}}$
$\displaystyle=\textbf{2}\mathbf{x}+\mathbf{\epsilon}+\textbf{2}.$

Note that the standard part of
$\textbf{2}\mathbf{x}+\mathbf{\epsilon}+\textbf{2}$ will depend on the domain
of $\mathbf{x}$. That is:

$\texttt{St}(\textbf{2}\mathbf{x}+\mathbf{\epsilon}+\textbf{2})=\begin{cases}2x+2,&\text{if
}x\in\mathbb{R}\\\ 2,&\text{else.}\end{cases}$

In other words, if (and only if) $\texttt{Nst}(\mathbf{x})=0$, i.e.
$x\in\mathbb{R}$, $Df$ can be permeated into $\Sigma_{T}$. Thus, if $x$ is a
real number, then we have the derivative of $f(x)=x^{2}+2x+3=2x+2$.

######  Example 5.8.

Suppose that $f(\mathbf{x})=\operatorname{sign}(\mathbf{x})$ is defined as:

$\operatorname{sign}(\mathbf{x})=\begin{cases}\mathbf{1},&\text{if
}\texttt{St}(\mathbf{x})>0\\\ \mathbf{0},&\text{if
}\texttt{St}(\mathbf{x})=0\\\ \mathbf{-1},&\text{if
}\texttt{St}(\mathbf{x})<0\end{cases}$

First, working within $\Sigma_{S}$, the operator $D$ is applied to $f$ so
that:

$\displaystyle Df$
$\displaystyle=\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon})$
$\displaystyle=\cfrac{\operatorname{sign}(\mathbf{x}+\mathbf{\epsilon})-\operatorname{sign}(\mathbf{x})}{\mathbf{\epsilon}}$
$\displaystyle=0\text{ (because $\forall x$
$\texttt{St}(\mathbf{x})=\texttt{St}(\mathbf{x}+\epsilon)$)}$

Note that
$\texttt{St}(\texttt{Der}_{f}(\mathbf{x},\mathbf{\epsilon}))=\texttt{St}(\mathbf{0})=0=f\textnormal{
}^{\prime}(x)$, and so $f(x)\in E$. Permeating the last equation of $Df$ above
to $\Sigma_{T}$ gives us:

$\displaystyle Df=0$

and so the derivative of $f(\mathbf{x})=\operatorname{sign}(\mathbf{x})$ is
$0$ for all $\mathbf{x}$. Notice that this is not the case in $\mathbb{R}$,
where the derivative of the $\operatorname{sign}$ function at $x=0$ is not
defined because of its discontinuity. However, this is not really a bizarre
behaviour because if we look very closely at the infinitesimal neighbourhood
of $\mathbf{x}$ when $\texttt{St}(\mathbf{x})=0$, the function
$\operatorname{sign}(\mathbf{x})$ will look like a straight horizontal line
and so it makes a perfect sense to have $0$ as the slope of the tangent line
there. Moreover, this phenomenon also happens in distribution theory where
$\operatorname{sign}$ function has its derivative everywhere.

###  5.2 Transcendental Functions in $\mathbb{R^{Z_{<}}}$

As we know, there are some special functions defined in real numbers and two
of them are the trigonometric and the exponential functions. How then are
these functions defined in $\mathbb{R^{Z_{<}}}$? Here we propose to define
them using power series.

The first two trigonometric functions that we are going to discuss are the
$\sin$ and $\cos$ functions. Using the MacLaurin power series, these two
functions are defined as follows:

$\displaystyle\sin(\mathbf{x})=\sum\limits_{n=0}\cfrac{(-1)^{n}}{(2n+1)!}\text{
}\mathbf{x}^{2n+1}$ (4)

and

$\displaystyle\cos(\mathbf{x})=\sum\limits_{n=0}\cfrac{(-1)^{n}}{(2n)!}\text{
}\mathbf{x}^{2n}.$ (5)

The exponential function is defined as:

$\displaystyle\exp(\mathbf{x})=\sum\limits_{n=0}\cfrac{1}{n!}\mathbf{x}^{n}.$
(6)

Note that the MacLaurin polynomial is just a special case of Taylor polynomial
with regards to how the function is approximated at $\mathbf{x}=\textbf{0}$.

######  Example 5.9.

Suppose that we have
$\mathbf{x}=x+a\mathbf{\epsilon}=\langle\widehat{x},a,0,0,\dots\rangle$ where
$x,a\in\mathbb{R}$. We want to know what $\sin(\mathbf{x})$ is. Based on
Equation 4,

$\displaystyle\sin(\mathbf{x})=\sin(x+\mathbf{\epsilon})=(x+\mathbf{\epsilon})-\tfrac{\textbf{1}}{\textbf{3}!}(x+\mathbf{\epsilon})^{\textbf{3}}+\tfrac{\textbf{1}}{\textbf{5}!}(x+\mathbf{\epsilon})^{\textbf{5}}-\tfrac{\textbf{1}}{\textbf{7}!}(x+\mathbf{\epsilon})^{\textbf{7}}+\dots$

Our task now is to find all the members of
$\texttt{Nst}_{\mathbf{\epsilon}}(\mathbf{\sin(x)})$ and also
$\texttt{St}(\mathbf{\sin(x)})$. These are shown in Table 2. Note that from
the way the $\sin$ function is defined, $x_{i}=0$ $\forall
x_{i}\in\texttt{Nst}_{\omega}(\sin(\mathbf{x}))$. Thus from Table 2, we get:

$\displaystyle\sin(\mathbf{x})$ $\displaystyle=\sin(x+a\mathbf{\epsilon})$
$\displaystyle=\langle\widehat{\sin(x)},a\cos(x),-\tfrac{a^{2}}{2!}\sin(x),-\tfrac{a^{\textbf{3}}}{\textbf{3}!}\cos(x),\tfrac{a^{4}}{4!}\sin(x),\tfrac{a^{\textbf{5}}}{\textbf{5}!}\cos(x),\dots\rangle,$

and we also get

$\displaystyle\sin(\mathbf{\epsilon})=\langle\widehat{0},\textbf{1},0,-\tfrac{\textbf{1}}{\textbf{3}!},0,\tfrac{\textbf{1}}{\textbf{5}!},\dots\rangle=\mathbf{\epsilon}-\tfrac{\textbf{1}}{\textbf{3}!}\mathbf{\epsilon}^{\textbf{3}}+\tfrac{\textbf{1}}{\textbf{5}!}\mathbf{\epsilon}^{\textbf{5}}-\dots$

for an infinitesimal angle $\mathbf{\epsilon}$.

Table 2: $\texttt{St}(\sin(\mathbf{x)})$ and the first four members of $\texttt{Nst}_{\epsilon}(\sin(\mathbf{x}))$ | Expanded Form | Simplified Form
---|---|---
real-part | $\begin{array}[]{ll}=&\mathbf{x}-\tfrac{1}{3!}\mathbf{x}^{3}+\tfrac{1}{\textbf{5}!}\mathbf{x}^{\textbf{5}}-\tfrac{1}{\textbf{7}!}\mathbf{x}^{\textbf{7}}+\dots\\\ =&\sum_{n=0}\tfrac{-1^{n}}{(2n+1)!}\text{ }\mathbf{x}^{2n+1}\end{array}$ | $=\sin(\mathbf{x})$
$\mathbf{\epsilon}$-part | $\begin{array}[]{ll}=&a\mathbf{\epsilon}-\tfrac{1}{2!}\mathbf{x}^{2}a\mathbf{\epsilon}+\tfrac{1}{4!}a\mathbf{\epsilon}\mathbf{x}^{4}-\tfrac{1}{6!}a\mathbf{\epsilon}\mathbf{x}^{6}+\dots\\\ =&\mathbf{\epsilon}(a-\tfrac{1}{2!}a\mathbf{x}^{2}+\tfrac{1}{4!}a\mathbf{x}^{4}-\tfrac{1}{6!}a\mathbf{x}^{6}+\dots)\\\ =&\mathbf{\epsilon}\sum_{n=0}\tfrac{-1^{n}}{(2n)!}\text{ }a\mathbf{x}^{2n}\end{array}$ | $=\mathbf{\epsilon}(a\cos(\mathbf{x}))$
$\mathbf{\epsilon}^{2}$-part | $\begin{array}[]{ll}=&-\tfrac{3}{3!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}+\tfrac{\textbf{10}}{\textbf{5}!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{3}-\tfrac{21}{\textbf{7}!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{\textbf{5}}+\dots\\\ =&-\tfrac{1}{2!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}+\tfrac{2}{4!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{3}-\tfrac{3}{6!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{\textbf{5}}+\dots\\\ =&\mathbf{\epsilon}^{2}(-\tfrac{1}{2!}\mathbf{x}+\tfrac{2}{4!}a^{2}\mathbf{x}^{3}-\tfrac{3}{6!}a^{2}\mathbf{x}^{\textbf{5}}+\dots)\\\ =&\mathbf{\epsilon}^{2}\sum_{n=0}\tfrac{-1^{n+1}(n+1)}{(2n+2)!}\text{ }a^{2}\mathbf{x}^{2n+1}\end{array}$ | $=\mathbf{\epsilon}^{2}(-\tfrac{a^{2}}{2}\sin(\mathbf{x}))$
$\mathbf{\epsilon}^{3}$-part | $\begin{array}[]{ll}=&-\tfrac{1}{3!}a^{3}\mathbf{\epsilon}^{3}+\tfrac{\textbf{10}}{\textbf{5}!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{2}-\tfrac{\textbf{3{5}}}{\textbf{7}!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{4}+\tfrac{\textbf{84}}{\textbf{9}!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{6}-\dots\\\ =&\mathbf{\epsilon}^{3}(-\tfrac{1}{3!}a^{3}\mathbf{x}^{0}+\tfrac{\textbf{10}}{\textbf{5}!}a^{3}\mathbf{x}^{2}-\tfrac{\textbf{3{5}}}{\textbf{7}!}a^{3}\mathbf{x}^{4}+\tfrac{\textbf{84}}{\textbf{9}!}a^{3}\mathbf{x}^{6}-\dots)\\\ =&\mathbf{\epsilon}^{3}\sum_{n=0}\tfrac{-1^{n+1}}{6(2n)!}\text{ }a^{3}\mathbf{x}^{2n}\end{array}$ | $=\mathbf{\epsilon}^{3}(-\tfrac{a^{3}}{6}\cos(\mathbf{x}))$
$\mathbf{\epsilon}^{4}$-part | $\begin{array}[]{ll}=&\tfrac{\textbf{5}}{\textbf{5}!}a^{4}\mathbf{\epsilon}^{4}\mathbf{x}-\tfrac{\textbf{3{5}}}{\textbf{7}!}a^{4}\mathbf{\epsilon}^{4}\mathbf{x}^{3}+\tfrac{\textbf{126}}{\textbf{9}!}a^{4}\mathbf{\epsilon}^{4}\mathbf{x}^{\textbf{5}}-\tfrac{\textbf{3{30}}}{11!}a^{4}\mathbf{\epsilon}^{4}\mathbf{x}^{\textbf{7}}+\dots\\\ =&\mathbf{\epsilon}^{4}(\tfrac{1}{4!}\mathbf{x}-\tfrac{\textbf{5}}{6!}a^{4}\mathbf{x}^{3}+\tfrac{\textbf{14}}{8!}a^{4}\mathbf{x}^{\textbf{5}}-\tfrac{\textbf{30}}{\textbf{10}!}a^{4}\mathbf{x}^{\textbf{7}}+\dots)\\\ =&\mathbf{\epsilon}^{4}\sum_{n=0}\tfrac{-1^{n}}{24(2n+1)!}\text{ }a^{4}\mathbf{x}^{2n+1}\end{array}$ | $=\mathbf{\epsilon}^{4}(\tfrac{a^{4}}{24}\sin(\mathbf{x}))$

######  Example 5.10.

Suppose that we have $\mathbf{x}=x+a\epsilon$ where $x,a\in\mathbb{R}$. Here
we try to find what $\cos\mathbf{x}$ is. With a similar method to the one used
in Example 5.9, we have a calculation like what is shown in Table 3.

