Naïve Infinitesimal Analysis: Its Construction and Its Properties
NNa¨ıve Infinitesimal Analysis
Its Construction and Its PropertiesAnggha Nugraha ∗ Maarten McKubre-JordensHannes Diener
Abstract
This paper aims to build a new understanding of the nonstandardmathematical analysis. The main contribution of this paper is the con-struction of a new set of numbers, R Z < , which includes infinities andinfinitesimals. The construction of this new set is done na¨ıvely in thesense that it does not require any heavy mathematical machinery, andso it will be much less problematic in a long term. Despite its na¨ıvetycharacter, the set R Z < is still a robust and rewarding set to work in. Wefurther develop some analysis and topological properties of it, where notonly we recover most of the basic theories that we have classically, but wealso introduce some new enthralling notions in them. The computabilityissue of this set is also explored. The works presented here can be seen asa contribution to bridge constructive analysis and nonstandard analysis,which has been extensively (and intensively) discussed in the past fewyears. There have been many attempts to rule out the existence of inconsistencies inmathematical and scientific theories. Since the 1930s, we have known (fromG¨odel’s results) that it is impossible to prove the consistency of any interestingsystem (in our case, this is a system capable of dealing with arithmetic andanalysis). One of the famous examples of inconsistency is as follows. Supposewe have a function f ( x ) = ax + bx + c and want to find its first derivative. Byusing Newton’s ‘definition of the derivative’: ∗ [email protected] a r X i v : . [ m a t h . L O ] S e p (cid:48) ( x ) = f ( x + h ) − f ( x ) h = a ( x + h ) + b ( x + h ) + c − ax − bx − ch = ax + 2 axh + 2 h + bx + bh + c − ax − bx − ch = 2 axh + 2 h + bhh = 2 ax + h + b (1)= 2 ax + 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 definitionthat h is a small but non-trivial neighbourhood around x and, because it isused as a divisor, cannot be zero. However, the fact that it is simply omitted atthe 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 th century bythe concept of limit, but its (intuitive) na¨ıve 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 reason-ing with prima facie inconsistent infinitesimals) is entrenched in our scientificcommunity and it means that inconsistency is something that, if unavoidable,should be handled appropriately. Actually, inconsistency would not have beensuch a problem if the logic used was not explosive [50]. The problem is thatour 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 andthis is the main reason for the birth of paraconsistent mathematics which usesparaconsistent logic as its base.Recent advances in paraconsistent mathematics have been built on develop-ments in set theory [51], geometry [29], arithmetic [28], and also the elementaryresearch at calculus [28] and [9]. A first thorough study to apply paraconsis-tent 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 useof inconsistent mathematics, McKubre-Jordens and Weber in [26] analysed anaxiomatic approach to the real line using paraconsistent logic. They succeedto show that basic field and also compactness theorems hold in that approach.They can also specify where the consistency requirement is necessary. Thesepreliminary works in [26] and [52] show how successful a paraconsistent setting If 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.
2o analysis can be. On the side of the non-standard analysis, it can be seen forexample 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: L , the language of real numbers R , and L ∗ , the languageof the set of hyperreal numbers ∗ R . The language L ∗ is an extension of thelanguage L . It can be shown that each of those two sets forms a model for theformulas in its respective language.Speaking about the hyperreals, the basic idea of this system is to extendthe set R to include infinitesimal and infinite numbers without changing any ofthe elementary axioms of algebra. Transfer principle holds an important rolein the formation of the set ∗ R in showing what are still preserved in spite ofthis extension. However, there are some problems with the transfer principle,notably its non-computability (see Subsection 2.2). To avoid its use, one canlogically think to simply collapsing the two languages into one language (cid:98) L whichcorresponds to the set (cid:98) R , a new set of numbers constructed by combining the twoset of axioms of R and ∗ 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 aremany paraconsistent logics that are available at the moment. This could be agood thing, or from another perspective, be an additional difficulty as we needto choose wisely which paraconsistent logic we want to use at first, i.e. whichone is the most appropriate or the best for our purpose. But then, to be ableto 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, whichwas introduced by Brown and Priest in 2003 [9]. Using this strategy, we divideour set (cid:98) R into some consistent chunks and build some permeability relationsbetween them. This process leads us to the creation of the sets R Z and R Z < .In our view, this idea is more sensible and promises to be more useful thanthe first. Moreover, after further analysing this idea, we produce some newinteresting and useful notions that will be worth to explore even further (seeSections 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 G¨odel’s second incomplete-ness theorem, if we can build a new structure for nonstandard mathematicalanalysis which is resilient to contradiction, we would open the door to havingnot just a sound, but a complete mathematical theory. To put it simply, like We-ber said, ‘In light of G¨odel’s result, an inconsistent foundation for mathematicsis the only remaining candidate for completeness’ [50].3 Some Preliminaries
To understand and carry our investigation, it is essential to have an accurategrasp of the received view about formal language, the reals and hyperreals, andthe paraconsistent logic. In turn, to analyse these matters, it will be useful tofix some terminologies.
Formal language is built by its syntax and semantics. Here are the symbols thatare used in our language:variables : a b c . . . x x . . . grammatical signs : ( ) ,connectives : ∧ ∨ ¬ → quantifiers : ∀ ∃ constant symbols : 1 − . π √ . . . function symbols : + − sin tan . . . relation symbols : = < > ≤ ≥ . . . Like in natural language, a sentence is built by its term. Terms and sentencesare defined as usual. Example 2.1 uses the simple language I to build somestatements about integer numbers, Z . Example 2.1.
In addition to our usual connectives, variables, quantifiers andgrammatical symbols in Z , I also contains:constant symbols : . . . , − , − , , , , . . . function symbols : q ( x ) = x add ( x, y ) = x + y mul ( x, y ) = x × y relation symbols : P ( x ) for “ x is positive” E ( x, y ) for “ x and y are equal” In this language I , one can translate an English statement ‘squaring any integernumber will give a positive number’ as ∀ x P ( s ( x )) . Now we can define a language for the set hyperreals. We define a language L whose every sentence, if true in reals, is also true in hyperreals. Definition 2.2 (Language L ) . The language L consists of the usual definedvariables, connectives, and grammatical signs in R , and the following:constant symbols : one symbol for every real numberfunction symbols : one symbol for every real-valued function of any finitenumber real variablesrelation symbols : one symbol for every relation on real numbers of any finitenumber real variables Definition 2.3 (Model of a Language) . Suppose that we have a language A . Amodel for A consists of:1. a set A so that each constant symbol in A corresponds to an element of A ,2. a set F of functions on A so that each function symbol in A correspondsto a function in F ,3. a set R of relations on A so that each relation symbol in A corresponds toa relation in R . Example 2.4.
For our language I over integer numbers, its model is the set A = Z with several functions and relations already well-defined in Z . Theorem 2.5 (Reals as a Model) . The real number system R is a model forthe language L .Proof. Take A = R and F and R as set of all functions and relations, respec-tively, which are already well-defined in R .Then, by using the definition of a model, we defined what hyperreal numbersystem is. Definition 2.6.
A hypereal number system is a model for the language L that,in addition to all real numbers, contains infinitesimal and infinite numbers. Now suppose that ∗ R is the set of all hyperreal numbers. Our goal now is toshow that ∗ R is a model for the language L . To show this, we need to extendthe definition of relations and functions on R into ∗ R . This extension can alsobe seen in [18]. Definition 2.7 (Extended Relation) . Let R be a k -variable relation on R , i.e.for every x , x , . . . , x k , R ( x , x , . . . , x k ) is a sentence that is either true orfalse. The extension of R to ∗ R is denoted by ∗ R . Suppose that x , x , . . . , x k are any hyperreal numbers whose form is { x n } , { x n } , . . . , { x kn } , respectively.We define ∗ R ( x , x , . . . , x k ) as true iff { n | R ( x n , x n , . . . , x kn ) is true in R } is big. Otherwise, R ( x , x , . . . , x k ) is false. Example 2.8.
Suppose that z ○ = { , , , , , ... } A ‘big set’ is a set of natural numbers so large that it includes all natural numbers withthe possible exception of finitely many [18]. y taking k = 1 in Definition 2.7, we are able to have a relation I ( x ) = “ x is an integer”. The relation I ( z ○ ) is true. This is because the set of indexeswhere relation I ( x ) is tru, is a big set. Thus, we conclude that z ○ is actually ahyperinteger. Definition 2.9 (Extended Function) . Let f be an k -variables function on R .The extension of f to ∗ R is denoted by ∗ f . Suppose that x , x , . . . , x k areany hyperreal numbers whose form is { x n } , { x n } , . . . , { x kn } , respectively. Wedefine ∗ f ( x , x , . . . , x k ) by ∗ f ( x , x , . . . , x k ) = { f ( x , x , . . . , x k ) , f ( x , x , . . . , x k ) , f ( x , x , . . . , x k ) , . . . } Example 2.10.
Suppose that e ○ = { , , , , ... } By taking k = 1 in Definition 2.9, we might have, for example, a well-definedhypersinus function: sin( e ○ ) = { sin(2) , sin(4) , sin(6) , sin(8) , ... } Theorem 2.11 (Hyperreals as a Model) . The set ∗ R is a model for the language L that contains infinitesimals and infinities.Proof. Take A = ∗ R in Definition 2.3 with all of the functions ∗ f defined inDefinition 2.9 and relations ∗ R defined in Definition 2.7. Definition 2.12 (Transfer Principle) . Let S be a sentence in L . The transferprinciple says that: S is true in the model R for L iff S is true in the model ∗ R for L . As Goldblatt said in [18], ‘The strength of nonstandard analysis lies in the abilityto transfer properties between R and ∗ R .’ But, there are some serious problemswith the transfer principle. Some of them are: it is non-computable in the senseof there is no good computable representation of the hyperreals to start with; itreally depends intrinsically on the mathematical model or language we use; weare prone to get things wrong when not handled correctly (especially becauseof human error).Furthermore, there is an all too often overlooked, yet major deficiency withthe transfer principle: it performs particularly poorly upon a rather simple“cost-return” analysis. Namely, on the one hand, the mathematical machinerywhich must be set up in advance in order to use the transfer principle is of such aconsiderable technical complication and strangeness with a lack of step-by-stepintuitive insight that, ever since 1966, when Abraham Robinson published thefirst major book on the subject — that is, for more than half a century by now6 none of the more major mathematicians ever chose to switch to the effectivedaily use of nonstandard analysis, except for very few among those who havedealt with time continuous stochastic processes, and decided to use the “LoebIntegral” introduced in 1975. On the other hand, relatively few properties ofimportance can ever be transferred, since they are not — and cannot be, withinusual nonstandard analysis — formulated in terms of first order logic.One possible solution for overcoming (some of) these problems is simplyby not using the transfer principle, by throwing together the two sets R and ∗ R , i,e. combining their languages and axioms. However, Example 2.13 showsthat if we just simply combine the two languages, it will pose one big problemof contradictions which can lead to absurdity if we use classical logic. Thatis the reason why we use paraconsistent logic as it is resilient against localcontradiction. Example 2.13.
Take the well-ordering principle for our example. This prin-ciple says that: “every non-empty set of natural numbers contains a least ele-ment”. Call a set S = { x ∈ N ∗ : x is infinite } . Let s be its least element. Notethat s is infinite and so s − . Thus, s − ∈ S and it makes s is not the leastelement. Therefore, there exists l such that l is the least element of S and thereis no l such that l is the east element of S . In addition to the problems with the transfer principle, there are also somedownsides of the construction of the set ∗ R itself. Indeed, the usual construc-tion of the hyperreal set R involves an ultrafilter on N , the existence of whichis justified by appealing to the full Axiom of Choice whose validity is still agreat deal to discuss [49, 22]. Moreover, it also relies on some heavy and non-constructive mathematical machineries such as Zorn’s lemma, the Hahn-Banachtheorem, Tychonoff’s theorem, the Stone-Cech compactification, or the booleanprime ideal theorem. On the side of the nonstandard analysis itself, there aresome critiques as can be seen in [7, 13, 14, 19, 45, 47]. Most of them are relatedwith its non-constructivism and its difficulties to be used in class teaching. Thisproblem can be solved by building a na¨ıve constructive non-standard set andmaking sure that it is still a useful set by redefining some well-known notionsin there. Generally, paraconsistent logics are logics which permit inference from incon-sistent information in a non-trivial fashion [36]. Paraconsistent logics are char-acterized by rejecting the universal validity of the principle ex contradictionequodlibet (ECQ) which is defined below.
Definition 2.14 (ECQ Principle) . The principle of explosion, ECQ, is the lawwhich states that any statement can be proven from a contradiction.
By admitting the ECQ principle in one theory, if that theory contains a singleinconsistency, it becomes absurd or trivial. This is something that, in paracon-sistent logics, does not follow necessarily.7araconsistent logicians believe that some contradictions does not necessar-ily make the theory absurd. It just means that one has to be very careful whendoing deductions so as to avoid falling from contradiction into an absurdity. Inother words, classical and paraconsistent logic treat contradiction in differentways. The former treats contradiction as a global contradiction (making thetheory absurd), while the latter treats some contradictions as a local contradic-tion. In other words, classical logic cannot recognise if there is an interestingstructure 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 ∃ A, B such that A ∧ ¬ A (cid:48) B . The symbol Γ (cid:96) A simply means that there exists a proof of A from set offormulas Γ, in a certain logic.There are at least two different approaches to paraconsistent logics. The firstis by adding another possible value, both true and false, to classical truth valueswhile 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, arePriest’s Paraconsistent Logic LP ⊃ and Relevant Logic R . More explanationson 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 beat least two terms that we have to be aware of: inconsistency and incoherence .The first term, inconsistency, is applicable if there occurs a contradiction ina system. Meanwhile, the second term, incoherence, is intended for a systemwhich proves anything (desired or not). In classical logic, there is no differencebetween these two terms because of the ECQ principle. Thus, if a contradictionarises inside a theory, anything that the author would like to say can be provedor inferred within that theory. This is something that likewise does not have tohappen in paraconsistent logic.In mathematical theory, foundation of mathematics is the study of the basicmathematical concepts and how they form more complex structures and con-cepts. This study is especially important for learning the structures that formthe language of mathematics (formulas, theories, definitions, etc.), structuresthat often called metamathematical concepts. A philosophical dimension ishence central to this study. One of the most interesting topics in the foundationof mathematics is the foundation of real structure, or analysis.Generally, it is known that the construction of real numbers is categorisedin classical logic — while there is an advancement in paraconsistent logic suchas in [4], this has not yet been extensively explored. However, it seems viable tomake a further study of real structure by developing paraconsistent foundationsof analysis. Note that in general, relevant logic differs from paraconsistent logic. When someoneclaims that they use relevant logic, it implies that they use paraconsistent logic, but not viceversa. Using paraconsistent logic does not necessarily mean using relevant logic, e.g. the logic LP ⊃ below is not relevant logic. The Creation of The New Sets
As noted before, the transfer principle is useful as well as fairly problematic atthe same time. One way to avoid the unnecessary complications of the transferprinciple is by collapsing the two languages involved into one language (cid:98) L . Simplycollapsing the two, however, causes additional problems. One of the problemsthat can be expected to appear is contradiction, but we can use a paraconsistentlogic to handle this when it arises.In this section, we construct a new number system (cid:98) R through its axiomati-sation by ‘throwing’ the axioms of R and ∗ R together. The set (cid:98) R is the numbersystem on which the language (cid:98) L will be based and it contains positive and neg-ative infinities, and also infinitesimals. It does make sense to insert infinities(and their reciprocals, infinitesimals) into (cid:98) R as some of the contradictions inmathematics come from their existence and also because they are still used intoday’s theory as can be seen in [48]. (cid:98) R For the sake of clarity, Axioms 3.1–3.7 give the axiomatisation of the numbersystem (cid:98) R . Axiom 3.1 (Additive Property of (cid:98) R ) . In the set (cid:98) R , there is an operator + thatsatisfies:A1: For any x, y ∈ (cid:98) R , x + y ∈ (cid:98) R .A2: For any x, y ∈ (cid:98) R , x + y = y + x .A3: For any x, y, z ∈ (cid:98) R , ( x + y ) + z = x + ( y + z ) .A4: There is ∈ (cid:98) R such that x + 0 = x for all x ∈ (cid:98) R .A5: For each x ∈ (cid:98) R , there is − x ∈ (cid:98) R such that x + ( − x ) = 0 . Axiom 3.2 (Multiplicative Property of (cid:98) R ) . In the set (cid:98) R , there is an operator · that satisfies:M1: For any x, y ∈ (cid:98) R , x · y ∈ (cid:98) R .M2: For any x, y ∈ (cid:98) R , x · y = y · x .M3: For any x, y, z ∈ (cid:98) R , ( x · y ) · z = x · ( y · z ) .M4: There is ∈ (cid:98) R such that (1 = 0) → ⊥ and ∀ x ∈ (cid:98) R x · x .M5: For each x ∈ (cid:98) R , if ( x = 0) → ⊥ , then there is y ∈ (cid:98) R such that x · y = 1 . Axiom 3.3 (Distributive Property of (cid:98) R ) . For all x, y, z ∈ (cid:98) R , x · ( y + z ) =( x · y ) + ( x · z ) . Axiom 3.4 (Total Partial Order Property of (cid:98) R ) . There is a relation ≤ in (cid:98) R ,such that for each x, y, z ∈ (cid:98) R : We are not necessarily expecting the resulting system to have contradictions, but we willmake sure that we maintain coherency by not allowing contradictions to become an absurdity.
1: Reflexivity: x ≤ x ,O2: Transitivity: ( x ≤ y ∧ y ≤ z ) → x ≤ z ,O3: Antisymmetry: ( x ≤ y ∧ y ≤ x ) ↔ x = y ,O4: Totality: ( x ≤ y → ⊥ ) → y ≤ x .O5: Addition order: x ≤ y → x + z ≤ y + z .O6: Multiplication order: ( x ≤ y ∧ z ≥ → xz ≤ yz . Axiom 3.5 (Completeness Property of (cid:98) R ) . Every non-empty bounded abovesubset of (cid:98) R has a least upper bound (see Definition ?? ). Axiom 3.6 (Infinitesimal Property of (cid:98) R ) . The set (cid:98) R has an infinitesimal (seeDefinition 3.8 for what infinitesimal is). Axiom 3.7 (Archimedean Property of (cid:98) R ) . For all x, y > , ∃ 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 (cid:98) R , causes computability issues inour set. The last two axioms, they cause the consistency trouble. Infinity andinfinitesimals are formally defined in Definition 3.8. Definition 3.8.
An element x ∈ (cid:98) R is • infinitesimal iff ∀ n ∈ N | x | < n ; • finite iff ∃ r ∈ R | x | < r ; • infinite iff ∀ r ∈ R | x | > r ; • appreciable iff x is finite but not an infinitesimal;By using notation (cid:15) as an infinitesimal, an infinity ω is defined as a reciprocalof (cid:15) , i.e. (cid:15) . Definition 3.9.
For any numbers x, y ∈ (cid:98) R ,1. x ≥ y := x < y → ⊥ x < y := x ≤ y ∧ ( x = y → ⊥ ) x > y := x ≥ y ∧ ( x = y → ⊥ )If we look further, the set (cid:98) R is actually an inconsistent set. Example 3.10 givesone of these contradictions. Example 3.10.
Suppose that we have a set S = { x ∈ (cid:98) R : | x | < n for all n ∈ N } .In other words, the set S consists of all infinitesimals in (cid:98) R . It is easily proventhat S is not empty and bounded above. So by Completeness Axiom, S has aleast upper bound. Suppose that z is its least upper bound (which also meansthat z must be an infinitesimal). Because z is an infinitesimal, z is also aninfinitesimal and this means z is also in (cid:98) R . By using Definition 3.9, z < z nd so, z is not the least upper bound of S . Now suppose that sup S = 2 z . Thesame argument can be used to show that z is not supremum of S but z . Wecan build this same argument infinitely to show that there does not exist s suchthat sup S = s . Thus, we have ∃ s : sup S = s and (cid:64) s : sup S = s . This kind of contradiction forces us to use a non-explosive logic such as para-consistent logic instead of classical logic to do our reasoning in (cid:98) 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 beseen in [9]. (cid:98) L and The Creation of The Set R Z < Using the C&P strategy, we divided (cid:98) L into some consistent chunks — naturally,there might be several ways to do it (e.g. one can have an idea to divide theoriginal set into two, three, or even more chunks). Nevertheless, we found outthat one particular way to have just two different chunks, as provided in here, isthe most interesting one as it leads to the creation of a new model. One chunk isa set which contains Axioms (3.1–3.4,3.6), while the other chunk is a set whichcontains Axioms (3.1–3.5,3.7). The consistency of each chunk was proved byproviding a model for each of them. One of the possible — and interesting — chunks is a set consists of Axioms3.1-3.4 and Axiom 3.6. Here we proved the consistency of this chunk and alsosome corollaries that we have.It is well-established that the set of hyperreals, ∗ R , clearly satisfied thoseaxioms [6]. Nevertheless, the construction of hyperreals ∗ R depends on highlynon-constructive arguments. In particular, it requires an axiom of set theory, thewell-ordering principle, which assumes into existence something that cannot beconstructed [21]. Here we proposed to take a look at a simpler set. Rememberthat our set has to contain not just R , but also infinitesimals and infinities(and the combinations of the two). We take R Z , functions from integers toreal numbers, as our base set. The member of R Z consists of standard and non-standard parts. The standard part of a certain number simply shows itsfinite element (the real part), while the non-standard part shows its infinite orinfinitesimal part (see Definition 3.11). Definition 3.11 (Member of R Z ) . A typical member of R Z has the form x = (cid:104) (cid:15) − i , (cid:98) x, (cid:15) j (cid:105) where (cid:98) x ∈ R and (cid:15) n denotes the sequence of the constant part ofinfininitesimals if n > , and infinities if n < . Notice that the symbol (cid:98) x in Definition 3.11 signs the standard part of a numberin R Z . Thus, the member of 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 R Z . Example 3.12 (Numbers in R Z ) . . The number 1 is written as = (cid:104) . . . , , (cid:98) , , , . . . (cid:105) .2. The number (cid:15) is written as (cid:15) = (cid:104) . . . , (cid:98) , , , . . . (cid:105) .3. The number ω (one of the infinities) is written as ω = (cid:104) . . . , , (cid:98) , , . . . (cid:105) .4. The number (cid:15) − ω is written as + (cid:15) − ω = (cid:104) . . . , , − , , (cid:98) , , , . . . (cid:105) . By using this form, all of the possible numbers can be written in R Z . How-ever, this infinite form is problematic in a number of ways. For example, multi-plication cannot be easily defined and there might exist multiple inverses if theset 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 R Z < . Theonly difference between R Z < and R is that, for any number x , we will not havean infinite sequence on the left side of its standard part. See example 3.13 andcompare to Example 3.12. Example 3.13 (Numbers in R Z < ) .