Table 3: $\texttt{St}(\cos(\mathbf{x)})$ and the first two members of $\texttt{Nst}_{\epsilon}(\cos(\mathbf{x}))$ real-part | $\begin{array}[]{ll}=&1-\tfrac{1}{2!}\mathbf{x}^{2}+\tfrac{1}{\textbf{4}!}\mathbf{x}^{\textbf{4}}-\tfrac{1}{\textbf{6}!}\mathbf{x}^{\textbf{6}}+\dots\\\ =&\sum_{n=0}\tfrac{-1^{n}}{(2n)!}\text{ }\mathbf{x}^{2n}\end{array}$ | $=\cos(\mathbf{x})$
---|---|---
$\mathbf{\epsilon}$-part | $\begin{array}[]{ll}=&-\tfrac{2}{2!}a\mathbf{\epsilon}\mathbf{x}+\tfrac{\textbf{4}}{\textbf{4}!}a\mathbf{\epsilon}\mathbf{x}^{3}-\tfrac{\textbf{6}}{\textbf{6}!}a\mathbf{\epsilon}\mathbf{x}^{5}+\dots\\\ =&\mathbf{\epsilon}(-a\mathbf{x}+\tfrac{1}{3!}a\mathbf{x}^{3}-\tfrac{1}{5!}a\mathbf{x}^{5}+\dots)\\\ =&\mathbf{\epsilon}\sum_{n=0}\tfrac{-1^{n+1}}{(2n+1)!}\text{ }a\mathbf{x}^{2n+1}\end{array}$ | $=\mathbf{\epsilon}(-a\sin(\mathbf{x}))$
$\mathbf{\epsilon}^{2}$-part | $\begin{array}[]{ll}=&-\tfrac{1}{2!}a^{2}\mathbf{\epsilon}^{2}+\tfrac{\textbf{6}}{\textbf{4}!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{2}-\tfrac{\textbf{15}}{\textbf{6}!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{\textbf{4}}+\tfrac{\textbf{28}}{8!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{\textbf{6}}-\dots\\\ =&\mathbf{\epsilon}^{2}\sum_{n=0}\tfrac{-1^{n+1}(n+1)(2n+1)}{(2n+2)!}\text{ }a^{2}\mathbf{x}^{2n}\\\ =&\mathbf{\epsilon}^{2}\sum_{n=0}\tfrac{1}{2}\tfrac{-1^{n+1}}{(2n)!}\text{ }\mathbf{x}^{2n}\\\ \end{array}$ | $=\mathbf{\epsilon}^{2}(-\tfrac{a^{2}}{2}\cos(\mathbf{x}))$

Thus from Table 3, we get:

$\displaystyle\cos(\mathbf{x})=\cos(x+a\mathbf{\epsilon})=\langle\widehat{\cos(x)},-a\sin(x),-\tfrac{a^{2}}{2!}\cos(x),\tfrac{a^{3}}{3!}\sin(x),\tfrac{a^{4}}{4!}\cos(x),\dots\rangle,$

and we also get

$\displaystyle\cos(\mathbf{\epsilon})=\langle\widehat{1},0,-\tfrac{1}{2},0,\dots\rangle=1-\tfrac{\textbf{1}}{\textbf{2}!}\mathbf{\epsilon}^{\textbf{2}}+\dots$

for an infinitesimal angle $\mathbf{\epsilon}$.

######  Example 5.11.

With the same $\mathbf{x}$ as in Examples 5.9 and 5.10, we try to know what
$\exp(\mathbf{x})$ is. Based on Equation 6,

$\displaystyle\exp(\mathbf{x})=\exp(x+a\mathbf{\epsilon})=\textbf{1}+(x+a\mathbf{\epsilon})-\tfrac{\textbf{1}}{\textbf{2}!}(x+a\mathbf{\epsilon})^{\textbf{2}}+\tfrac{\textbf{1}}{\textbf{3}!}(x+a\mathbf{\epsilon})^{\textbf{3}}+\dots$

Our task now is to find all the members of
$\texttt{Nst}_{\epsilon}(\exp(\mathbf{x}))$ and also
$\texttt{St}(\exp(\mathbf{x}))$, which are shown in Table 4. Note that from
the way we define the function $\exp$, $\forall
x_{i}\in\texttt{Nst}_{\omega}(\exp(\mathbf{x}))$ $x_{i}=0$. Thus from Table 4,
we get:

$\displaystyle\exp(\mathbf{x})=\exp(x+a\mathbf{\epsilon})=\langle\widehat{\exp(x)},a\exp(x),\tfrac{a^{2}}{2!}\exp(x),\tfrac{a^{3}}{3!}\exp(x),\tfrac{a^{4}}{4!}\exp(x),\dots\rangle,$

and we also get

$\displaystyle\exp(\mathbf{\epsilon})=\langle\widehat{1},1,\tfrac{1}{2!},\tfrac{1}{3!},\tfrac{1}{4!},\dots\rangle=1+\mathbf{\epsilon}+\tfrac{\textbf{1}}{\textbf{2}!}\mathbf{\epsilon}^{\textbf{2}}+\tfrac{\textbf{1}}{\textbf{3}!}\mathbf{\epsilon}^{\textbf{3}}-\dots$

for an infinitesimal angle $\mathbf{\epsilon}$.

Table 4: $\texttt{St}(\exp(\mathbf{x)})$ and The First Three Members of $\texttt{Nst}_{\epsilon}(\exp(\mathbf{x}))$ | Expanded Form | Simplified Form
---|---|---
real-part | $\begin{array}[]{ll}=&1+\mathbf{x}+\tfrac{1}{2!}\mathbf{x}^{2}+\tfrac{1}{3!}\mathbf{x}^{3}+\dots\\\ =&\sum_{n=0}\tfrac{1}{n!}\text{ }\mathbf{x}^{n}\end{array}$ | $=\exp(\mathbf{x})$
$\mathbf{\epsilon}$-part | $\begin{array}[]{ll}=&a\mathbf{\epsilon}+\tfrac{2}{2!}a\mathbf{\epsilon}\mathbf{x}+\tfrac{3}{3!}a\mathbf{\epsilon}\mathbf{x}^{2}+\tfrac{\textbf{4}}{\textbf{4}!}a\mathbf{\epsilon}\mathbf{x}^{3}\mathbf{\epsilon}+\dots\\\ =&\mathbf{\epsilon}(a+a\mathbf{x}+\tfrac{1}{2!}a\mathbf{x}^{2}+\tfrac{1}{3!}a\mathbf{x}^{3}-\tfrac{1}{\textbf{4}!}a\mathbf{x}^{\textbf{4}}+\dots)\\\ =&\mathbf{\epsilon}\sum_{n=0}\tfrac{1}{n!}\text{ }a\mathbf{x}^{n}\end{array}$ | $=\mathbf{\epsilon}(a\exp(\mathbf{x}))$
$\mathbf{\epsilon}^{2}$-part | $\begin{array}[]{ll}=&\tfrac{1}{2!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}+\tfrac{3}{3!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}+\tfrac{6}{\textbf{4}!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{2}+\tfrac{\textbf{10}}{5!}a^{2}\mathbf{\epsilon}^{2}\mathbf{x}^{3}+\dots\\\ =&\mathbf{\epsilon}^{2}(\tfrac{1}{2!}a^{2}+\tfrac{3}{3!}a^{2}\mathbf{x}+\tfrac{6}{\textbf{4}!}a^{2}\mathbf{x}^{2}+\tfrac{\textbf{10}}{5!}a^{2}\mathbf{x}^{3}+\dots)\\\ =&\mathbf{\epsilon}^{2}\sum_{n=0}\tfrac{(n+1)(n+2)}{2(n+2)!}\text{ }a^{2}\mathbf{x}^{n}\\\ =&\mathbf{\epsilon}^{2}\sum_{n=0}\tfrac{1}{2(n!)}\text{ }a^{2}\mathbf{x}^{n}\end{array}$ | $=\mathbf{\epsilon}^{2}(\tfrac{a^{2}}{2}\exp(\mathbf{x}))$
$\mathbf{\epsilon}^{3}$-part | $\begin{array}[]{ll}=&\tfrac{1}{3!}a^{3}\mathbf{\epsilon}^{3}+\tfrac{\textbf{4}}{\textbf{4}!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}+\tfrac{\textbf{10}}{5!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{2}+\tfrac{\textbf{20}}{6!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{3}+\tfrac{\textbf{35}}{7!}a^{3}\mathbf{\epsilon}^{3}\mathbf{x}^{\textbf{4}}+\dots\\\ =&\mathbf{\epsilon}^{3}(\tfrac{1}{3!}a^{3}+\tfrac{\textbf{4}}{\textbf{4}!}a^{3}\mathbf{x}-\tfrac{\textbf{10}}{5!}a^{3}\mathbf{x}^{2}+\tfrac{\textbf{20}}{6!}a^{3}\mathbf{x}^{3}+\tfrac{\textbf{35}}{7!}a^{3}\mathbf{x}^{\textbf{4}}+\dots)\\\ =&\mathbf{\epsilon}^{3}\sum_{n=0}\tfrac{(n+1)(n+2)(n+3)}{6(n+3)!}\text{ }a^{3}\mathbf{x}^{n}\\\ =&\mathbf{\epsilon}^{3}\sum_{n=0}\tfrac{1}{6(n!)}\text{ }a^{3}\mathbf{x}^{n}\end{array}$ | $=\mathbf{\epsilon}^{3}(\tfrac{a^{3}}{6}\exp(\mathbf{x}))$

From the preceding discussion, we have the following proposition.

######  Proposition 5.12.

For the $\sin$, $\cos$, and $\exp$ functions:

  1. 1.

$\textnormal{{Der}}_{\sin{\mathbf{x}}}(\mathbf{x},\mathbf{\epsilon})=\tfrac{\sin(\mathbf{x}+\mathbf{\epsilon})-\sin(\mathbf{x})}{\mathbf{\epsilon}}=\langle\widehat{\cos(x)},-\tfrac{1}{2!}\sin(x),-\tfrac{1}{3!}\cos(x),\dots\rangle$,
and so we have:

$\textnormal{{St}}(\textnormal{{Der}}_{\sin(\mathbf{x})}(\mathbf{x},\mathbf{\epsilon}))=\cos(\mathbf{x})$

  2. 2.

$\textnormal{{Der}}_{\cos{\mathbf{x}}}(\mathbf{x},\mathbf{\epsilon})=\tfrac{\cos(\mathbf{x}+\mathbf{\epsilon})-\cos(\mathbf{x})}{\mathbf{\epsilon}}=\langle\widehat{-\sin(x)},-\tfrac{1}{2!}\cos(x),\tfrac{1}{3!}\sin(x),\dots\rangle$,
and so

$\textnormal{{St}}(\textnormal{{Der}}_{\cos(\mathbf{x})}(\mathbf{x},\mathbf{\epsilon}))=-\sin(\mathbf{x})$

  3. 3.

$\textnormal{{Der}}_{\exp{\mathbf{x}}}(\mathbf{x},\mathbf{\epsilon})=\tfrac{\exp(\mathbf{x}+\mathbf{\epsilon})-\exp(\mathbf{x})}{\mathbf{\epsilon}}=\langle\widehat{\exp(x)},\tfrac{1}{2!}\exp(x),\tfrac{1}{3!}\exp(x),\dots\rangle$,
and so

$\textnormal{{St}}(\textnormal{{Der}}_{\exp(\mathbf{x})}(\mathbf{x},\mathbf{\epsilon}))=\exp(\mathbf{x})$

###  5.3 Continuity

In this subsection, we try to pinpoint what the good definition for continuous
functions is. We also decide whether we can permeate it between $\mathbb{R}$
and $\mathbb{R^{Z_{<}}}$. Note that if the domain and codomain of a function
is not explicitly stated, they will be determined from the specified model.

######  Definition 5.13 ($\text{ED}_{\text{CLASS}}$).

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous at a point
$a\in\mathbb{R}$ if, given $n\in\mathbb{N}$, there exists a $m\in\mathbb{N}$
such that

$|f(x)-f(a)|<\frac{1}{n}$ whenever $|x-a|<\frac{1}{m}$.

The function $f$ is called continuous on an interval $I$ iff $f$ is continuous
at every point in $I$.

######  Definition 5.14 (ED).

A function $f$ is continuous at a point $\mathbf{c}\in\mathbb{R^{Z_{<}}}$ if,
given $\mathbf{\mathfrak{e}_{1}}\in\mathbb{R^{Z_{<}}}>\textbf{0}$, there
exists a $\mathfrak{e}_{2}\in\mathbb{R^{Z_{<}}}>\textbf{0}$ such that

$|\mathbf{f(x)}-\mathbf{f(c)}|<\mathbf{\mathfrak{e}_{1}}$ whenever
$|\mathbf{x}-\mathbf{c}|<\mathbf{\mathfrak{e}_{2}}$.

That function $\mathbf{f}$ is called continuous function over an interval $I$
iff $\mathbf{f}$ is continuous at every point in $I$.

######  Proposition 5.15.

There exists a function $f$ in $\mathbb{R^{Z_{<}}}$ which is continuous under
Definition 5.14, but discontinuous under Definition 5.13, i.e.

$\mathbb{R^{Z_{<}}}\models\exists f$ s.t.
$(\textnormal{ED}_{\textnormal{CLASS}}(f)\land\neg\textnormal{ED}(f))$.

###### Proof.

Suppose that $\Delta=\\{x\mid\forall n\in\mathbb{N},\left\lvert
x\right\rvert<\frac{1}{n}\\}$ – in other words, $\Delta$ is a set of all
infinitesimals – and consider the indicator function around $\Delta$, that is

$\displaystyle\mathds{1}_{\Delta}(x)=\begin{cases}1,&x\in\Delta\\\
0,&\textnormal{otherwise.}\end{cases}$

Then $\textnormal{ED}_{\textnormal{CLASS}}(\mathds{1}_{\Delta})$ but
$\neg\textnormal{ED}(\mathds{1}_{\Delta})$. ∎

######  Remark 5.16.

Note that:

  1. 1.

The set $\Delta$ in $\mathbb{R}$ only has 0 as its member. That is
$\mathbb{R}\models\Delta=\\{0\\}$.

  2. 2.

In $\mathbb{R}$, both Definitions 5.13 and 5.14 are equivalent, that is for
any function $f$,
$\mathbb{R}\models\textnormal{ED}_{\textnormal{CLASS}}(f)\leftrightarrow\textnormal{ED}(f)$.

######  Property 5.17 (EVP).

If $I$ is an interval and $f:I\rightarrow J$, we say that $f$ has the extreme
value property iff $f$ has its maximum value on $I$. That is,

$\forall a\leq b\in I$, $\exists x\in[a,b]$ s.t. $\forall y\in[a,b](f(y)\leq
f(x))$.

######  Property 5.18 (IVP).

If $I$ is an interval, and $f:I=[a,b]\rightarrow J$, we say that $f$ has the
intermediate value property iff

$\forall c^{\prime}\in(f(a),f(b))$, $\exists c\in(a,b)$ s.t.
$f(c)=c^{\prime}$.

######  Theorem 5.19.

$\mathbb{R}\models\textnormal{ED}\rightarrow\textnormal{EVP}$

###### Proof.

The proof of this theorem can be found in any standard book for Analysis
course (in [3] for example). ∎

######  Theorem 5.20.

There is a function $f$ such that

$\mathbb{R^{Z_{<}}}\models\textnormal{ED}(f)\land\lnot\textnormal{EVP}(f)$.

###### Proof.

Take the function $f$ on $[1,2]$ as defined below:

$\displaystyle f(x)=\begin{cases}\frac{1}{n}&x\sim\frac{m}{n}\textnormal{
(reduced fraction)}\\\ 0&\text{otherwise}.\end{cases}$

∎

######  Remark 5.21.

This research now reaches an especially engrossing object. The function $f(x)$
in Theorem 5.20 can be used to construct a fractal-like object. Fractals are
classically defined as geometric objects that exhibit some form of self-
similarity. Figure 1 shows what the function $f(x)$ in Theorem 5.20 looks
like, and also what occurs when we zoom in on a particular point. In this
sense, the function from Theorem 5.20 is an infinitesimal fractal. Formally
speaking, suppose that we have a function
$f:\mathbb{R^{Z_{<}}}\rightarrow\mathbb{R}$ and let us define another function

$\displaystyle F:\mathbb{R^{Z_{<}}}\rightarrow\mathbb{R^{Z_{<}}}$

by

$\displaystyle\begin{aligned} F(x)&=f(x)+\epsilon
f(x)+\epsilon^{2}f(x)+\dots\\\
&=\langle\widehat{f(x)},f(x),f(x),\dots\rangle.\end{aligned}$

Then, that function $F(x)$ will define an infinite fractal (if
$\texttt{ni}_{\omega}(\frac{1}{\epsilon}\widehat{\times}F(x))=F(x)$) or
infinitesimal fractals (if
$\texttt{ni}_{\omega}(\epsilon\widehat{\times}F(x))=F(x)$), where
$\texttt{ni}_{\omega}(\mathbf{x})$ denotes the non-infinity part of
$\mathbf{x}$.

Figure 1: Illustration of the infinitesimal fractal from the function defined
in Theorem 5.20

Figure 2 shows the relationship among the three definitions of continuity in
both $\mathbb{R}$ and $\mathbb{R^{Z_{<}}}$. The proof of each of the relations
there can be seen in [32, p. 94–98].

EVPIVPED (a) In $\mathbb{R}$

EVPIVPED (b) In $\mathbb{R^{Z_{<}}}$

Figure 2: Relationship among the three definitions of continuity

The obvious question worth asking is how do we define continuity in our set
$\mathbb{R^{Z_{<}}}$. As seen before, there are three possible ways to define
it, namely: with the $\epsilon$-$\delta$ definition (ED), with extreme value
property (EVP), or with the intermediate value property (IVP). We will now
discuss them one by one. Firstly, through IVP. The IVP basically says that for
every value within the range of the given function, we can find a point in the
domain corresponding to that value. Will this work in our set
$\mathbb{R^{Z_{<}}}$? Let us consider the $\mathbb{R^{Z_{<}}}$-valued function
$f(x)=x^{2}$ on $[a,b]$ for any $a,b\in\mathbb{R^{Z_{<}}}$ and let us assume
that IVP holds. It follows that for every $c^{\prime}$ between $f(a)=a^{2}$
and $f(b)=b^{2}$, $\exists c\in(a,b)$ such that $f(c)=c^{2}=c^{\prime}$. The
only $c$ which satisfies that last equation is $c=\sqrt{c^{\prime}}$, which
cannot be defined in our set $\mathbb{R^{Z_{<}}}$. Thus, IVP, even though it
is somehow intuitively “obvious”, it does not work in $\mathbb{R^{Z_{<}}}$.
This phenomenon is actually not uncommon if we want to have a world with
infinitesimals (or infinities) in it. See [5, p. 107] for example.

However, note that in $\mathbb{R}$, the function $x^{2}$ still satisfies IVP.
Now, is there a function in $\mathbb{R^{Z_{<}}}$ that satisfies IVP? Consider
the identity function $f(x)=x$. This function clearly satisfies IVP in both
domains, and so we have the following theorem.

######  Theorem 5.22.

There exists a function $f$ such that
$(\mathbb{R}\models\textnormal{IVP}(f)\land\mathbb{R^{Z_{<}}}\models\lnot\textnormal{IVP}(f))$,
and there exists a function $g$ s.t.
$\mathbb{R},\mathbb{R^{Z_{<}}}\models\textnormal{IVP}(g)$.

Hence, from the argument above we also argued that defining continuity in our
set with IVP is not really useful.

Secondly, in regards to EVP. This is clearly not a good way to define
continuity in our set because even in the set of real numbers, there are some
continuous functions which do not satisfy EVP themselves. So the last
available option now is the third one, which is the $\epsilon$-$\delta$ (ED)
definition. We argued that this definition is the best way to define
continuity in $\mathbb{R^{Z_{<}}}$. Moreover, in this way, it preserves much
of the spirit of classical analysis on $\mathbb{R}$ while retaining the
intuition of infinitesimals.

It is important to note that in the ED definition of continuity (Definition
5.14), there are two variables which are in play, i.e. $\mathfrak{e}_{1}$ and
$\mathfrak{e}_{2}$. When we applied this definition on our set, these two
variables hold important (or rather, very interesting) roles where we will
have different levels of continuity from the same function. What we mean is
that these two variables can greatly vary depending on how far (‘deep’) we
want to push (observe) them, e.g. $\mathfrak{e}_{2}$ can be a real number
($\mathfrak{e}_{2}\in\Delta^{0}$), or it can be in $\Delta^{4}$, $\Delta^{8}$
and so on. Remember that these two numbers, $\mathfrak{e}_{1}$ and
$\mathfrak{e}_{2}$, will determine how subtle we want our intervals to be (see
Figure 3 for illustration).