1. A number 1 is written as = (cid:104) (cid:98) , , , . . . (cid:105) .2. A number (cid:15) is written as (cid:15) = (cid:104) (cid:98) , , , . . . (cid:105) .3. A number ω (one of the infinities) is written as ω = (cid:104) , (cid:98) , , . . . (cid:105) .4. A number (cid:15) − ω is written as + (cid:15) − ω = (cid:104)− , , (cid:98) , , , · · · (cid:105) . Some of the properties of the set R Z < are as follows, while their proof (whenneeded) and some examples of them can be seen in [32, p. 51–57]. Definition 3.14 (Addition and Multiplication in R Z < ) . For any number x = (cid:104) x z (cid:105) and y = (cid:104) y z (cid:105) in R Z < , define: x + y = (cid:104) x z + y z : z ∈ Z (cid:105) and x × y is calculated by: x × y = (cid:32) (cid:88) i = − m a i (cid:15) i (cid:33) (cid:98) × (cid:88) j = − n b j (cid:15) j = (cid:32)(cid:88) k ∈ Z c k (cid:15) k (cid:33) , where c k = (cid:80) i + j = k a i b j . Definition 3.15 (Order in R Z < ) . The set R Z < is endowed with (cid:98) ≤ , the lexico-graphical ordering. Proposition 3.16.
The set R Z < satisfies the additive property in Axiom 3.1. Proposition 3.17.
The set R Z < satisfies the multiplicative property in Axiom3.2. roposition 3.18. The set R Z < satisfies the distributive property in Axiom3.3. Proposition 3.19.
The set R Z < satisfies the total order property in Axiom 3.4. The following results show how to find an inverse of any members of R Z < and its uniqueness property. Proposition 3.20.
The number + ω has a unique inverse. Proposition 3.21.
The number ω (cid:98) + (cid:15) has a unique inverse. Lemma 3.22.
For any r, s ∈ R , a number r(cid:15) (cid:98) + sω has a unique inverse. Lemma 3.23.
For any r ∈ R , a number r (cid:98) + ω (or r (cid:98) + (cid:15) ) has a unique inverse. Theorem 3.24.
For any number x ∈ R Z < , x has a unique inverse. The most evident model for this second chunk is the set of real numbers, R .Thus, so far, we have already had two chunks in (cid:98) L and we proved their consis-tencies by providing a model for each of them. R Z < Theories that contain infinities have always been an issue and have attractedmuch research, for example [10, 17, 20, 25, 37]. Note that the arithmetic devel-oped for infinite numbers was quite different with respect to the finite arithmeticthat we are used to dealing with. For example, Sergeyev in [44] created theGrossone 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 num-ber
Grossone and denoted it with (cid:172) . The four axioms that form this Grossonetheory and its details can be seen [44].It is very important to emphasis that (cid:172) is a number, and so it works as ausual number. For example, there exist numbers such as (cid:172) − , (cid:172) + 16 , ln (cid:172) ,and etc. Also for instance, (cid:172) − < (cid:172) . The introduction for this new number (cid:172) makes us able to rewrite the set of natural numbers N as: N = { , , , , , , . . . , (cid:172) − , (cid:172) − , (cid:172) } .Furthermore, adding the Infinite Unit Axiom (IUA) to the axioms of naturalnumbers will define the set of extended natural numbers ∗ N : ∗ N = { , , , , , , . . . , (cid:172) − , (cid:172) (cid:172) + 1 , . . . , (cid:172) − , (cid:172) , . . . } ,and the set ∗ Z , extended integer numbers, can be defined from there.Here we argued that our new set R Z < provides the model of Grossone theoryand therefore proves its consistency rather in a deftly way. This is unlike the way the usual infinity, ∞ , behaves where for example, ∞ − ∞ . Italso differs from how Cantors cardinal numbers behave. efinition 3.25. In our system R Z < , the number (cid:172) is written as: (cid:172) = (cid:104) , (cid:98) , , . . . (cid:105) . Proposition 3.26.
For every finite number r ∈ R Z < , r < (cid:172) .Proof. The order in set R Z < is defined lexicographically. Now suppose that r = (cid:104) , (cid:98) r, , , . . . (cid:105) where (cid:98) r ∈ R . Then it is clear that r < (cid:172) . Proposition 3.27.
All of the equations in the Identity Axiom of Grossonetheory are also hold in R Z < . The fractional form of (cid:172) can also be defined in R Z < as:for any n ∈ N , (cid:172) n = (cid:104) n , (cid:98) , , . . . (cid:105) .Speaking about the inverse, one of the advantages of having the set R Z < is tobe able to see what the inverse of a number looks like, not like in the Grossonetheory. See Example 3.28 for more details. Example 3.28.
In Grossone theory, the inverse of (cid:172) + (cid:172) is just (cid:172) + (cid:172) . Whilein our set R Z < , (cid:172) + (cid:172) is written as (cid:15) + ω and its inverse is (cid:104) (cid:98) , , , − , , , , − , , . . . (cid:105) = (cid:15) − (cid:15) + (cid:15) − (cid:15) + . . . .In other words, the more explicit form of (cid:172) + (cid:172) is a series ( − n +1 (cid:172) − (2 n − for n = 1 , , , · · · ∈ N . In [23], Gabriele Lolli analysed and built a formal foundation of the Grossonetheory based on Peano’s second order arithmetic. He also gave a slightly differ-ent notion of some axioms that Sergeyev used. One of the important theoremsin 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 isof course conditional, as apparent from the proof, upon the consistency of PA µ ’while ‘its model theoretic proof is technically rather demanding’.Thus, through what presented in this subsection, we have proposed a newway to prove the consistency of Grossone theory by providing a straightforward model of it. There is no need for complicated model-theoretic proofs. Theset 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 inpart, as a contribution to the development of Grossone theory. R Z < Some topological properties of the set R Z < are discussed in this section. How-ever, there are a number of definitions and issues that should be addressed firstin order to understand how those properties will be applied to our set properly.14 .1 Metrics in R Z < We defined what is meant by a distance (metric) between each pair of elementsof R Z < . Definition 4.1.
A metric ρ in a set X is a function ρ : X × X → [0 , ∞ ) where for all x, y, z ∈ X , these four conditions are satisfied:1. ρ ( x, y ) ≥ ,2. ρ ( x, y ) = 0 if and only if x = y ,3. ρ ( x, y ) = d ( y, x ) ,4. ρ ( x, z ) ≤ ρ ( x, y ) + ρ ( y, z ) .When that function ρ satisfies all of the four conditions above except the secondone, ρ is called a pseudo-metric on X . Now we define two functions d and d ψ in R Z < as follows: Definition 4.2.
For all x , y ∈ R Z < , d : R Z < × R Z < → R Z < and d ψ : R Z < × R Z < → R where d ( x , y ) = | y − x | and d ψ ( x , y ) = St ( | y − x | ) . It can be easily verified that d is a metric in R Z < (and so (cid:0) R Z < , d (cid:1) formsa metric space) and d ψ is a pseudo-metric in R Z < (and so (cid:0) R Z < , d ψ (cid:1) forms apseudo-metric space). R Z < Now that we have the notion of distance in R Z < , we can define what it meansto be an open set in R Z < by first defining what a ball is in R Z < . Definition 4.3.
A ball of radius y around the point x ∈ R Z < is B x ( y ) = { z ∈ R Z < | d ( x , z ) < y } , where d ( x , y ) is either d ( x , y ) or d ψ ( x , y ) . We require this additional definition in order to set forth our explanation aboutballs properly:
Definition 4.4.
The sets ∆ m and ∆ m ↓ are defined as follows: It is not without reason that we introduced the concept of the pseudo-metric here. Thiskind of metric will make sense when we are in R . For example, the distance between 0 and (cid:15) is 0 as our lens is not strong enough to distinguish those two numbers in R . R and R Z < Balls in R and R Z < The Set The Metric The Form of The Balls R ρ ( x, y ) = | x − y | B x ( r ) = ( x − r, x + r ) R Z < d ( x , y ) = | x − y | B x ( r ) = ( x − r , x + r ) R Z < d ( x , y ) = | x − y | B x ( / n ) = ( x − / n , x + / n ) R Z < d ψ ( x , y ) = St ( | x − y | ) B x ( / n ) = B St ( x ) ( / n ) = { y | St ( y ) ∈ ( St ( x ) − / n , St ( x ) + / n ) } ∆ m = { x : x = a m (cid:15) m } and ∆ m ↓ = (cid:83) n ≥ m ∆ n ,where m ∈ N ∪ { } , a m ∈ R and a m (cid:54) = 0 whenever m ≥ . We have to be careful here as unlike in classical topology, there are differentnotions of balls that can be described as follows. The first possible notion ofballs is when we use d as our metric and having y > R Z < , the ball around a point x with y radius is an interval ( x − y , x + y ).Note that by using y as the radius, beside having the usual balls with “real”radius ( St-balls ) (that is when y = r ∈ R ), we also have some infinitesimallysmall balls ( e-balls ) when y = e ∈ ∆ m ↓ for any given m . The second possiblenotion is while we use the same metric d , we have / n for some n ∈ N as itsradius. This produces balls ( rat-balls ) in the form of ( x − / n , x + / n ). The third possibility is by using d ψ as our metric. In this case, interestingly, theballs around a point x with / n radius will be in the form of the following set: { y | St ( y ) ∈ ( St ( x ) − / n , St ( x ) + / n ) } .We call this kind of balls as psi-balls . See Table 1 for the summary of thesepossibilities of balls in our sets. Remark 4.5. In R , the e -ball does not exist, whereas in R Z < there are infinitelymany e -balls around every point there. Finally, we defined what it means to be an open set in R Z < . Notice thatbecause we had two notions of ball in our set, i.e. St -balls and e -ball, it led usto two different notions of openness as follows. Definition 4.6 ( St -open) . A subset O ⊆ R Z < is St-open iff ∀ x ∈ O ∃ n ∈ N s.t. B x (cid:0) n (cid:1) ⊆ O . Definition 4.7 ( e -open) . A subset O ⊆ R Z < is e-open iff ∀ x ∈ O ∃ e ∈ ∆ m ↓ s.t. B x ( e ) ⊆ O . m ↓ is defined in Definition 4.4. Example 4.8.
The interval ( , ) in R Z < is St -open and also e -open. Example 4.9.
The interval ( , e ) in R Z < is e -open, but not St -open. Example 4.9 gives us the theorem below:
Theorem 4.10.
For any set U ⊆ R Z < , R Z < (cid:54)| = If U is St -open, then U is e -open . Using the two definition of openness given in Definitions 4.6 and 4.7, wedefined what it means by two points are topologically distinguishable. There arealso two different notions of distinguishable points as can be seen in Definitions4.11 and 4.12.
Definition 4.11 ( St -distinguishable) . Any two points in R Z < are St-distinguishableif and only if there is a St -open set containing precisely one of the two points. Definition 4.12 ( e -distinguishable) . Any two points in R Z < are e-distinguishableif and only if there is an e -open set containing precisely one of the two points. Definition 4.13.
Let X be a non-empty set and τ a collection of subsets of X such that:1. X ∈ τ ,2. ∅ ∈ τ ,3. If O , O , . . . , O n ∈ τ , then (cid:84) nk =1 O k ∈ τ ,4. If O α ∈ τ for all α ∈ A , then (cid:83) α ∈ A O α ∈ τ .The pair of objects ( X, τ ) is called a topological space where X is called theunderlying set, the collection τ is called the topology in X , and the members of τ are called open sets. Note that if τ is the collection of open sets of a metric space ( X , ρ ), then( X , τ ) is a topological metric space , i.e. a topological space associated with themetric space ( X, ρ ). There are at least three interesting topologies in R Z < ascan be seen in Definition 4.14 below. Definition 4.14.
The standard topology τ St on the set R Z < is the topology gen-erated by all unions of St -balls. The e -topology in R Z < , τ e , is the topology gen-erated by all unions of e -balls and the third topology in R Z < is pseudo-topology, τ ψ , when it is induced by d ψ . Axiom 4.15. ( R Z < , τ n ) , ( R Z < , τ e ) , and ( R Z < , τ ψ ) form topological metric spacewith d as their metrics (for the first two) and d ψ for the third one. heorem 4.16. ( R Z < , τ n ) is not a Hausdorff space but it is a preregular space. Proof. R Z < does not form a Hausdorff space because under the topology τ St ,there are two distinct points, (cid:15) = (cid:104) (cid:98) , (cid:105) and (cid:15) + = (cid:104) (cid:98) , (cid:105) for example, which arenot neighbourhood-separable. It is impossible to separate those two points with St -ballss as / n > e for every n ∈ N and e ∈ ∆ m ↓ . However, it is a preregularspace as every pair of two St -distinguishable points in R Z < can be separated bytwo disjoint neighbourhoods. This follows directly from Definition 4.11. Theorem 4.17. ( R Z < , τ e ) is a non-connected space and it forms a Hausdorffspace.Proof. We observe that for all x ∈ R Z < and e ∈ ∆ m ↓ , the balls B e ( x ) are e -open and so is the whole space. To show that R Z < is not connected, let S = { x ∈ R Z < | ( x ≤ ) or ( x > and x ∈ ∆ m ↓ ) } and S = { x ∈ R Z < | ( x > ) and x / ∈ ∆ m ↓ } . The sets S and S are e -open, disjoint and moreover, we have that R Z < = S ∪ S (and so R Z < is not connected). For any x , y ∈ R Z < , B x ( d ( x,y ) / ) and B y ( d ( x,y ) / )are open and disjoint. Thus, R Z < forms a Hausdorff space.We will now state the usual definition of the basis of a topology τ . Definition 4.18.
Let ( X, τ ) be a topological space. A basis for the topology τ is a collection B of subsets from τ such that every U ∈ τ is the union of somecollections of sets in B , i.e. ∀ U ∈ τ , ∃B ∗ ⊆ B s.t. U = (cid:83) B ∈B ∗ B Example 4.19. On R with its usual topology, the set B = { ( a, b ) : a < b } is atopological basis. Definition 4.20.
Let ( X, τ ) be a topological space and let x ∈ X . A local basisof x is a collection of open neighbourhoods of x , B x , such that for all U ∈ τ with x ∈ U , ∃ B ∈ B x such that x ∈ B ⊂ U . Definition 4.21.
Let ( X, τ ) be a topological space. Then ( X, τ ) is first-countableif every point x ∈ X has a countable local basis. Definition 4.22.
Let ( X, τ ) be a topological space. Then ( X, τ ) is second-countable if there exists a basis B of τ that is countable. Theorem 4.23. ( R Z < , τ e ) is first countable but not second-countable. Proof.
From Axiom 4.15 and because every metric space is first-countable, itfollows that ( R Z < , τ e ) is first-countable. However, there cannot be any countablebases in τ e as the uncountably many open sets O x = ( x − e , x + e ) are disjoint. Hausdorff space and preregular space are defined as usual. Note that the space ( R Z < , τ n ) is still second-countable. Calculus on R Z < It has been proved previously that the set R Z < forms a field. Remember thatfor any x ∈ R Z < , x = (cid:104) x − n , x − ( n − , . . . , x − , x − , (cid:98) x, x , x , x , . . . (cid:105) where St ( x ) = (cid:98) x , Nst (cid:15) ( x ) = { x , x , x , . . . } , and Nst ω ( x ) = { x − n , x − ( n − , . . . x − } .In other words, for every x ∈ R Z < , x = Nst ω ( x ) + St ( x ) + Nst (cid:15) ( x ).Note that we can think of St (), Nst (cid:15) (), and
Nst ω () as linear functions – thatis for any x , y ∈ R Z < and a constant c ∈ R , St ( x + y ) = St ( x )+ St ( y ), St ( c x ) = c St ( x ), Nst (cid:15) ( x + y ) = Nst (cid:15) ( x )+ Nst (cid:15) ( y ), Nst (cid:15) ( c x ) = c NSt (cid:15) ( x ), Nst ω ( x + y ) = Nst ω ( x )+ Nst ω ( y ), and Nst ω ( c x ) = c Nst ω ( x ). Definition 5.1.
Suppose that ni (cid:15) ( x ) denotes the non-infinitesimal part of x ∈ R Z < , i.e. ni (cid:15) ( x ) = Nst ω ( x ) + St ( x ) and function in R Z < be defined in the usualway. Then a function f in R Z < is microstable if and only if ni (cid:15) ( f ( x + (cid:15) )) = ni (cid:15) ( f ( x )) , Example 5.2.
Suppose that a function f in R Z < is defined as follows: f ( x ) = (cid:40) , if St ( x ) > , else.Then f ( x ) is a microstable function. Theorem 5.3.
Microstability is closed under addition, multiplication, and com-position. Now for every function f defined in R Z < , we are going to have the operator Der f which takes a 2-tuple in (cid:0) R × R Z < (cid:1) as its input and returns a member of R Z < as the output, i.e.: Der f : R × R Z < → R Z < . The proof of this theorem can be seen in [32, p. 73]. derivative of f .Using Newton’s original definition (and a slight change of notation), if afunction f ( x ) is differentiable, then its derivative is given by: Der f ( x , (cid:15) ) = f ( x + (cid:15) ) − f ( x ) (cid:15) . (3)Now suppose that we want to find a derivative of f where f is a function de-fined in R Z < . We can certainly use Equation (3) to calculate it as that equationholds for any function f . But how is this calculation related to the calculus prac-tised in classical mathematics? Note that using Newton’s definition to calculatethe derivative will necessarily involve an inconsistent step. This inconsistency islocated in the treatment given to the infinitesimal number. Thus it makes sensethat in order to explore the problem posed above, we will use a paraconsistentreasoning strategy which is called Chunk and Permeate. R Z < Details on the Chunk & Permeate reasoning strategy can be seen in [9]. Beforeapplying this strategy for the derivative in R Z < , define a set E which consistsof any algebraic terms such that they satisfy: St ( Der f ( x , (cid:15) )) = f (cid:48) ( x ),where f (cid:48) ( x ) denotes the usual derivative of f in 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 clearthat the identity function id ( x ) = x is in E because for all (cid:15) : St ( Der x ( x , (cid:15) )) = St (cid:32) x + (cid:15) − x (cid:15) (cid:33) = St (cid:32) (cid:15)(cid:15) (cid:33) = St ( ) = 1 = f (cid:48) ( x ) . Theorem 5.5. If f and g are microstable functions in E and c is any realconstant, then1. f ± g are in E ,2. cf is in E ,3. f g is in E , . fg is in E , and5. f ◦ 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 (cid:98) R ,which is formally written as (cid:98) R = (cid:104){ Σ S , Σ T } , ρ, T (cid:105) where the source chunk Σ S is the language of R Z < , the target chunk Σ T is the language of R , and ρ is thepermeability relation between S and T . The source chunk Σ S As stated before, this chunk is actually the languageof the set R Z < and therefore, it consists of all six of its axioms. The sourcechunk requires one additional axiom to define what it means by derivative.This additional axiom can be stated as:S1: Df = Der f ( x , (cid:15) )where Der f ( x , (cid:15) ) is defined in Equation (3). The target chunk Σ T Again, the target chunk contains the usual axiom forthe set or real numbers, R . There is only one additional axiom needed for thischunk: T1: ∀ x x = St ( x ).Note that the axiom T1 above is actually equivalent to saying that ∀ x Nst ( x ) =0. The permeability relation
The permeability relation ρ ( S, T ) is the set ofequations of the form Df = g where f ∈ E . The function g which is permeated by this permeability relationwill be the first derivative of f in R . This permeability relation shows that thederivative notion is permeable to the set R . Example 5.6.
Suppose that f ( x ) = x for all x . First, working within Σ S , theoperator D is applied to f such that: Df = Der f ( x , (cid:15) )= ( x + (cid:15) ) − x (cid:15) = (cid:15)(cid:15) = . ote that St ( Der f ( x , (cid:15) )) = St ( ) = 3 = f (cid:48) ( x ) , and so f ( x ) ∈ E . Permeatingthe last equation of Df above to Σ T gives us: Df = 3 and so the derivative of f ( x ) = 3 x is . Example 5.7.
Suppose that f ( x ) = x + x + for all x . First, working within Σ S , the operator D is applied to f such that: Df = Der f ( x , (cid:15) )= ( x + (cid:15) ) + ( x + (cid:15) ) + − x − x − (cid:15) = x (cid:15) + (cid:15) + (cid:15)(cid:15) = x + (cid:15) + . Note that the standard part of x + (cid:15) + will depend on the domain of x . Thatis: St ( x + (cid:15) + ) = (cid:40) x + 2 , if x ∈ R , else.In other words, if (and only if ) Nst ( x ) = 0 , i.e. x ∈ R , Df can be permeatedinto Σ T . Thus, if x is a real number, then we have the derivative of f ( x ) = x + 2 x + 3 = 2 x + 2 . Example 5.8.
Suppose that f ( x ) = sign( x ) is defined as: sign( x ) = , if St ( x ) > , if St ( x ) = 0 − , if St ( x ) < First, working within Σ S , the operator D is applied to f so that: Df = Der f ( x , (cid:15) )= sign( x + (cid:15) ) − sign( x ) (cid:15) = 0 (because ∀ x St ( x ) = St ( x + (cid:15) ) )Note that St ( Der f ( x , (cid:15) )) = St ( ) = 0 = f (cid:48) ( x ) , and so f ( x ) ∈ E . Permeatingthe last equation of Df above to Σ T gives us: Df = 0 and so the derivative of f ( x ) = sign( x ) is for all x . Notice that this is notthe case in R , where the derivative of the sign function at x = 0 is not definedbecause of its discontinuity. However, this is not really a bizarre behaviour ecause if we look very closely at the infinitesimal neighbourhood of x when St ( x ) = 0 , the function sign( x ) will look like a straight horizontal line and so itmakes a perfect sense to have as the slope of the tangent line there. Moreover,this phenomenon also happens in distribution theory where sign function has itsderivative everywhere. R Z < As we know, there are some special functions defined in real numbers and two ofthem are the trigonometric and the exponential functions. How then are thesefunctions defined in R Z < ? Here we propose to define them using power series.The first two trigonometric functions that we are going to discuss are thesin and cos functions. Using the MacLaurin power series, these two functionsare defined as follows: sin( x ) = (cid:88) n =0 ( − n (2 n + 1)! x n +1 (4)and cos( x ) = (cid:88) n =0 ( − n (2 n )! x n . (5)The exponential function is defined as:exp( x ) = (cid:88) n =0 n ! x n . (6)Note that the MacLaurin polynomial is just a special case of Taylor polynomialwith regards to how the function is approximated at x = . Example 5.9.