(a) An $\mathfrak{e}_{1}$ bound & its $\mathfrak{e}_{2}$ neighbourhood
fulfilling Definition 5.14

(b) A smaller bound & its neighbourhood

Figure 3: Illustration of $\mathfrak{e}_{1}$ and $\mathfrak{e}_{2}$ intervals

Thus this definition of continuity works as follows. Suppose that we have a
function $f$ and we want to decide whether it is continuous or not. With this
concept of two variables, we will have what we call as $(k,n)$-continuity
where $k,n\in\mathbb{N}\cup\\{0\\}$.

######  Definition 5.23 ($(k,n)$-Continuity).

A function $f$ is $(k,n)$-continuous at a point $\mathbf{c}$ iff
$\forall\mathbf{\mathfrak{e}_{1_{k}}}>0$, $\exists\mathfrak{e}_{2_{n}}>0$ such
that

if $|\mathbf{x}\widehat{-}\mathbf{c}|<\mathbf{\mathfrak{e}_{2_{n}}}$, then
$|\mathbf{f(x)}\widehat{-}\mathbf{f(c)}|<\mathbf{\mathfrak{e}_{1_{k}}}$

where $\mathfrak{e}_{1_{p}},\mathfrak{e}_{2_{p}}\in\Delta^{p}$.

######  Definition 5.24.

A function $f$ is said to be $(k,n)$-continuous iff it is $(k,n)$-continuous
at every point in the given domain.

######  Remark 5.25.

From the definition of the set $\Delta^{m}$, note that for any
$r\in\mathbb{R^{Z_{<}}},$ $d\in\Delta^{p},$ and $e\in\Delta^{p+1},$

$\displaystyle(r-e,r+e)\subseteq(r-d,r+d).$

To be able to grasp a better understanding of Definition 5.23, see the
examples below.

######  Example 5.26.

Consider the $\mathbb{R^{Z_{<}}}$-valued function $f(x)$ defined as follows:

$f(\mathbf{x})=\begin{cases}\mathbf{x}&\texttt{St}(\mathbf{x})\leq 1\\\
\mathbf{x}\widehat{+}\mathbf{1}&\textnormal{otherwise.}\end{cases}$

First we need to understand clearly how this function actually works. Figure
4, where $i$ denotes an arbitrary infinitesimal number, illustrates to us what
the function $f(\mathbf{x})$ looks like. Notice that at
$\mathbf{x}=\mathbf{1}$, what looks like a point in real numbers is actually a
(constant) line when we zoom in deep enough into $\mathbb{R^{Z_{<}}}$101010We
have to be really careful here because if the first condition there was
$\mathbf{x}\leq 1$ (instead of $\texttt{St}(\mathbf{x})\leq 1$), then there
would be no line there — it will be _exactly_ one point.. So how about the
continuity of this function? It is obvious that $f(\mathbf{x})$ is not
$0,0$-continuous (by taking, for example, $\mathfrak{e}_{1_{0}}=\frac{1}{2}$
and $\mathbf{x}=\mathbf{1.5}$). However, interestingly enough, it is
$\mathit{(0,1)}$-continuous by taking $\mathfrak{e}_{2_{1}}\in\Delta^{1}$. Why
was that? The fact that $\mathfrak{e}_{2_{1}}\in\Delta^{1}$ and that it has to
depend on $\mathfrak{e}_{1_{0}}$ means that $\mathfrak{e}_{1_{0}}$ has to be
in $\Delta^{1}$ as well. Now, assigning
$\mathfrak{e}_{2_{1}}=\mathfrak{e}_{1_{0}}$ is sufficient to prove its
$\mathit{0,1}$-continuity.

Figure 4: Illustration of Function $f(x)$ in Example 5.26

######  Example 5.27.

The identity function $f(\mathbf{x})=\mathbf{x}$ for all
$\mathbf{x}\in\mathbb{R^{Z_{<}}}$ is $\mathit{(0,0)}$-continuous, just like in
reals. However, it is not $\mathit{(1,0)}$-continuous because for any point
$\mathbf{c}$, there is an $\mathfrak{e}_{1_{1}}=\mathbf{\epsilon}$ such that
for every $\mathfrak{e}_{2_{0}}=r$ where $r\in\mathbb{R}$,
$|\mathbf{x}\widehat{-}\mathbf{c}|<r$ but
$f(\mathbf{x})-f(c)\geq\mathbf{\epsilon}$. In fact, identity function is
$(k,n)$-continuous only when $k\leq n$, but not otherwise.

The next theorem below is very interesting in as much as it enables us to
classify whether a function is a constant function or not by using
$(k,n)$-continuity.

######  Theorem 5.28.

For any function $f$, if $f$ is $(k,n)$-continuous for any
$k,n\in\mathbb{N}\cup\\{0\\}$, then $f$ is a constant function.

###### Proof.

Here we want to prove its contrapositive, in other words, if $f$ is not
constant, then there exist $(k,n)$ such that $f$ is not $(k,n)$-continuous.
Because $f$ is not constant, there will be $a,b$ in the domain such that
$f(a)\neq f(b)$ and suppose that $|f(a)-f(b)|\in\Delta^{m}$ such that
$|a-b|\in\Delta^{l}$. By this construction, $f$ will not be
$(m,l-1)$-continuous and so we can take $k=m$ and $n=l-1$. ∎

The theorem below is a generalisation of Theorem 5.28.

######  Theorem 5.29.

If there exists $m$ for all $k$ such that a function $f$ is
$(k,m)$-continuous, then $f$ will be constant in $\Delta^{m}$-neighbourhood.

The next interesting question is: what is the relation between, for example,
$\mathit{(0,1)}$-continuity and $\mathit{(0,2)}$-continuity? In general, what
is the relation between $(k,n)$-continuity and $(k,(n+1))$-continuity? And
also between $(k,n)$-continuity and $((k+1),n)$-continuity? See these two
theorems below.

######  Theorem 5.30.

For any function $f$, if $f$ is $(k,n)$-continuous, then $f$ is also
$(k,(n+1))$-continuous.

###### Proof.

Suppose that a function $f$ is $(k,n)$-continuous at point $\mathbf{c}$. This
would mean that $\forall\mathfrak{e}_{1_{k}}$, $\exists\mathfrak{e}_{2_{n}}$
such that if $|\mathbf{x}\widehat{-}\mathbf{c}|<\mathfrak{e}_{2_{n}}$, then
$|f(\mathbf{x})\widehat{-}f(\mathbf{c})|<\mathfrak{e}_{1_{k}}$. By using the
same $\mathfrak{e}_{1_{k}}$ and from Remark 5.25, we can surely find
$\mathfrak{e}_{2_{(n+1)}}=\mathfrak{e}_{2_{n}}\widehat{\times}\mathbf{\epsilon}$
such that for all
$x\in(\mathbf{c}\widehat{-}\mathfrak{e}_{2_{(n+1)}},\mathbf{c}\widehat{+}\mathfrak{e}_{2_{(n+1)}})$,
$f(x)\in(f(\mathbf{c})\widehat{-}\mathfrak{e}_{1_{k}},f(\mathbf{c})\widehat{+}\mathfrak{e}_{1_{k}})$.
∎

######  Example 5.31.

By Theorem 5.30, the function $f(x)$ in Example 5.26 is also
$\mathit{(0,2)}$-continuous, and also $\mathit{(0,3)}$-continuous, and so on.

######  Theorem 5.32.

For any function $f$, if $f$ is $((k+1),n)$-continuous, then $f$ is also
$(k,n)$-continuous.

###### Proof.

Suppose that a function $f$ is $((k+1),n)$-continuous at point $\mathbf{c}$
and the set $\Delta^{m}$ defined as in Theorem 5.30. The fact that $f$ is
$((k+1),n)$-continuous means that $\forall\mathfrak{e}_{1_{(k+1)}}$,
$\exists\mathfrak{e}_{2_{n}}$ such that if
$|\mathbf{x}\widehat{-}\mathbf{c}|<\mathfrak{e}_{2_{n}}$, then
$|f(\mathbf{x})\widehat{-}f(\mathbf{c})|<\mathfrak{e}_{1_{(k+1)}}$ is hold.
Here we want to prove that $\forall\mathfrak{e}_{1_{k}}$,
$\exists\mathfrak{e}_{2_{n}}$ such that if
$|\mathbf{x}\widehat{-}\mathbf{c}|<\mathfrak{e}_{2_{n}}$, then
$|f(\mathbf{x})\widehat{-}f(\mathbf{c})|<\mathfrak{e}_{1_{k}}$. This actually
follows directly from Remark 5.25 as
$(f(\mathbf{c})\widehat{-}\mathfrak{e}_{1_{(k+1)}},f(\mathbf{c})\widehat{+}\mathfrak{e}_{1_{(k+1)}})\subseteq(f(\mathbf{c})\widehat{-}\mathfrak{e}_{1_{k}},f(\mathbf{c})\widehat{+}\mathfrak{e}_{1_{k}})$.
∎

Now suppose that $f$ and $g$ are two $(k,n)$-continuous functions in
$\mathbb{R^{Z_{<}}}$. We will examine how the arithmetic of those two
continuous functions works. It is clear that $(k,n)$-continuity is closed
under addition and subtraction, i.e. $f+g$ and $f-g$ are both
$(k,n)$-continuous. The composition and multiplication of two continuous
functions are particularly interesting as can be seen in Theorem 5.33 and
Theorem 5.34, respectively.

######  Theorem 5.33.

If $f$ is a $(k,n)$-continuous function and $g$ is an $(n,q)$-continuous
function, then $f\circ g$ will be $(k,q)$-continuous.

###### Proof.

Since $f$ is $(k,n)$-continuous at $g(c)$, our definition of continuity tells
us that for all $\mathfrak{e}_{1_{k}}>0$, there exists $\mathfrak{e}_{2_{n}}$
such that

if $\left\lvert g(x)-g(a)\right\rvert<\mathfrak{e}_{2_{n}}$, then $\left\lvert
f(g(x))-f(g(a))\right\rvert<\mathfrak{e}_{1_{k}}$.

Also since $g$ is $(n,q)$-continuous at $c$, there exists
$\mathfrak{e}_{2_{q}}$ such that

if $\left\lvert x-a\right\rvert<\mathfrak{e}_{2_{q}}$, then $\left\lvert
g(x)-g(a)\right\rvert<\mathfrak{e}_{2_{n}}$.