Suppose that we have x = x + a(cid:15) = (cid:104) (cid:98) x, a, , , . . . (cid:105) where x, a ∈ R .We want to know what sin( x ) is. Based on Equation (4) , sin( x ) = sin( x + (cid:15) ) = ( x + (cid:15) ) − ! ( x + (cid:15) ) + ! ( x + (cid:15) ) − ! ( x + (cid:15) ) + . . . Our task now is to find all the members of
Nst (cid:15) (sin( x )) and also St (sin( x )) .These are shown in Table 2. Note that from the way the sin function is defined, x i = 0 ∀ x i ∈ Nst ω (sin( x )) . Thus from Table 2, we get: sin( x ) = sin( x + a(cid:15) )= (cid:104) (cid:92) sin( x ) , a cos( x ) , − a sin( x ) , − a ! cos( x ) , a sin( x ) , a ! cos( x ) , . . . (cid:105) , and we also get sin( (cid:15) ) = (cid:104) (cid:98) , , , − ! , , ! , . . . (cid:105) = (cid:15) − ! (cid:15) + ! (cid:15) − . . . for an infinitesimal angle (cid:15) . St (sin( x )) and the first four members of Nst (cid:15) (sin( x ))Expanded Form Simplified Formreal-part = x − x + ! x − ! x + . . . = (cid:80) n =0 − n (2 n +1)! x n +1 = sin( x ) (cid:15) -part = a(cid:15) − x a(cid:15) + a(cid:15) x − a(cid:15) x + . . . = (cid:15) ( a − a x + a x − a x + . . . )= (cid:15) (cid:80) n =0 − n (2 n )! a x n = (cid:15) ( a cos( x )) (cid:15) -part = − a (cid:15) x + ! a (cid:15) x − ! a (cid:15) x + . . . = − a (cid:15) x + a (cid:15) x − a (cid:15) x + . . . = (cid:15) ( − x + a x − a x + . . . )= (cid:15) (cid:80) n =0 − n +1 ( n +1)(2 n +2)! a x n +1 = (cid:15) ( − a sin( x )) (cid:15) -part = − a (cid:15) + ! a (cid:15) x − ! a (cid:15) x + ! a (cid:15) x − . . . = (cid:15) ( − a x + ! a x − ! a x + ! a x − . . . )= (cid:15) (cid:80) n =0 − n +1 n )! a x n = (cid:15) ( − a cos( x )) (cid:15) -part = ! a (cid:15) x − ! a (cid:15) x + ! a (cid:15) x − a (cid:15) x + . . . = (cid:15) ( x − a x + a x − ! a x + . . . )= (cid:15) (cid:80) n =0 − n n +1)! a x n +1 = (cid:15) ( a sin( x ))Table 3: St (cos( x )) and the first two members of Nst (cid:15) (cos( x ))real-part = 1 − x + ! x − ! x + . . . = (cid:80) n =0 − n (2 n )! x n = cos( x ) (cid:15) -part = − a(cid:15) x + ! a(cid:15) x − ! a(cid:15) x + . . . = (cid:15) ( − a x + a x − a x + . . . )= (cid:15) (cid:80) n =0 − n +1 (2 n +1)! a x n +1 = (cid:15) ( − a sin( x )) (cid:15) -part = − a (cid:15) + ! a (cid:15) x − ! a (cid:15) x + a (cid:15) x − . . . = (cid:15) (cid:80) n =0 − n +1 ( n +1)(2 n +1)(2 n +2)! a x n = (cid:15) (cid:80) n =0 12 − n +1 (2 n )! x n = (cid:15) ( − a cos( x ))24 xample 5.10. Suppose that we have x = x + a(cid:15) where x, a ∈ R . Here we tryto find what cos 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.Thus from Table 3, we get: cos( x ) = cos( x + a(cid:15) ) = (cid:104) (cid:92) cos( x ) , − a sin( x ) , − a cos( x ) , a sin( x ) , a cos( x ) , . . . (cid:105) , and we also get cos( (cid:15) ) = (cid:104) (cid:98) , , − , , . . . (cid:105) = 1 − ! (cid:15) + . . . for an infinitesimal angle (cid:15) . Example 5.11.
With the same x as in Examples 5.9 and 5.10, we try to knowwhat exp( x ) is. Based on Equation (6) , exp( x ) = exp( x + a(cid:15) ) = + ( x + a(cid:15) ) − ! ( x + a(cid:15) ) + ! ( x + a(cid:15) ) + . . . Our task now is to find all the members of
Nst (cid:15) (exp( x )) and also St (exp( x )) ,which are shown in Table 4. Note that from the way we define the function exp , ∀ x i ∈ Nst ω (exp( x )) x i = 0 . Thus from Table 4, we get: exp( x ) = exp( x + a(cid:15) ) = (cid:104) (cid:92) exp( x ) , a exp( x ) , a exp( x ) , a exp( x ) , a exp( x ) , . . . (cid:105) , and we also get exp( (cid:15) ) = (cid:104) (cid:98) , , , , , . . . (cid:105) = 1 + (cid:15) + ! (cid:15) + ! (cid:15) − . . . for an infinitesimal angle (cid:15) . From the preceding discussion, we have the following proposition.
Proposition 5.12.
For the sin , cos , and exp functions:1. Der sin x ( x , (cid:15) ) = sin( x + (cid:15) ) − sin( x ) (cid:15) = (cid:104) (cid:92) cos( x ) , − sin( x ) , − cos( x ) , . . . (cid:105) , andso we have: St ( Der sin( x ) ( x , (cid:15) )) = cos( x ) Der cos x ( x , (cid:15) ) = cos( x + (cid:15) ) − cos( x ) (cid:15) = (cid:104) (cid:92) − sin( x ) , − cos( x ) , sin( x ) , . . . (cid:105) , andso St ( Der cos( x ) ( x , (cid:15) )) = − sin( x ) Der exp x ( x , (cid:15) ) = exp( x + (cid:15) ) − exp( x ) (cid:15) = (cid:104) (cid:92) exp( x ) , exp( x ) , exp( x ) , . . . (cid:105) , andso St ( Der exp( x ) ( x , (cid:15) )) = exp( x )25able 4: St (exp( x )) and The First Three Members of Nst (cid:15) (exp( x ))Expanded Form Simplified Formreal-part = 1 + x + x + x + . . . = (cid:80) n =0 1 n ! x n = exp( x ) (cid:15) -part = a(cid:15) + a(cid:15) x + a(cid:15) x + ! a(cid:15) x (cid:15) + . . . = (cid:15) ( a + a x + a x + a x − ! a x + . . . )= (cid:15) (cid:80) n =0 1 n ! a x n = (cid:15) ( a exp( x )) (cid:15) -part = a (cid:15) x + a (cid:15) x + ! a (cid:15) x + a (cid:15) x + . . . = (cid:15) ( a + a x + ! a x + a x + . . . )= (cid:15) (cid:80) n =0 ( n +1)( n +2)2( n +2)! a x n = (cid:15) (cid:80) n =0 12( n !) a x n = (cid:15) ( a exp( x )) (cid:15) -part = a (cid:15) + ! a (cid:15) x + a (cid:15) x + a (cid:15) x + a (cid:15) x + . . . = (cid:15) ( a + ! a x − a x + a x + a x + . . . )= (cid:15) (cid:80) n =0 ( n +1)( n +2)( n +3)6( n +3)! a x n = (cid:15) (cid:80) n =0 16( n !) a x n = (cid:15) ( a exp( x )) In this subsection, we try to pinpoint what the good definition for continuousfunctions is. We also decide whether we can permeate it between R and 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 (ED
CLASS ) . A function f : R → R is continuous at a point a ∈ R if, given n ∈ N , there exists a m ∈ N such that | f ( x ) − f ( a ) | < n whenever | x − a | < m .The function f is called continuous on an interval I iff f is continuous at everypoint in I . Definition 5.14 (ED) . A function f is continuous at a point c ∈ R Z < if, given e ∈ R Z < > , there exists a e ∈ R Z < > such that | f ( x ) − f ( c ) | < e whenever | x − c | < e .That function f is called continuous function over an interval I iff f is contin-uous at every point in I . Proposition 5.15.
There exists a function f in R Z < which is continuous underDefinition 5.14, but discontinuous under Definition 5.13, i.e. R Z < | = ∃ f s.t. (ED CLASS ( f ) ∧ ¬ ED( f )) .Proof. Suppose that ∆ = { x | ∀ n ∈ N , | x | < n } – in other words, ∆ is a set ofall infinitesimals – and consider the indicator function around ∆, that is ∆ ( x ) = (cid:40) , x ∈ ∆0 , otherwise.26hen ED CLASS ( ∆ ) but ¬ ED( ∆ ). Remark 5.16.
Note that:1. The set ∆ in R only has 0 as its member. That is R | = ∆ = { } .2. In R , both Definitions 5.13 and 5.14 are equivalent, that is for any function f , R | = ED CLASS ( f ) ↔ ED( f ) . Property 5.17 (EVP) . If I is an interval and f : I → J , we say that f hasthe extreme value property iff f has its maximum value on I . That is, ∀ a ≤ b ∈ I , ∃ x ∈ [ a, b ] s.t. ∀ y ∈ [ a, b ]( f ( y ) ≤ f ( x )) . Property 5.18 (IVP) . If I is an interval, and f : I = [ a, b ] → J , we say that f has the intermediate value property iff ∀ c (cid:48) ∈ ( f ( a ) , f ( b )) , ∃ c ∈ ( a, b ) s.t. f ( c ) = c (cid:48) . Theorem 5.19. R | = ED → EVP
Proof.
The proof of this theorem can be found in any standard book for Analysiscourse (in [3] for example).
Theorem 5.20.
There is a function f such that R Z < | = ED( f ) ∧ ¬ EVP( f ) .Proof. Take the function f on [1 ,
2] as defined below: f ( x ) = (cid:40) n x ∼ mn (reduced fraction)0 otherwise . Remark 5.21.
This research now reaches an especially engrossing object. Thefunction 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 ofself-similarity. Figure 1 shows what the function f ( x ) in Theorem 5.20 lookslike, 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 : R Z < → R and let us define another function F : R Z < → R Z < by F ( x ) = f ( x ) + (cid:15)f ( x ) + (cid:15) f ( x ) + . . . = (cid:104) (cid:100) f ( x ) , f ( x ) , f ( x ) , . . . (cid:105) . Then, that function F ( x ) will define an infinite fractal (if ni ω ( (cid:15) (cid:98) × F ( x )) = F ( x ) )or infinitesimal fractals (if ni ω ( (cid:15) (cid:98) × F ( x )) = F ( x ) ), where ni ω ( x ) denotes thenon-infinity part of x . R and R Z < . The proof of each of the relations there can be seen in [32,p. 94–98]. EVP IVPED (a) In R EVP IVPED (b) In R Z < Figure 2: Relationship among the three definitions of continuityThe obvious question worth asking is how do we define continuity in our set R Z < . As seen before, there are three possible ways to define it, namely: withthe (cid:15) - δ definition (ED), with extreme value property (EVP), or with the inter-mediate 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 ofthe given function, we can find a point in the domain corresponding to thatvalue. Will this work in our set R Z < ? Let us consider the R Z < -valued function28 ( x ) = x on [ a, b ] for any a, b ∈ R Z < and let us assume that IVP holds. Itfollows that for every c (cid:48) between f ( a ) = a and f ( b ) = b , ∃ c ∈ ( a, b ) suchthat f ( c ) = c = c (cid:48) . The only c which satisfies that last equation is c = √ c (cid:48) ,which cannot be defined in our set R Z < . Thus, IVP, even though it is somehowintuitively “obvious”, it does not work in R Z < . This phenomenon is actuallynot uncommon if we want to have a world with infinitesimals (or infinities) init. See [5, p. 107] for example.However, note that in R , the function x still satisfies IVP. Now, is therea function in 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 followingtheorem. Theorem 5.22.
There exists a function f such that ( R | = IVP( f ) ∧ R Z < | = ¬ IVP( f )) , and there exists a function g s.t. R , R Z < | = IVP( g ) . Hence, from the argument above we also argued that defining continuity in ourset with IVP is not really useful.Secondly, in regards to EVP. This is clearly not a good way to define continu-ity in our set because even in the set of real numbers, there are some continuousfunctions which do not satisfy EVP themselves. So the last available option nowis the third one, which is the (cid:15) - δ (ED) definition. We argued that this definitionis the best way to define continuity in R Z < . Moreover, in this way, it preservesmuch of the spirit of classical analysis on R while retaining the intuition ofinfinitesimals.It is important to note that in the ED definition of continuity (Definition5.14), there are two variables which are in play, i.e. e and e . When we appliedthis definition on our set, these two variables hold important (or rather, veryinteresting) roles where we will have different levels of continuity from the samefunction. What we mean is that these two variables can greatly vary dependingon how far (‘deep’) we want to push (observe) them, e.g. e can be a real number( e ∈ ∆ ), or it can be in ∆ , ∆ and so on. Remember that these two numbers, e and e , will determine how subtle we want our intervals to be (see Figure 3for illustration).Thus this definition of continuity works as follows. Suppose that we have afunction f and we want to decide whether it is continuous or not. With thisconcept of two variables, we will have what we call as ( k, n )-continuity where k, n ∈ N ∪ { } . Definition 5.23 (( k, n )-Continuity) . A function f is ( k, n ) -continuous at apoint c iff ∀ e k > , ∃ e n > such thatif | x (cid:98) − c | < e n , then | f ( x ) (cid:98) − f ( c ) | < e k where e p , e p ∈ ∆ 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. a) An e bound & its e neighbourhood fulfilling Definition 5.14(b) A smaller bound & its neighbourhood Figure 3: Illustration of e and e intervals30igure 4: Illustration of Function f ( x ) in Example 5.26 Remark 5.25.
From the definition of the set ∆ m , note that for any r ∈ R Z < ,d ∈ ∆ p , and e ∈ ∆ p +1 , ( r − e, r + e ) ⊆ ( r − d, r + d ) . To be able to grasp a better understanding of Definition 5.23, see the exam-ples below.
Example 5.26.
Consider the R Z < -valued function f ( x ) defined as follows: f ( x ) = (cid:40) x St ( x ) ≤ x (cid:98) + otherwise. First we need to understand clearly how this function actually works. Figure4, where i denotes an arbitrary infinitesimal number, illustrates to us what thefunction f ( x ) looks like. Notice that at x = , what looks like a point in realnumbers is actually a (constant) line when we zoom in deep enough into R Z < .So how about the continuity of this function? It is obvious that f ( x ) is not , -continuous (by taking, for example, e = and x = . ). However, inter-estingly enough, it is ( , ) -continuous by taking e ∈ ∆ . Why was that? Thefact that e ∈ ∆ and that it has to depend on e means that e has to be in ∆ as well. Now, assigning e = e is sufficient to prove its 0 , Example 5.27.
The identity function f ( x ) = x for all x ∈ R Z < is ( , ) -continuous, just like in reals. However, it is not ( , ) -continuous because for We have to be really careful here because if the first condition there was x ≤ St ( x ) ≤ exactly one point. ny point c , there is an e = (cid:15) such that for every e = r where r ∈ R , | x (cid:98) − c | < r but f ( x ) − f ( c ) ≥ (cid:15) . In fact, identity function is ( k, n ) -continuousonly when k ≤ n , but not otherwise. The next theorem below is very interesting in as much as it enables usto 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 ∈ N ∪ { } , then f is a constant function.Proof. Here we want to prove its contrapositive, in other words, if f is notconstant, 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 ) (cid:54) = f ( b ) andsuppose that | f ( a ) − f ( b ) | ∈ ∆ m such that | a − b | ∈ ∆ l . By this construction, f will not be ( m, l − k = m and n = l − 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 ∆ m -neighbourhood. The next interesting question is: what is the relation between, for example,( , )-continuity and ( , )-continuity? In general, what is the relation between( k, n )-continuity and ( k, ( n + 1))-continuity? And also between ( k, n )-continuityand (( 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 c . This wouldmean that ∀ e k , ∃ e n such that if | x (cid:98) − c | < e n , then | f ( x ) (cid:98) − f ( c ) | < e k . Byusing the same e k and from Remark 5.25, we can surely find e ( n +1) = e n (cid:98) × (cid:15) such that for all x ∈ ( c (cid:98) − e ( n +1) , c (cid:98) + e ( n +1) ), f ( x ) ∈ ( f ( c ) (cid:98) − e k , f ( c ) (cid:98) + e k ). Example 5.31.
By Theorem 5.30, the function f ( x ) in Example 5.26 is also ( , ) -continuous, and also ( , ) -continuous, and so on. Theorem 5.32.
For any function f , if f is (( k + 1) , n ) -continuous, then f isalso ( k, n ) -continuous.Proof. Suppose that a function f is (( k + 1) , n )-continuous at point c and theset ∆ m defined as in Theorem 5.30. The fact that f is (( k + 1) , n )-continuousmeans that ∀ e ( k +1) , ∃ e n such that if | x (cid:98) − c | < e n , then | f ( x ) (cid:98) − f ( c ) | < e ( k +1) is hold. Here we want to prove that ∀ e k , ∃ e n such that if | x (cid:98) − c | < e n ,then | f ( x ) (cid:98) − f ( c ) | < e k . This actually follows directly from Remark 5.25 as( f ( c ) (cid:98) − e ( k +1) , f ( c ) (cid:98) + e ( k +1) ) ⊆ ( f ( c ) (cid:98) − e k , f ( c ) (cid:98) + e k ).32ow suppose that f and g are two ( k, n )-continuous functions in R Z < . Wewill examine how the arithmetic of those two continuous functions works. It isclear 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 twocontinuous functions are particularly interesting as can be seen in Theorem 5.33and Theorem 5.34, respectively. Theorem 5.33. If f is a ( k, n ) -continuous function and g is an ( n, q ) -continuousfunction, then f ◦ g will be ( k, q ) -continuous.Proof. Since f is ( k, n )-continuous at g ( c ), our definition of continuity tells usthat for all e k >
0, there exists e n such thatif | g ( x ) − g ( a ) | < e n , then | f ( g ( x )) − f ( g ( a )) | < e k .Also since g is ( n, q )-continuous at c , there exists e q such thatif | x − a | < e q , then | g ( x ) − g ( a ) | < e n .I have taken e n = e n here. Now this tells us that for all e k >
0, there exists e q (and an e n ) such thatif | x − a | < e q , then | g ( x ) − g ( a ) | < e n which implies that | f ( g ( x )) − f ( g ( c )) | < e 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 · 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 ∈ R Z < , for every e max { k,l } >
0, there exists e min { n,o } > x ∈ R Z < with | x − c | < e min { n,o } , | H ( x ) − H ( c ) | < e max { k,l } holds.Now let c and e max { k,l } be given and we choose e such that e ∈ ∆ min { n,o } , i.e. e = e min { n,o } . Then for all x ∈ R Z < with | x − c | < e min { n,o } , | H ( x ) − H ( c ) | = | f ( x ) g ( x ) − f ( c ) g ( c ) | = | f ( x ) g ( x ) − f ( x ) g ( a ) + f ( x ) g ( a ) − f ( c ) g ( c ) |≤ | f ( x ) g ( x ) − f ( c ) g ( c ) | + | f ( x ) g ( a ) − f ( c ) g ( c ) | = | f ( x )( g ( x ) − g ( a )) | + | g ( a )( f ( x ) f ( c )) | < | f ( x ) | e l + | g ( a ) | e k (7)Note that because f and g are limited-valued function, then | f ( x ) | e l and | g ( a ) | e k are still in ∆ l and ∆ k , respectively. This means that the right sideof Inequality 7 will be in ∆ max { k,l } and so H ( x ) − H ( c ) < e max { k,l } is hold.33t is worth pointing out here that the definition of ( k, n )-continuity is a muchmore fine-grained notion than the classical continuity. This is self-explanatoryby the use of those two variables k and n which makes us able to take muchmore infinitesimals — in other words, we will be able to examine a far greaterdepth — than in the classical definition. Furthermore, there might be somepossible connections to one of the quantum phenomenons in physics: ‘actionat a distance’. This concept is typically characterized in terms of some causeproducing a spatially separated effect in the absence of any medium by whichthe causal interaction is transmitted [16] and closely connected to the questionof what the deepest level of physical reality is [30, pg. 168]. Note that researchon this phenomenon is still being conducted up until now, as can be seen forexample [33], [46] and [53]. Definition 5.35.
A function f from a topological space ( X, τ ) to a topologicalspace ( Y, τ ) is a function f : X → Y . From now on, we will abbreviate this function notation by f : X → Y or simply f every time the topologies in X and Y need not be explicitly mentioned. Also, f − denotes the inverse image of f as usual. Definition 5.36 (St-continuous) . A function f : X → Y between topologicalspaces is standard topologically continuous, denoted by St-continuous, if f − ( U ) ⊆ X is St-open whenever U ⊆ Y is St-open. Definition 5.37 ( e -continuous) . A function f : X → Y between topologicalspaces is infinitesimally topologically continuous, denoted by e -continuous, if f − ( U ) ⊆ X is e -open whenever U ⊆ Y is e -open. Theorem 5.38.
Suppose that
X, Y ⊆ R Z < . Under the metric d , a function f : X → Y is St-continuous if and only if f satisfies ED CLASS definition (Definition5.13).Proof.
We need to prove the implication both ways.1. We want to prove that if f is St-continuous, then f satisfies ED CLASS .Suppose that f is St-continuous and let x o ∈ X and n ∈ N >
0. Then,the ball B f ( x ) ( / n ) = { y ∈ Y | d ( y , f ( x )) < / n } is open in Y , and hence f − ( B f ( x ) ) is open in X . Since x ∈ f − ( B f ( x ) ),there exists some balls of radius / m for some m ∈ N such that B x ( / m ) ⊆ f − ( B f ( x ) ) . This is exactly what the ED
CLASS says.34. We want to prove that if f satisfies ED CLASS , then f is St-continuous.Suppose that f satisfies ED CLASS and let U ⊆ Y is open. By Definition4.6, for all y ∈ U there exists some d y = / n y where n y ∈ N such that B y ( d y ) ⊆ U and in fact, U = (cid:91) y ∈ U B d y ( y ) . (8)Now we claim that f − ( U ) is open in X and suppose that x ∈ f − ( U ).Then f ( x ) ∈ U and so from Equation 8, f ( x ) ∈ B d y ( y ) for some y ∈ U and d y = / n y for some n y ∈ N , i.e. d ( f ( x ) , y ) < d y . Nowdefine e = d y − d ( f ( x ) , y ) > . (9)By Definition 5.13, there exists some m ∈ N such thatif x ∈ X and d ( x , x ) < / m , then d ( f ( x ) , f ( x )) < e. (10)Now we claim that B x ( / m ) ⊆ f − ( U ) , (11)which will actually show that f − ( U ) is indeed open. To this end, let x ∈ B x ( / m ), i.e. d ( x , x ) < / m . Then from (10), we have d ( f ( x ) , f ( x )) < e .Then, the triangle inequality and (9) imply that d ( f ( x ) , y ) ≤ d ( f ( x ) , f ( x )) + d ( f ( x ) , y ) < e + d ψ ( f ( x ) , y ) = d y . This means that f ( x ) ∈ B y ( d y ) ⊆ U , so that x ∈ f − ( 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 ⊆ R Z < . Under the metric d , a function f : X → Y is 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 withsome slight modifications in the distances (from / n for some n ∈ N into e ∈ R Z < ). When we are talking about sequences, it is necessary to talk also about what itmeans when we say that a sequence is convergent to a particular number. Thissubsection presents not only some possible definitions that can be used to defineconvergence in R Z < , but also the problems which occur when we apply them in R Z < . 35 efinition 5.40 (Classical Convergence) . A sequence s n converges to s iff, ∀ m ∈ N , ∃ N such that ∀ n > N , | s n − s | < m .We write 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 ofconvergent classically.