I have taken $\mathfrak{e}_{1_{n}}=\mathfrak{e}_{2_{n}}$ here. Now this tells
us that for all $\mathfrak{e}_{1_{k}}>0$, there exists $\mathfrak{e}_{2_{q}}$
(and an $\mathfrak{e}_{2_{n}}$) such that

if $\left\lvert x-a\right\rvert<\mathfrak{e}_{2_{q}}$, then $\left\lvert
g(x)-g(a)\right\rvert<\mathfrak{e}_{2_{n}}$ which implies that $\left\lvert
f(g(x))-f(g(c))\right\rvert<\mathfrak{e}_{1_{k}}$,

which is what we wanted to show. ∎

######  Theorem 5.34.

Suppose that $f,g$ are finite-valued functions. If $f$ is a $(k,n)$-continuous
function and $g$ is an $(l,o)$-continuous function, then the function
$H=f\cdot g$ will be $(\max\\{k,l\\},\min\\{n,o\\})$-continuous.

###### Proof.

Let $f,g$ be given such that $f$ is $(k,n)$-continuous and $g$ is
$(l,o)$-continuous. Now let $H$ be defined by $H(x)=f(x)g(x)$ and so we want
to show that $H$ is $(\max\\{k,l\\},\min\\{n,o\\})$-continuous, that is, for
all $c\in\mathbb{R^{Z_{<}}}$, for every $\mathfrak{e}_{1_{\max\\{k,l\\}}}>0$,
there exists $\mathfrak{e}_{2_{\min\\{n,o\\}}}>0$ such that for all
$x\in\mathbb{R^{Z_{<}}}$ with $\left\lvert
x-c\right\rvert<\mathfrak{e}_{2_{\min\\{n,o\\}}}$, $\left\lvert
H(x)-H(c)\right\rvert<\mathfrak{e}_{1_{\max\\{k,l\\}}}$ holds.
Now let $c$ and $\mathfrak{e}_{1_{\max\\{k,l\\}}}$ be given and we choose
$\mathfrak{e}_{2}$ such that $\mathfrak{e}_{2}\in\Delta^{\min\\{n,o\\}}$, i.e.
$\mathfrak{e}_{2}=\mathfrak{e}_{2_{\min\\{n,o\\}}}$. Then for all
$x\in\mathbb{R^{Z_{<}}}$ with $\left\lvert
x-c\right\rvert<\mathfrak{e}_{2_{\min\\{n,o\\}}}$,

$\displaystyle\left\lvert H(x)-H(c)\right\rvert$ $\displaystyle=\left\lvert
f(x)g(x)-f(c)g(c)\right\rvert$ $\displaystyle=\left\lvert
f(x)g(x)-f(x)g(a)+f(x)g(a)-f(c)g(c)\right\rvert$ $\displaystyle\leq\left\lvert
f(x)g(x)-f(c)g(c)\right\rvert+\left\lvert f(x)g(a)-f(c)g(c)\right\rvert$
$\displaystyle=\left\lvert f(x)(g(x)-g(a))\right\rvert+\left\lvert
g(a)(f(x)f(c))\right\rvert$ $\displaystyle<\left\lvert
f(x)\right\rvert\mathfrak{e}_{1_{l}}+\left\lvert
g(a)\right\rvert\mathfrak{e}_{1_{k}}$ (7)

Note that because $f$ and $g$ are limited-valued function, then $\left\lvert
f(x)\right\rvert\mathfrak{e}_{1_{l}}$ and $\left\lvert
g(a)\right\rvert\mathfrak{e}_{1_{k}}$ are still in $\Delta^{l}$ and
$\Delta^{k}$, respectively. This means that the right side of Inequality 7
will be in $\Delta^{\max\\{k,l\\}}$ and so
$H(x)-H(c)<\mathfrak{e}_{1_{\max\\{k,l\\}}}$ is hold. ∎

It is worth pointing out here that the definition of $(k,n)$-continuity is a
much more fine-grained notion than the classical continuity. This is self-
explanatory by the use of those two variables $k$ and $n$ which makes us able
to take much more infinitesimals — in other words, we will be able to examine
a far greater depth — than in the classical definition. Furthermore, there
might be some possible connections to one of the quantum phenomenons in
physics: ‘action at a distance’. This concept is typically characterized in
terms of some cause producing a spatially separated effect in the absence of
any medium by which the causal interaction is transmitted [16] and closely
connected to the question of what the deepest level of physical reality is
[30, pg. 168]. Note that research on this phenomenon is still being conducted
up until now, as can be seen for example [33], [46] and [53].

###  5.4 Topological Continuity

######  Definition 5.35.

A function $f$ from a topological space $(X,\uptau_{1})$ to a topological
space $(Y,\uptau_{2})$ is a function $f:X\rightarrow Y$.

From now on, we will abbreviate this function notation by $f:X\rightarrow\ Y$
or simply $f$ every time the topologies in $X$ and $Y$ need not be explicitly
mentioned. Also, $f^{-1}$ denotes the inverse image of $f$ as usual.

######  Definition 5.36 (St-continuous).

A function $f:X\rightarrow Y$ between topological spaces is standard
topologically continuous, denoted by St-continuous, if

$f^{-1}(U)\subseteq X$ is St-open whenever $U\subseteq Y$ is St-open.

######  Definition 5.37 ($\mathfrak{e}$-continuous).

A function $f:X\rightarrow\ Y$ between topological spaces is infinitesimally
topologically continuous, denoted by $\mathfrak{e}$-continuous, if

$f^{-1}(U)\subseteq X$ is $\mathfrak{e}$-open whenever $U\subseteq Y$ is
$\mathfrak{e}$-open.

######  Theorem 5.38.

Suppose that $X,Y\subseteq\mathbb{R^{Z_{<}}}$. Under the metric
$\operatorname{\texttt{d}}$, a function $f:X\rightarrow Y$ is St-continuous if
and only if $f$ satisfies $\textnormal{ED}_{\textnormal{CLASS}}$ definition
(Definition 5.13).

###### Proof.

We need to prove the implication both ways.

  1. 1.

We want to prove that if $f$ is St-continuous, then $f$ satisfies
ED${}_{\textnormal{CLASS}}$. Suppose that $f$ is St-continuous and let
$\mathbf{x_{o}}\in X$ and $n\in\mathbb{N}>0$. Then, the ball

$\displaystyle B_{f(\mathbf{x_{0}})}(\nicefrac{{1}}{{n}})=\\{\mathbf{y}\in
Y\mid\operatorname{\texttt{d}}(\mathbf{y},f(\mathbf{x_{0}}))<\nicefrac{{1}}{{n}}\\}$

is open in $Y$, and hence $f^{-1}(B_{f(\mathbf{x_{0}})})$ is open in $X$.
Since $\mathbf{x_{0}}\in f^{-1}(B_{f(\mathbf{x_{0}})})$, there exists some
balls of radius $\nicefrac{{1}}{{m}}$ for some $m\in\mathbb{N}$ such that

$\displaystyle B_{\mathbf{x_{0}}}(\nicefrac{{1}}{{m}})\subseteq
f^{-1}(B_{f(\mathbf{x_{0}})}).$

This is exactly what the ED${}_{\textnormal{CLASS}}$ says.

  2. 2.

We want to prove that if $f$ satisfies ED${}_{\textnormal{CLASS}}$, then $f$
is St-continuous. Suppose that $f$ satisfies ED${}_{\textnormal{CLASS}}$ and
let $U\subseteq Y$ is open. By Definition 4.6, for all $\mathbf{y}\in U$ there
exists some $d_{y}=\nicefrac{{1}}{{n_{y}}}$ where $n_{y}\in\mathbb{N}$ such
that

$\displaystyle B_{\mathbf{y}}(d_{y})\subseteq U$

and in fact,

$\displaystyle U=\bigcup_{y\in U}B_{d_{y}}(\mathbf{y}).$ (8)

Now we claim that $f^{-1}(U)$ is open in $X$ and suppose that
$\mathbf{x_{0}}\in f^{-1}(U)$. Then $f(\mathbf{x_{0}})\in U$ and so from
Equation 8, $f(\mathbf{x_{0}})\in B_{d_{y_{0}}}(\mathbf{y_{0}})$ for some
$\mathbf{y_{0}}\in U$ and $d_{y_{0}}=\nicefrac{{1}}{{n_{y_{0}}}}$ for some
$n_{y_{0}}\in\mathbb{N}$, i.e.
$\operatorname{\texttt{d}}(f(\mathbf{x_{0}}),\mathbf{y_{0}})<d_{y_{0}}$. Now
define

$\displaystyle
e=d_{y_{0}}-\operatorname{\texttt{d}}(f(\mathbf{x_{0}}),\mathbf{y_{0}})>0.$
(9)

By Definition 5.13, there exists some $m\in\mathbb{N}$ such that

$\displaystyle\textnormal{if }\mathbf{x}\in X\textnormal{ and
}\operatorname{\texttt{d}}(\mathbf{x},\mathbf{x_{0}})<\nicefrac{{1}}{{m}},\textnormal{
then }\operatorname{\texttt{d}}(f(\mathbf{x}),f(\mathbf{x_{0}}))<e.$ (10)

Now we claim that

$\displaystyle B_{\mathbf{x_{0}}}(\nicefrac{{1}}{{m}})\subseteq f^{-1}(U),$
(11)

which will actually show that $f^{-1}(U)$ is indeed open. To this end, let
$x\in B_{\mathbf{x_{0}}}(\nicefrac{{1}}{{m}})$, i.e.
$\operatorname{\texttt{d}}(\mathbf{x},\mathbf{x_{0}})<\nicefrac{{1}}{{m}}$.
Then from (10), we have
$\operatorname{\texttt{d}}(f(\mathbf{x}),f(\mathbf{x_{0}}))<e$. Then, the
triangle inequality and (9) imply that

$\displaystyle\operatorname{\texttt{d}}(f(\mathbf{x}),\mathbf{y_{0}})\leq\operatorname{\texttt{d}}(f(\mathbf{x}),f(\mathbf{x_{0}}))+\operatorname{\texttt{d}}(f(\mathbf{x_{0}}),\mathbf{y_{0}})<e+\operatorname{\texttt{d}_{\texttt{$\psi$}}}(f(\mathbf{x_{0}}),\mathbf{y_{0}})=d_{y_{0}}.$

This means that $f(x)\in B_{\mathbf{y_{0}}}(d_{y_{0}})\subseteq U$, so that
$x\in f^{-1}(U)$. Therefore, (11) holds, as claimed.

And so from those two points above, we have proved what we want. ∎

######  Theorem 5.39.

Suppose that $X,Y\subseteq\mathbb{R^{Z_{<}}}$. Under the metric
$\operatorname{\texttt{d}}$, a function $f:X\rightarrow Y$ is
$\mathfrak{e}$-continuous if and only if $f$ satisfies ED definition.

###### Proof.

The proof of this theorem is similar with the one in Theorem 5.38 with some
slight modifications in the distances (from $\nicefrac{{1}}{{n}}$ for some
$n\in\mathbb{N}$ into $\mathbf{\mathfrak{e}}\in\mathbb{R^{Z_{<}}}$). ∎

###  5.5 Convergence

When we are talking about sequences, it is necessary to talk also about what
it means when we say that a sequence is convergent to a particular number.
This subsection presents not only some possible definitions that can be used
to define convergence in $\mathbb{R^{Z_{<}}}$, but also the problems which
occur when we apply them in $\mathbb{R^{Z_{<}}}$.