Definition 5.41 (Hyperconvergence) . A sequence s n converges to s iff, ∀ r > , ∃ N such that ∀ n > N , | s n − s | < r .We write HC( s n , s ) to denote that a sequence s n is hyperconvergent to s . Theinterpretation of r can be either in R or in R Z < . Example 5.42.
Suppose that we have a sequence s n = (cid:15) n as follows: S = (cid:15) = (cid:104) (cid:98) , , , . . . (cid:105) S = (cid:15) = (cid:104) (cid:98) , , , , . . . (cid:105) S = (cid:15) = (cid:104) (cid:98) , , , , , . . . (cid:105) ...This sequence s n will hyperconverge to (cid:104) (cid:98) , , , . . . (cid:105) , i.e. s n satisfies HC ( s n , ) . Theorem 5.43.
For any sequence s n , R | = CC( s n , s ) ↔ 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 between0 and 1. From the Archimedean property of reals we know that for every0 < r <
1, we can find an m ∈ N such that m < r , and so because of CC( s n ),we have | s n − s | < m < r . Theorem 5.44.
For any sequence s n in R Z < , HC ( s n , s ) always implies CC ( s n , s ).However, there exists a sequence ( t n ) such that R Z < | = CC( t n , s ) (cid:54)→ HC( t n , s ) .Proof.
1. To prove the first clause, suppose that a sequence s n satisfiesHC( s n , s ). This means that we are able to find a number N such that ∀ n ≥ N , | s n − s | < r for any r ∈ R Z < which includes infinitesimals. Byusing the same N , s n will satisfy CC( s n , s ) .2. To prove the second clause, take the sequence t n = n where n ∈ N . Thissequence satisfies CC( t n , t n , s ) for any s (asany r ∈ ∆ will satisfy the negation of Definition 5.41). Lemma 5.45.
Let ( s n ) be a sequence in R Z < such that HC( s n , s ) is hold. Then,HC( | s n | , | s | ) is hold. roof. Let r > ∈ R Z < be given. Then this means that there exists N ∈ N such that ∀ m > N , | s m − s | < r . Therefore, we also have ∀ m > N , || s m | − | s || ≤ | s m − s | < r. Hence, HC( | s n | , | s | ) is true. Theorem 5.46.
Let X ⊂ R Z < and f : X → R Z < . Then f is e -continuous at x ∈ X iff for any sequence x n in X that satisfies HC ( x n , x ) , the sequence f ( x n ) satisfies HC ( f ( x n ) , f ( x )) .Proof. Suppose that f is e -continuous at x and let the sequence x n be definedin X and that x n hyper converges to x . Now let e > e > ∈ R Z < such thatif x ∈ X and | x − x | < e , then | f ( x ) − f ( x ) | < e .Now since x n hyper converges to x , then there exists N ∈ N such that ∀ n ≥ N | x n − x | < e . Thus we have ∀ n ≥ N | f ( x n ) − f ( x ) | < e . and so the sequence f ( x n ) hyper converges to f ( x ).For the converse, we will prove the contrapositive. Suppose that f is not e -continuous at x . Then it means that there exists e > ∈ R Z < such that forall e > ∈ R Z < , there exists x ∈ X such that | x − x | < e but | f ( x ) − f ( x ) | > e . In particular, for all n ∈ N , there exists x n ∈ X such that | x n − x | < e and | f ( x n ) − f ( x ) | > e . Thus x n is a sequence in X that hyper converges to x , but the sequence f ( x n ) does not hyper converge to f ( x ). Definition 5.47.
Let s n be a sequence in R Z < . Then we say that s n is ahyper-Cauchy sequence iff ∀ e ∈ R Z < , ∃ N ∈ N such that ∀ l, m ≥ N | s l − s m | < e . Theorem 5.48.
Every hyper convergent sequence in R Z < is a hyper-Cauchysequence.Proof. Let s n be a sequence in R Z < that satisfies HC( s n , s ). We want to showthat s n is hyper-Cauchy. Let e ∈ R Z < be given. Then there exists N ∈ N suchthat ∀ n > N , | s n − s | < e . Then for all l, m > N , we have | s l − s m | = | s l − s − ( s m − s ) | ≤ | s l − s | + | s m − s | < e e e and so s n is hyper-Cauchy. Conjecture 5.49.
The set R Z < is hyper-Cauchy complete with respect to the e -topology. emma 5.50. Let s n be a sequence in R Z < whose members are just real numbers– that is, for all s ∈ s n , Nst (cid:15) ( s ) = Nst ω ( s ) = ∅ . Then s n is hyper-Cauchy ifand only if there exists N ∈ N such that s m = s N for all m ≥ N .Proof. Let s n be a hyper-Cauchy sequence in R Z < whose members are realnumbers. Then there exists N ∈ N such that | s m − s l | < (cid:15) for all m, l ≥ N. (12)Since s n is a sequence of real numbers, we obtain from Inequality 12 that forall m, l ≥ N , | s m − s l | = 0 and so s m = s N for all m ≥ N .Conversely, let s n be a sequence in R Z < whose members are real numbers andassume that there exists N ∈ N such that s m = s N for all m ≥ N . Now let e > l, m ≥ N , | s m − s l | = 0 < e and so s n ishyper-Cauchy.Another possible way to define convergence in our set is through the conceptof (cid:96) ∞ as follows: Definition 5.51 ( R Z < -Convergence) . Suppose that s n is a sequence where everymember of it is another sequence itself, i.e. s n = ( s n ) , ( s n ) , ( s n ) , . . . , ( s n ) i , . . . .Then, s n converges to s iff ∀ m ∈ N , ∃ N such that ∀ n ≥ N , ∀ i | ( s n ) i − s i | < m .We write RC ( s n , s ) to denote that a sequence s n is R Z < -convergent to s . Example 5.52.
The sequence s n = (cid:104) (cid:98) n , , , . . . (cid:105) is R Z < -convergent to . The next interesting question is which of the three definitions above can beused to define convergence in R Z < ? Unfortunately, neither of them is adequateto serve as the definition of convergence in our set. The three examples belowdemonstrate the reason. The first example shows that when Classical Conver-gence is adopted in R Z < , convergence is no longer unique. While the second oneshows how adopting Definition 5.41 gave something unexpected occurs in ourset, the last example shows why R Z < -convergence is not adequate. Example 5.53.
Suppose that s n is a sequence defined by: s n = (cid:104) (cid:98) , n, , . . . (cid:105) .Then by using Definition 5.40 above and the fact that any infinitesimals are lessthan any rational numbers, s n classically converges to (cid:15) , (cid:15) , (cid:15) , and soon. In other words, the sequence s n satisfies (CC ( s n , (cid:15) ) ), (CC ( s n , (cid:15) ) ),(CC ( s n , (cid:15) ) ), and so on. Example 5.54.
Using Definition 5.41, the sequence s n = (cid:104) (cid:98) n , , , . . . (cid:105) does notconverge in the usual sense to 0, i.e. s n does not satisfy HC ( s n , ) . Taking r = (cid:15) = (cid:104) (cid:98) , , , . . . (cid:105) and n = N + 1 will show this. xample 5.55. The sequence s n = (cid:15) n does not R Z < -converge to , as it shoulddo intuitively. Thus, this leaves us with the three definitions of convergence used in R Z < .There is no one definition of convergence in our set. This is not necessarily abad thing, it simply means that our notion of convergence will differ from thatof classical analysis.Note that our attempts to have a proper notion of continuity and convergencein R Z < can be used in the area of reverse mathematics. From what we havedone here, it can help us to gain a better understanding about some necessarycondition , for example, for a function f to be continuous or for a sequence tobe convergent. R Z < A computable function is a function f which could, in principle, be calculatedusing a mechanical calculation tool and given a finite amount of time. In thelanguage of computer science, we would say that there is an algorithm com-puting the function. A computable real number is, in essence, a number whoseapproximations are given by a computable function.The notion of a function N → N being computable is well understood. Infact, all definitions that so far capturing this idea (such as Turing Machines,Markov Algorithms, Lambda Calculus, the (partial) recursive functions, andmany more) have all led to the same class of functions. This, in turn, has led tothe so called Church-Markov-Turing thesis, which says that this class is exactlywhat computable intuitively means. Given computable pairing functions also,immediately, lead to a notion of computability for other function types such as N k → N m , N → Z or N → Q . If we see a real number as a sequence of rationalapproximation, we also get a definition of a computable real number.However, we have to be a bit careful. There are many equivalent formulationsfor when a real number r is computable, that work well in practice. This happenssuch as when • there is a finite machine that computes a quickly converging Cauchysequence that converges to r , or • it can be approximated by some computable function f : N → Z suchthat: given any positive integer n , the function produces an integer f ( n )such that f ( n ) − n ≤ a ≤ f ( n )+1 n .We denote the set of all computable real numbers by R c . It is well known (andalso well studied) that many real numbers, such as π or e , are computable.However, not every real number is computable. That is with a fixed modulus of Cauchyness. The set of all real numbers thathave a computable decimal representation is denoted by R d . Remark 6.1.
Although the set R d is closed under the usual arithmetic opera-tions, we have to be careful of what it really means. Take, for example, addition.We know that if x and y are in R d , then x + y is also in R d . However, it does not mean that the addition operator itself is computable. These ideas of computability can be extended to infinitesimals. In R Z < , wedefine its member to be computable if it satisfies the condition as stated inDefinition 6.2. Definition 6.2.
A number z ∈ R Z < is computable iff there is a computablefunction f such that f ( n, · ) are computable numbers and z = (cid:104) f (1 , · ) , f (2 , · ) , . . . , (cid:92) f ( l, · ) , . . . (cid:105) where l = f (0 , denotes the index where the St ( z ) is. We denote the set of allcomputable members of R Z < by R Z < c . In this section, we showed that the standard arithmetic operations (func-tions) in R Z < are computable (provided that the domain and codomain of thosefunctions are (in) R Z < c ). This was done by explicitly showing the program foreach one of them. We actually uses a concrete implementation of these ideasin the programming language Python, whose syntax should be intuitively un-derstandable even by those not familiar with it. There is also no need to showthat our programs are correct, since they are so short that such a proof wouldbe trivial.Assuming that we already had a working implementation of R c , our class R Z < c could be implemented as in Listing 1. There we defined the members ofour set R Z < c (basically just a container for the index l as in Definition 6.2 andthe sequence of digits) and how their string representation would look like. c l a s s i n f r e a l :def i n i t ( s e l f , digits , k=0) :s e l f . k = ks e l f . d i g i t s = d i g i t sdef r e p r ( s e l f ) :i f s e l f . k == 0:return ”ˆ” + ” , ” . join ( [ s t r ( s e l f . d i g i t s ( i ) ) for i in range ( s e l f . k , s e l f . k+7) ] )+ ” , . . . ”e l s e :return ” , ” . join ( [ s t r ( s e l f . d i g i t s ( i ) ) for i in range ( s e l f . k) ] ) + ” , ˆ” + ” , ” .join ( [ s t r ( s e l f . d i g i t s ( i ) ) for i in range ( s e l f . k , s e l f . k+7) ] ) + ” , . . . ”def g e t i t e m ( s e l f , key ) : return s e l f . d i g i t s ( key ) Listing 1: How to define the members of R Z < c . Example 6.3.