######  Definition 5.40 (Classical Convergence).

A sequence $s_{n}$ converges to $s$ iff,

$\forall m\in\mathbb{N}$, $\exists N$ such that $\forall n>N$, $\left\lvert
s_{n}-s\right\rvert<\frac{1}{m}$.

We write $\textnormal{CC}(s_{n},s)$ to denote that a sequence $s_{n}$ is
classically convergent to $s$.

Definition 5.40 above is the standard definition of how we define the notion
of convergent classically.

######  Definition 5.41 (Hyperconvergence).

A sequence $s_{n}$ converges to $s$ iff,

$\forall r>0$, $\exists N$ such that $\forall n>N$, $\left\lvert
s_{n}-s\right\rvert<r$.

We write $\textnormal{HC}(s_{n},s)$ to denote that a sequence $s_{n}$ is
hyperconvergent to $s$. The interpretation of $r$ can be either in
$\mathbb{R}$ or in $\mathbb{R^{Z_{<}}}$.

######  Example 5.42.

Suppose that we have a sequence $s_{n}=\epsilon^{n}$ as follows:

$S_{1}=\epsilon=\langle\widehat{0},1,0,\dots\rangle$

$S_{2}=\epsilon^{2}=\langle\widehat{0},0,1,0,\dots\rangle$

$S_{3}=\epsilon^{3}=\langle\widehat{0},0,0,1,0,\dots\rangle$

⋮

This sequence $s_{n}$ will hyperconverge to
$\langle\widehat{0},0,0,\dots\rangle$, i.e. $s_{n}$ satisfies
HC$(s_{n},\textbf{0})$.

######  Theorem 5.43.

For any sequence $s_{n}$,
$\mathbb{R}\models\textnormal{CC}(s_{n},s)\leftrightarrow\textnormal{HC}(s_{n},s)$.

###### Proof.

The proof from HC to CC is obvious. Now suppose that a sequence $s_{n}$
satisfies CC$(s_{n})$ and w.l.o.g. we assume that the $r$ in HC definition is
between 0 and 1. From the Archimedean property of reals we know that for every
$0<r<1$, we can find an $m\in\mathbb{N}$ such that $\frac{1}{m}<r$, and so
because of CC$(s_{n})$, we have $\left\lvert
s_{n}-s\right\rvert<\frac{1}{m}<r$. ∎

######  Theorem 5.44.

For any sequence $s_{n}$ in $\mathbb{R^{Z_{<}}}$, HC($s_{n},s$) always implies
CC($s_{n},s$). However, there exists a sequence $(t_{n})$ such that

$\mathbb{R^{Z_{<}}}\models\textnormal{CC}(t_{n},s)\not\rightarrow\textnormal{HC}(t_{n},s)$.

###### Proof.

  1. 1.

To prove the first clause, suppose that a sequence $s_{n}$ satisfies
HC$(s_{n},s)$. This means that we are able to find a number $N$ such that
$\forall n\geq N$, $\left\lvert s_{n}-s\right\rvert<r$ for any
$r\in\mathbb{R^{Z_{<}}}$ which includes infinitesimals. By using the same $N$,
$s_{n}$ will satisfy CC$(s_{n},s)$ .

  2. 2.

To prove the second clause, take the sequence $t_{n}=\frac{1}{n}$ where
$n\in\mathbb{N}$. This sequence satisfies CC$(t_{n},0)$, but it does not
satisfy HC$(t_{n},s)$ for any $s$ (as any $r\in\Delta$ will satisfy the
negation of Definition 5.41).

∎

######  Lemma 5.45.

Let $(s_{n})$ be a sequence in $\mathbb{R^{Z_{<}}}$ such that HC($s_{n},s$) is
hold. Then, HC($\left\lvert s_{n}\right\rvert,\left\lvert s\right\rvert$) is
hold.

###### Proof.

Let $r>0\in\mathbb{R^{Z_{<}}}$ be given. Then this means that there exists
$N\in\mathbb{N}$ such that $\forall m>N$, $\left\lvert s_{m}-s\right\rvert<r$.
Therefore, we also have

$\displaystyle\forall m>N\textnormal{, }\left\lvert\left\lvert
s_{m}\right\rvert-\left\lvert s\right\rvert\right\rvert\leq\left\lvert
s_{m}-s\right\rvert<r.$

Hence, HC($\left\lvert s_{n}\right\rvert,\left\lvert s\right\rvert$) is true.
∎

######  Theorem 5.46.

Let $X\subset\mathbb{R^{Z_{<}}}$ and $f:X\rightarrow\mathbb{R^{Z_{<}}}$. Then
$f$ is $\mathfrak{e}$-continuous at $x_{0}\in X$ iff for any sequence $x_{n}$
in $X$ that satisfies HC$(x_{n},x_{0})$, the sequence $f(x_{n})$ satisfies
HC$(f(x_{n}),f(x_{0}))$.

###### Proof.

Suppose that $f$ is $\mathfrak{e}$-continuous at $x_{0}$ and let the sequence
$x_{n}$ be defined in $X$ and that $x_{n}$ hyper converges to $x_{0}$. Now let
$\mathfrak{e}>0$ be given. Then from Theorem 5.39, there exists
$\mathfrak{e}_{2}>0\in\mathbb{R^{Z_{<}}}$ such that

if $x\in X$ and $\left\lvert x-x_{0}\right\rvert<\mathfrak{e}_{2}$, then
$\left\lvert f(x)-f(x_{0})\right\rvert<\mathfrak{e}$.

Now since $x_{n}$ hyper converges to $x_{0}$, then there exists
$N\in\mathbb{N}$ such that $\forall n\geq N$ $\left\lvert
x_{n}-x_{0}\right\rvert<\mathfrak{e}_{2}$. Thus we have

$\displaystyle\forall n\geq N\textnormal{ }\left\lvert
f(x_{n})-f(x_{0})\right\rvert<\mathfrak{e}.$

and so the sequence $f(x_{n})$ hyper converges to $f(x_{0})$.
For the converse, we will prove the contrapositive. Suppose that $f$ is not
$\mathfrak{e}$-continuous at $x_{0}$. Then it means that there exists
$\mathfrak{e}_{0}>0\in\mathbb{R^{Z_{<}}}$ such that for all
$\mathfrak{e}_{2}>0\in\mathbb{R^{Z_{<}}}$, there exists $x\in X$ such that
$\left\lvert x-x_{0}\right\rvert<\mathfrak{e}_{2}$ but $\left\lvert
f(x)-f(x_{0})\right\rvert>\mathfrak{e}_{0}$. In particular, for all
$n\in\mathbb{N}$, there exists $x_{n}\in X$ such that $\left\lvert
x_{n}-x_{0}\right\rvert<\mathfrak{e}_{2}$ and $\left\lvert
f(x_{n})-f(x_{0})\right\rvert>\mathfrak{e}_{0}$. Thus $x_{n}$ is a sequence in
$X$ that hyper converges to $x_{0}$, but the sequence $f(x_{n})$ does not
hyper converge to $f(x_{0})$. ∎

######  Definition 5.47.

Let $s_{n}$ be a sequence in $\mathbb{R^{Z_{<}}}$. Then we say that $s_{n}$ is
a hyper-Cauchy sequence iff $\forall\mathfrak{e}\in\mathbb{R^{Z_{<}}}$,
$\exists N\in\mathbb{N}$ such that

$\displaystyle\forall l,m\geq N\textnormal{ }\left\lvert
s_{l}-s_{m}\right\rvert<\mathfrak{e}.$

######  Theorem 5.48.

Every hyper convergent sequence in $\mathbb{R^{Z_{<}}}$ is a hyper-Cauchy
sequence.

###### Proof.

Let $s_{n}$ be a sequence in $\mathbb{R^{Z_{<}}}$ that satisfies
HC$(s_{n},s)$. We want to show that $s_{n}$ is hyper-Cauchy. Let
$\mathfrak{e}\in\mathbb{R^{Z_{<}}}$ be given. Then there exists
$N\in\mathbb{N}$ such that $\forall n>N$, $\left\lvert
s_{n}-s\right\rvert<\frac{\mathfrak{e}}{2}$. Then for all $l,m>N$, we have

$\displaystyle\left\lvert s_{l}-s_{m}\right\rvert=\left\lvert
s_{l}-s-(s_{m}-s)\right\rvert\leq\left\lvert s_{l}-s\right\rvert+\left\lvert
s_{m}-s\right\rvert<\frac{\mathfrak{e}}{2}+\frac{\mathfrak{e}}{2}=\mathfrak{e}$

and so $s_{n}$ is hyper-Cauchy. ∎

######  Conjecture 5.49.

The set $\mathbb{R^{Z_{<}}}$ is hyper-Cauchy complete with respect to the
$\mathfrak{e}$-topology.

######  Lemma 5.50.

Let $s_{n}$ be a sequence in $\mathbb{R^{Z_{<}}}$ whose members are just real
numbers – that is, for all $s\in s_{n}$,
$\texttt{Nst}_{\epsilon}(s)=\texttt{Nst}_{\omega}(s)=\emptyset$. Then $s_{n}$
is hyper-Cauchy if and only if there exists $N\in\mathbb{N}$ such that
$s_{m}=s_{N}$ for all $m\geq N$.

###### Proof.

Let $s_{n}$ be a hyper-Cauchy sequence in $\mathbb{R^{Z_{<}}}$ whose members
are real numbers. Then there exists $N\in\mathbb{N}$ such that

$\displaystyle\left\lvert s_{m}-s_{l}\right\rvert<\epsilon\textnormal{ for all
}m,l\geq N.$ (12)

Since $s_{n}$ is a sequence of real numbers, we obtain from Inequality 12 that
for all $m,l\geq N$, $\left\lvert s_{m}-s_{l}\right\rvert=0$ and so
$s_{m}=s_{N}$ for all $m\geq N$.
Conversely, let $s_{n}$ be a sequence in $\mathbb{R^{Z_{<}}}$ whose members
are real numbers and assume that there exists $N\in\mathbb{N}$ such that
$s_{m}=s_{N}$ for all $m\geq N$. Now let $\mathfrak{e}>0$ be given. We have
that for all $l,m\geq N$, $\left\lvert s_{m}-s_{l}\right\rvert=0<\mathfrak{e}$
and so $s_{n}$ is hyper-Cauchy. ∎

Another possible way to define convergence in our set is through the concept
of $\ell^{\infty}$ as follows:

######  Definition 5.51 ($\mathbb{R^{Z_{<}}}$-Convergence).

Suppose that $s_{n}$ is a sequence where every member of it is another
sequence itself, i.e.

$s_{n}=(s_{n})_{1},(s_{n})_{2},(s_{n})_{3},\dots,(s_{n})_{i},\dots$.

Then, $s_{n}$ converges to $\mathbf{s}$ iff $\forall m\in\mathbb{N}$, $\exists
N$ such that

$\forall n\geq N$, $\forall i$
$\left\lvert(s_{n})_{i}-s_{i}\right\rvert<\frac{1}{m}$.

We write RC$(s_{n},s)$ to denote that a sequence $s_{n}$ is
$\mathbb{R^{Z_{<}}}$-convergent to $s$.

######  Example 5.52.

The sequence $s_{n}=\langle\widehat{\frac{1}{n}},0,0,\dots\rangle$ is
$\mathbb{R^{Z_{<}}}$-convergent to 0.

The next interesting question is which of the three definitions above can be
used to define convergence in $\mathbb{R^{Z_{<}}}$? Unfortunately, neither of
them is adequate to serve as the definition of convergence in our set. The
three examples below demonstrate the reason. The first example shows that when
Classical Convergence is adopted in $\mathbb{R^{Z_{<}}}$, convergence is no
longer unique. While the second one shows how adopting Definition 5.41 gave
something unexpected occurs in our set, the last example shows why
$\mathbb{R^{Z_{<}}}$-convergence is not adequate.

######  Example 5.53.

Suppose that $s_{n}$ is a sequence defined by:

$s_{n}=\langle\widehat{0},n,0,\dots\rangle$.

Then by using Definition 5.40 above and the fact that any infinitesimals are
less than any rational numbers, $s_{n}$ classically converges to
$100\epsilon$, $200\epsilon$, $300\epsilon$, and so on. In other words, the
sequence $s_{n}$ satisfies (CC$(s_{n},100\epsilon)$),
(CC$(s_{n},200\epsilon)$), (CC$(s_{n},300\epsilon)$), and so on.

######  Example 5.54.

Using Definition 5.41, the sequence
$s_{n}=\langle\widehat{\frac{1}{n}},0,0,\dots\rangle$ does not converge in the
usual sense to 0, i.e. $s_{n}$ does not satisfy HC$(s_{n},\textbf{0})$. Taking
$r=\epsilon=\langle\widehat{0},1,0,\dots\rangle$ and $n=N+1$ will show this.

######  Example 5.55.

The sequence $s_{n}=\epsilon^{n}$ does not $\mathbb{R^{Z_{<}}}$-converge to 0,
as it should do intuitively.

Thus, this leaves us with the three definitions of convergence used in
$\mathbb{R^{Z_{<}}}$. There is no one definition of convergence in our set.
This is not necessarily a bad thing, it simply means that our notion of
convergence will differ from that of classical analysis.

Note that our attempts to have a proper notion of continuity and convergence
in $\mathbb{R^{Z_{<}}}$ can be used in the area of reverse mathematics. From
what we have done here, it can help us to gain a better understanding about
some necessary condition, for example, for a function $f$ to be continuous or
for a sequence to be convergent.

##  6 Some Notes on The Computability in $\mathbb{R^{Z_{<}}}$

A computable function is a function $f$ which could, in principle, be
calculated using a mechanical calculation tool and given a finite amount of
time. In the language of computer science, we would say that there is an
algorithm computing the function. A computable real number is, in essence, a
number whose approximations are given by a computable function.

The notion of a function $\mathbb{N}\to\mathbb{N}$ being computable is well
understood. In fact, all definitions that so far capturing this idea (such as
Turing Machines, Markov Algorithms, Lambda Calculus, the (partial) recursive
functions, and many more) have all led to the same class of functions. This,
in turn, has led to the so called Church-Markov-Turing thesis, which says that
this class is exactly what computable intuitively means. Given computable
pairing functions also, immediately, lead to a notion of computability for
other function types such as $\mathbb{N}^{k}\to\mathbb{N}^{m}$,
$\mathbb{N}\to\mathbb{Z}$ or $\mathbb{N}\to\mathbb{Q}$. If we see a real
number as a sequence of rational approximation, we also get a definition of a
computable real number.

However, we have to be a bit careful. There are many equivalent formulations
for when a real number $r$ is computable, that work well in practice. This
happens such as when

  * •

there is a finite machine that computes a quickly converging111111That is with
a fixed modulus of Cauchyness. Cauchy sequence that converges to $r$, or

  * •

it can be approximated by some computable function
$f:\mathbb{N}\rightarrow\mathbb{Z}$ such that: given any positive integer $n$,
the function produces an integer $f(n)$ such that

$\frac{f(n)-1}{n}\leq a\leq\frac{f(n)+1}{n}$.

We denote the set of all computable real numbers by $\mathbb{R}_{c}$. It is
well known (and also well studied) that many real numbers, such as $\pi$ or
$e$, are computable. However, not every real number is computable.

One possibility that does not turn out to be useful is to write down a real
number by using its decimal representation.121212Consider a number $r$ such
that there is an algorithm whose input is $n$, and it will give the
$n^{\textnormal{th}}$-digit of $r$’s decimal representation. The set of all
real numbers that have a computable decimal representation is denoted by
$\mathbb{R}_{d}$.

######  Remark 6.1.

Although the set $\mathbb{R}_{d}$ is closed under the usual arithmetic
operations, we have to be careful of what it really means. Take, for example,
addition. We know that if $x$ and $y$ are in $\mathbb{R}_{d}$, then $x+y$ is
also in $\mathbb{R}_{d}$. However, it does _not_ mean that the addition
operator itself is computable.

These ideas of computability can be extended to infinitesimals. In
$\mathbb{R^{Z_{<}}}$, we define its member to be computable if it satisfies
the condition as stated in Definition 6.2.

######  Definition 6.2.

A number $\mathbf{z}\in\mathbb{R^{Z_{<}}}$ is computable iff there is a
computable function $f$ such that $f(n,\cdot)$ are computable numbers and

$\mathbf{z}=\langle
f(1,\cdot),f(2,\cdot),\dots,\widehat{f(l,\cdot)},\dots\rangle$

where $l=f(0,0)$ denotes the index where the $\texttt{St}(\mathbf{z})$ is. We
denote the set of all computable members of $\mathbb{R^{Z_{<}}}$ by
$\mathbb{R}_{c}^{Z_{<}}$.

In this section, we showed that the standard arithmetic operations (functions)
in $\mathbb{R^{Z_{<}}}$ are computable (provided that the domain and codomain
of those functions are (in) $\mathbb{R}_{c}^{Z_{<}}$). This was done by
explicitly showing the program for each one of them. We actually uses a
concrete implementation of these ideas in the programming language Python,
whose syntax should be intuitively understandable even by those not familiar
with it. There is also no need to show that our programs are correct, since
they are so short that such a proof would be trivial.

Assuming that we already had a working implementation of $\mathbb{R}_{c}$, our
class $\mathbb{R}_{c}^{Z_{<}}$ could be implemented as in Listing
LABEL:mycode. There we defined the members of our set $\mathbb{R}_{c}^{Z_{<}}$
(basically just a container for the index $l$ as in Definition 6.2 and the
sequence of digits) and how their string representation would look like.

⬇

class infreal:

def __init__(self, digits, k=0):

self.k = k

self.digits = digits

def __repr__(self):

if self.k == 0:

return ”^” + ”, ”.join([str(self.digits(i)) for i in range(self.k,self.k+7)])
+ ”, …”

else:

return ”, ”.join([str(self.digits(i)) for i in range(self.k)]) + ”, ^” + ”,
”.join([str(self.digits(i)) for i in range(self.k,self.k+7)]) + ”, …”

def __getitem__(self, key): return self.digits(key)

Listing 1: How to define the members of $\mathbb{R}_{c}^{Z_{<}}$.

######  Example 6.3.

Suppose that we want to write the number
$\mathbf{1}=\langle\widehat{1},0,0,\dots\rangle$. Then by writing

One=infreal(lambda n:one if n==0 else zero, 0)

where zero and one are the real numbers $0$ and $1$, respectively, we just
created the number $\mathbf{1}$ in our system. The second argument of the
function infreal is just to give how many digits we want to have before the
real part of our number (the number with a hat). Its input and output will
look like as follows:

⬇

>>> zero = real(0)

>>> one = real(1)

>>> One = infreal(lambda n: one if n==0 else zero, 0)

>>> One

^1, 0, 0, 0, 0, 0, 0, …\end{lstlisting}

Furthermore, we will also be able to know what is its $n^{\textnormal{th}}$
digit for any $n\in\mathbb{N}$. See the code below:

\begin{lstlisting}[language=Python, numbers=none]

>>> One

^1, 0, 0, 0, 0, 0, 0, …

>>> One[0]

1

>>> One[-56]

0

>>> One[2454]

0\end{lstlisting}

\end{example}

\begin{example}

Similar to \Cref{ex:6.4}, the numbers $\mathbf{\epsilon}$ and
$\mathbf{\omega}$ could also be defined in our system.

\begin{lstlisting}[language=Python, numbers=none]

>>> Epsilon = infreal(lambda n: one if n == 1 else zero, 0)

>>> Epsilon

^0, 1, 0, 0, 0, 0, 0, …

>>> Omega = infreal(lambda n: one if n == 0 else zero, 1)

>>> Omega

1, ^0, 0, 0, 0, 0, 0, 0, … \end{lstlisting}

Also, we will be able to have exotic numbers such as
$\mathbf{\me}+\mathbf{2\me\epsilon}+\mathbf{3\me\epsilon}^2+\mathbf{4\me\epsilon}^3+\dots$
and its code will be as follows:

\begin{lstlisting}[language=Python, numbers=none]

>>> e = exp(rational(1,1))

>>> Funny = infreal(lambda n: real(rational(n + 1, 1)) * e if n > -1 else
zero, 0)

>>> Funny

^2.71828, 5.43656, 8.15485, 10.8731, 13.5914, 16.3097, 19.028, …

>>> Funny[43532]

118335

>>> Funny[-12964]

0\end{lstlisting}

\end{example}

\Cref{thm:addIsComputable}-\ref{thm:mulIsComputable} show that addition,
subtraction, and multiplication in $\Rzlc$ are computable.

\begin{theorem}

\label{thm:addIsComputable}

Suppose that we have $x,y\in\Rzlc$. Then the function $\widehat{+}_c$ defined
by

\begin{eqnarray*}

\widehat{+}_c: & \Rzlc & \rightarrow\Rzlc\\\

& (\mathbf{x},\mathbf{y})& \mapsto \mathbf{x}\widehat{+}\mathbf{y}

\end{eqnarray*}

is computable.

\end{theorem}

\begin{proof}

The following code shows that the function $\widehat{+}_c$ defined above is
computable.

\begin{lstlisting}[language=Python, numbers=none]

def __add__(self, other):

k = max(self.k, other.k)

return infreal(lambda n: self.digits(n - (k-self.k)) + other.digits(n -
(k-other.k)), k)\end{lstlisting}

\end{proof}

\begin{example}

Suppose that we want to add $\mathbf{\epsilon}$, $\mathbf{\omega}$, and
$\mathbf{1}$. Then we will have:

\begin{lstlisting}[language=Python, numbers=none]

>>> Omega

1, ^0, 0, 0, 0, 0, 0, 0, …

>>> Epsilon

^0, 1, 0, 0, 0, 0, 0, …

>>> Epsilon + Omega + One

1, ^1, 1, 0, 0, 0, 0, 0, …\end{lstlisting}

\end{example}

\begin{theorem}

\label{thm:minusIsComputable}

Suppose that we have $x,y\in\Rzlc$. Then the function $\widehat{-}_c$ defined
by

\begin{eqnarray*}

\widehat{-}_c: & \Rzlc & \rightarrow\Rzlc\\\

& (\mathbf{x},\mathbf{y})& \mapsto \mathbf{x}\widehat{-}\mathbf{y}

\end{eqnarray*}

is computable.

\end{theorem}

\begin{proof}

The following code shows that the function $\widehat{-}_c$ defined above is
computable.

\begin{lstlisting}[language=Python,escapeinside={(*}{*)}]

def __neg__(self): return infreal(lambda n: -self[n], self.k)
(*\label{line8}*)

def __sub__(self, other): return (self + (-other))\end{lstlisting}

The definition in line \ref{line8} shows that the additive inverse function is
computable.

\end{proof}

\begin{example}

Suppose that we want to add $\mathbf{\epsilon}\widehat{-}\mathbf{\omega}$ to
$\mathbf{1}$. Then we will have:

\begin{lstlisting}[language=Python, numbers=none]

>>> Epsilon - Omega + One

-1, ^1, 1, 0, 0, 0, 0, 0, …\end{lstlisting}

\end{example}

\begin{theorem}

\label{thm:mulIsComputable}

Suppose that we have $x,y\in\Rzlc$. Then the function $\widehat{\times}_c$
defined by

\begin{eqnarray*}

\widehat{\times}_c: & \Rzlc & \rightarrow\Rzlc\\\

& (\mathbf{x},\mathbf{y})& \mapsto \mathbf{x}\widehat{\times}\mathbf{y}

\end{eqnarray*}

is computable.

\end{theorem}

\begin{proof}

The following code shows that the function $\widehat{\times}_c$ defined above
is computable.

\begin{lstlisting}[language=Python,escapeinside={(*}{*)}, numbers=none]

def __mul__(self, other):

k = self.k + other.k

def digits(n):

if n < 0:

return zero

else:

return reduce((lambda x,y:x+y), [self.digits(i) * other.digits(n - i) for i in
range(n + 1)])

return infreal(digits, k) \end{lstlisting}

\end{proof}

\begin{example}

Suppose that we want to add $\mathbf{\epsilon}\widehat{\times}\mathbf{\omega}$
to $\mathbf{1}$ and also $-\mathbf{\epsilon}^2$ to $\mathbf{1}$. Then we will
have:

\begin{lstlisting}[language=Python, numbers=none]

>>> Epsilon * Omega + One

0, ^2, 0, 0, 0, 0, 0, 0, …

>>> Epsilon * -Epsilon + One

^1, 0, -1, 0, 0, 0, 0, …\end{lstlisting}

\end{example}

\subsection{Some Remarks on Non-Computability in $\Rzl$}

\begin{remark}

Even though division on $\R_c$ is computable (assuming the input does not
equal $0$), the same can not be said of $\Rzlc$.

\end{remark}

\begin{remark}

Suppose that we have a number $\mathbf{x}\in\Rzl$. Then without any further
information, the process of finding $\mathbf{x}^{-1}$ (the multiplicative
inverse of $\mathbf{x}$) is not computable. One extra information needed to
make it computable in $\Rzl$ is how many digits we want to have in
$\mathbf{x}^{-1}$, which of course will affect the accuracy of our result.
More precisely, it is known that it is not possible to give an algorithm that,
a given number $a \in \R_c$, decides whether $a = 0$ or $\lnot a =0$. So let
$a \in \R_c$ and consider $\mathbf{z} = a + \mathbf{\epsilon}$. We have $
\mathbf{\epsilon} \neq 0$. If $a = 0$ then $\mathbf{z}^{-1} =
\mathbf{\omega}$. If $a \neq 0$ then $z^{-1} < \mathbf{\omega}$. Thus by
checking whether the $\omega$-part of $z^{-1}$ is less than $1$ or greater
than $0$, we would be able to decide whether $a = 0$ or $\lnot a =0$.

\end{remark}

\begin{remark}

Similarly surprising, we can show that the absolute value function, which is
computable for $\R_c$, is not computable for $\Rzlc$. Here the absolute value
function is the function

\\[ \left| \mathbf{z} \right| = \begin{cases}

\mathbf{z} & \text{if } \mathbf{z} \geq 0 \\\ -\mathbf{z} & \text{if }
\mathbf{z} < 0

\end{cases} \\]

Similar to the above, take $a \in \R_c$ and consider $z = |a| -
\mathbf{\epsilon}$. If $a =0$ then $|z| =- |a| + \mathbf{\epsilon}$. If $a
\neq 0$ then $|z| = |a| - \mathbf{\epsilon}$. Thus by checking the
$\epsilon$-part of $|z|$, we would be able to decide whether $a = 0$ or $\lnot
a =0$.

\end{remark}

This also leads to comparison between numbers not being computable. Now this
is also the case in $\R_c$. However, for numbers $x,y \in \R_c $ such that $x
\neq y$, we can decide whether $x<y$ or $x > y$. This does not extend to
$\Rzlc$:

\begin{remark}

Comparison among the members in $\Rzlc$ is not computable.

Again, let $a \in \R_c$ and consider $\mathbf{x} = |a| + \mathbf{\epsilon}$
and $\mathbf{y} = \textbf{2}|a|$. If $a = 0$ then $\mathbf{x} > \mathbf{y}$,
and if $a \neq 0$ then $\mathbf{x} <\mathbf{y}$. Thus, once again we would be
able to decide whether $a = 0$ or $\lnot a =0$.

\end{remark}

##  7 Conclusion and Suggestions for Further Research

In this article we treated the nonstandard real numbers in the spirit of the
‘Chunk and Permeate’ approach. The sets $\mathbb{R}$ and $\mathbb{{}^{*}R}$
were thrown together (that means: combining their languages and axioms) and
arising inconsistency issues were dealt with using paraconsistent reasoning
strategy. In this case — and this is one of the interesting novelties — the
procedure was made explicit by introducing new sets $\mathbb{\widehat{R}}$ and
in turn $\mathbb{R^{Z_{<}}}$, the separate ‘chunks’. In a sense, we
transferred the ‘Chunk and Permeate’ approach from the theoretical level to
the (explicit) model level. Whereas the impact on infinitesimal mathematics
was only sketched in [9], it was worked out in detail here, using the sets
$\mathbb{\widehat{R}}$ and $\mathbb{R^{Z_{<}}}$. After introducing the
theoretical background, we constructed the new model of nonstandard analysis
in detail. The remaining part of this paper lies in an extensive discussion of
topological, applied (in the sense of calculus), and computability issues of
the obtained model. A side result of the constructed set $\mathbb{R^{Z_{<}}}$
was a direct consistency proof of the Grossone theory, see [23].

On the topological aspect in Section 4, we introduced some new notions of
metrics, balls, open sets, and etc in $\mathbb{R^{Z_{<}}}$ together with their
properties. On the applied aspect in Section 5, some new concepts on the
calculus in $\mathbb{R^{Z_{<}}}$ were discussed, e.g. derivative (we
successfully developed a permeability relation such that the derivative
function in $\mathbb{R^{Z_{<}}}$ can be permeated to $\mathbb{R}$),
continuity, and convergence. For the two last issues, some new notions were
introduced in this article. First of all, we discussed the three possible
notions of continuity that can be applied to either $\mathbb{R}$ or
$\mathbb{R^{Z_{<}}}$. We also determined how they relate to each other in
their respective model. While doing that, we discovered a new kind of fractals
— infinitesimal fractals. After analysing three possible notions of
continuity, we decided that the best notion that can be used in our setting
$\mathbb{R^{Z_{<}}}$ is the $\epsilon$-$\delta$ definition and by doing that,
we do not only preserve much of the spirit of classical analysis but also
retain the intuition of infinitesimals. After establishing our position, we
introduced a more detailed notion of continuity which is called
($k,n$)-continuity (as can be seen in Definition 5.23). We explored how this
new notion of continuity behaves, e.g. what happens with the composition of
two continuous functions and also how this notion behaves under
multiplication. It is worth pointing out here that this new notion of
continuity is a much more fine-grained notion than the classical continuity.
Last but not least, we showed that the set $\mathbb{R^{Z_{<}}}$ has nice
computability features. We succeeded in building a program, in Python, to show
that we can have a computable number $\mathbb{R^{Z_{<}}}$. The set of all
these computable numbers is denoted by $\mathbb{R}_{c}^{Z_{<}}$. We also
showed some interesting remarks regarding this computability issue.

In term of further research, we indicated some possible areas of further
development as follows. First, one could try to do infinitesimal analysis
using the relevant logic R. The comparison between the results (perhaps)
gotten in R and the one described here might be interesting, especially in
term of usefulness and simplicity. Secondly, regarding the ‘transfer
principle’, our intuition says that it is equivalent to the notion of
permeability in the Cchunk & Permeate strategy. One could try to formally
prove it, or disprove it. Thirdly, in term of computability issue, using the
calculus on $\mathbb{R^{Z_{<}}}$, one could try to formulate the necessary and
sufficient conditions for the derivatives of functions, for example, on a
computer to exist. And perhaps, showing also how to find these derivatives
whenever they exist. This, of course, can also be applied to the other
notions. Fourthly, as been said in the previous sections, some results
described in this article could help us to gain a better understanding in
another area of research (the two that were mentioned in Section 5 are reverse
mathematics and quantum physics). One could try to work out the details on
this.

In general, with the new consistent sets created in this work, new
opportunities awaits mathematicians. One of the joys of mathematics is to
explore a world which has no physical substance, and yet is everywhere in
every aspect of our lives. Infinities and infinitesimals offer ways to explore
hitherto unseen aspects of our world and our universe, by giving us the vision
to see the greatest and smallest aspects of life. Even a naïve set, when it
demonstrates harmony, offer another dimension of even clearer precision. In a
wide sense, the work on this article can also be seen as a contribution to
bridge (the antipodes) constructive analysis and nonstandard analysis. This
problem has been extensively (and intensively) discussed in the past few years
(see for example [40, 41, 42, 8, 43, 31]).

## References

  * [1] Arkeryd, L.O., Cutland, N.J., Henson, C.W.: Nonstandard analysis: Theory and applications, vol. 493.  Springer Science & Business Media (2012)
  * [2] Avron, A.: Natural 3-valued logics-characterization and proof theory.  The Journal of Symbolic Logic 56(01), 276–294 (1991)
  * [3] Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis, vol. 2.  Wiley (1992)
  * [4] Batens, D., Mortensen, C., Priest, G., Bendegem, J.P.V.: Frontiers in Paraconsistent Logic.  Research Studies Press (2000)
  * [5] Bell, J.L.: A primer of infinitesimal analysis.  Cambridge University Press (1998)
  * [6] Benci, V., Di Nasso, M.: Alpha-theory: an elementary axiomatics for nonstandard analysis.  Expositiones Mathematicae 21(4), 355–386 (2003)