Suppose that we want to write the number = (cid:104) (cid:98) , , , . . . (cid:105) .Then by writing One=infreal(lambda n:one if n==0 else zero, 0) Consider a number r such that there is an algorithm whose input is n , and it will givethe n th -digit of r ’s decimal representation. here zero and one are the real numbers and , respectively, we just createdthe number in our system. The second argument of the function infreal isjust 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 = i n f r e a l ( lambda n : one i f n==0 e l s e zero , 0) >>>
Oneˆ1 , 0 , 0 , 0 , 0 , 0 , 0 , . . .
Furthermore, we will also be able to know what is its n th digit for any n ∈ N .See the code below: >>> Oneˆ1 , 0 , 0 , 0 , 0 , 0 , 0 , . . . >>>
One [ 0 ]1 >>>
One[ − >>> One[2454]0
Example 6.4.
Similar to Example 6.3, the numbers (cid:15) and ω could also bedefined in our system. >>> Epsilon = i n f r e a l ( lambda n : one i f n == 1 e l s e zero , 0) >>>
Epsilonˆ0 , 1 , 0 , 0 , 0 , 0 , 0 , . . . >>>
Omega = i n f r e a l ( lambda n : one i f n == 0 e l s e zero , 1) >>>
Omega1 , ˆ0 , 0 , 0 , 0 , 0 , 0 , 0 , . . .
Also, we will be able to have exotic numbers such as e + e (cid:15) + e (cid:15) + e (cid:15) + . . . and its code will be as follows: >>> e = exp ( rational (1 ,1) ) >>> Funny = i n f r e a l ( lambda n : real ( rational (n + 1 , 1) ) ∗ e i f n > − >>> Funnyˆ2.71828 , 5.43656 , 8.15485 , 10.8731 , 13.5914 , 16.3097 , 19.028 , . . . >>>
Funny [43532]118335 >>>
Funny[ − Theorem 6.5-6.9 show that addition, subtraction, and multiplication in R Z < c are computable. Theorem 6.5.
Suppose that we have x, y ∈ R Z < c . Then the function (cid:98) + c definedby (cid:98) + c : R Z < c → R Z < c ( x , y ) (cid:55)→ x (cid:98) + y is computable.Proof. The following code shows that the function (cid:98) + c defined above is com-putable. def add ( s e l f , other ) :k = max( s e l f . k , other . k)return i n f r e a l ( lambda n : s e l f . d i g i t s (n − (k − s e l f . k) ) + other . d i g i t s (n − (k − other . k) ) , k) xample 6.6. Suppose that we want to add (cid:15) , ω , and . Then we will have: >>> Omega1 , ˆ0 , 0 , 0 , 0 , 0 , 0 , 0 , . . . >>>
Epsilonˆ0 , 1 , 0 , 0 , 0 , 0 , 0 , . . . >>>
Epsilon + Omega + One1 , ˆ1 , 1 , 0 , 0 , 0 , 0 , 0 , . . .
Theorem 6.7.
Suppose that we have x, y ∈ R Z < c . Then the function (cid:98) − c definedby (cid:98) − c : R Z < c → R Z < c ( x , y ) (cid:55)→ x (cid:98) − y is computable.Proof. The following code shows that the function (cid:98) − c defined above is com-putable.
47 def n e g ( s e l f ) : return i n f r e a l ( lambda n : − s e l f [ n ] , s e l f . k)48 def s u b ( s e l f , other ) : return ( s e l f + ( − other ) ) The definition in line 47 shows that the additive inverse function is computable.
Example 6.8.
Suppose that we want to add (cid:15) (cid:98) − ω to . Then we will have: >>> Epsilon − Omega + One −
1, ˆ1 , 1 , 0 , 0 , 0 , 0 , 0 , . . .
Theorem 6.9.
Suppose that we have x, y ∈ R Z < c . Then the function (cid:98) × c definedby (cid:98) × c : R Z < c → R Z < c ( x , y ) (cid:55)→ x (cid:98) × y is computable.Proof. The following code shows that the function (cid:98) × c defined above is com-putable. def mul ( s e l f , other ) :k = s e l f . k + other . kdef d i g i t s (n) :i f n < ∗ other . d i g i t s (n − i ) for iin range (n + 1) ] )return i n f r e a l ( digits , k) Example 6.10.
Suppose that we want to add (cid:15) (cid:98) × ω to and also − (cid:15) to . Thenwe will have: >>> Epsilon ∗ Omega + One0 , ˆ2 , 0 , 0 , 0 , 0 , 0 , 0 , . . . >>>
Epsilon ∗ − Epsilon + Oneˆ1 , 0 , −
1, 0 , 0 , 0 , 0 , . . . .1 Some Remarks on Non-Computability in R Z < Remark 6.11.
Even though division on R c is computable (assuming the inputdoes not equal ), the same can not be said of R Z < c . Remark 6.12.
Suppose that we have a number x ∈ R Z < . Then without anyfurther information, the process of finding x − (the multiplicative inverse of x ) is not computable. One extra information needed to make it computable in R Z < is how many digits we want to have in x − , which of course will affect theaccuracy of our result. More precisely, it is known that it is not possible to givean algorithm that, a given number a ∈ R c , decides whether a = 0 or ¬ a = 0 .So let a ∈ R c and consider z = a + (cid:15) . We have (cid:15) (cid:54) = 0 . If a = 0 then z − = ω .If a (cid:54) = 0 then z − < ω . Thus by checking whether the ω -part of z − is less than or greater than , we would be able to decide whether a = 0 or ¬ a = 0 . Remark 6.13.
Similarly surprising, we can show that the absolute value func-tion, which is computable for R c , is not computable for R Z < c . Here the absolutevalue function is the function | z | = (cid:40) z if z ≥ − z if z < Similar to the above, take a ∈ R c and consider z = | a | − (cid:15) . If a = 0 then | z | = −| a | + (cid:15) . If a (cid:54) = 0 then | z | = | a | − (cid:15) . Thus by checking the (cid:15) -part of | z | , wewould be able to decide whether a = 0 or ¬ a = 0 . This also leads to comparison between numbers not being computable. Nowthis is also the case in R c . However, for numbers x, y ∈ R c such that x (cid:54) = y , wecan decide whether x < y or x > y . This does not extend to R Z < c : Remark 6.14.
Comparison among the members in R Z < c is not computable.Again, let a ∈ R c and consider x = | a | + (cid:15) and y = | a | . If a = 0 then x > y ,and if a (cid:54) = 0 then x < y . Thus, once again we would be able to decide whether a = 0 or ¬ a = 0 . In this article we treated the nonstandard real numbers in the spirit of the‘Chunk and Permeate’ approach. The sets R and ∗ R were thrown together(that means: combining their languages and axioms) and arising inconsistencyissues were dealt with using paraconsistent reasoning strategy. In this case —and this is one of the interesting novelties — the procedure was made explicitby introducing new sets (cid:98) R and in turn R Z < , the separate ‘chunks’. In a sense,we transferred the ‘Chunk and Permeate’ approach from the theoretical levelto the (explicit) model level. Whereas the impact on infinitesimal mathematics43as only sketched in [9], it was worked out in detail here, using the sets (cid:98) R and R Z < . After introducing the theoretical background, we constructed the newmodel of nonstandard analysis in detail. The remaining part of this paper liesin an extensive discussion of topological, applied (in the sense of calculus), andcomputability issues of the obtained model. A side result of the constructed set 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 ofmetrics, balls, open sets, and etc in R Z < together with their properties. On the applied aspect in Section 5, some new concepts on the calculus in R Z < werediscussed, e.g. derivative (we successfully developed a permeability relationsuch that the derivative function in R Z < can be permeated to R ), continuity,and convergence. For the two last issues, some new notions were introduced inthis article. First of all, we discussed the three possible notions of continuitythat can be applied to either R or R Z < . We also determined how they relateto each other in their respective model. While doing that, we discovered a newkind of fractals — infinitesimal fractals. After analysing three possible notions ofcontinuity, we decided that the best notion that can be used in our setting R Z < isthe (cid:15) - δ definition and by doing that, we do not only preserve much of the spirit ofclassical analysis but also retain the intuition of infinitesimals. After establishingour 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 newnotion of continuity behaves, e.g. what happens with the composition of twocontinuous functions and also how this notion behaves under multiplication. Itis worth pointing out here that this new notion of continuity is a much morefine-grained notion than the classical continuity. Last but not least, we showedthat the set R Z < has nice computability features. We succeeded in building aprogram, in Python, to show that we can have a computable number R Z < . Theset of all these computable numbers is denoted by R Z < c . We also showed someinteresting remarks regarding this computability issue.In term of further research, we indicated some possible areas of further de-velopment as follows. First , one could try to do infinitesimal analysis using therelevant logic R . The comparison between the results (perhaps) gotten in R and the one described here might be interesting, especially in term of usefulnessand simplicity. Secondly , regarding the ‘transfer principle’, our intuition saysthat it is equivalent to the notion of permeability in the Cchunk & Permeatestrategy. One could try to formally prove it, or disprove it.
Thirdly , in term ofcomputability issue, using the calculus on R Z < , one could try to formulate thenecessary and sufficient conditions for the derivatives of functions, for example,on a computer to exist. And perhaps, showing also how to find these derivativeswhenever 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 thisarticle 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 quantumphysics). One could try to work out the details on this.In general, with the new consistent sets created in this work, new opportuni-ties awaits mathematicians. One of the joys of mathematics is to explore a world44hich has no physical substance, and yet is everywhere in every aspect of ourlives. Infinities and infinitesimals offer ways to explore hitherto unseen aspectsof our world and our universe, by giving us the vision to see the greatest andsmallest aspects of life. Even a na¨ıve set, when it demonstrates harmony, offeranother dimension of even clearer precision. In a wide sense, the work on thisarticle can also be seen as a contribution to bridge (the antipodes) constructiveanalysis and nonstandard analysis. This problem has been extensively (and in-tensively) discussed in the past few years (see for example [40, 41, 42, 8, 43, 31]).
Acknowledgements
The first author received financial support from Indonesia EndowmentFund for Education that enables the research of this article. We wouldalso like to thank Professor Elem´er Rosinger and Dr. Josef Berger for theirinvaluable inputs.
References [1] Arkeryd, L.O., Cutland, N.J., Henson, C.W.: Nonstandard analysis: The-ory and applications, vol. 493. Springer Science & Business Media (2012)[2] Avron, A.: Natural 3-valued logics-characterization and proof theory. TheJournal of Symbolic Logic (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 inParaconsistent 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 non-standard analysis. Expositiones Mathematicae (4), 355–386 (2003)[7] Bishop, E., et al.: H. jerome keisler, elementary calculus. Bulletin of theAmerican Mathematical Society (2), 205–208 (1977)[8] Bournez, O., Ouazzani, S.: Cheap non-standard analysis and computabil-ity. arXiv preprint arXiv:1804.09746 (2018)[9] Brown, B., Priest, G.: Chunk and permeate, a paraconsistent inferencestrategy. part i: The infinitesimal calculus. Journal of Philosophical Logic (4), 379–388 (2004)[10] Cantor, G.: Contributions to the Founding of the Theory of TransfiniteNumbers. 1. Open Court Publishing Company (1915)4511] Da Costa, N.C., et al.: On the theory of inconsistent formal systems. NotreDame Journal of Formal Logic (4), 497–510 (1974)[12] Dunn, J.M., Restall, G.: Relevance logic. In: Handbook of philosophicallogic, pp. 1–128. Springer (2002)[13] Earman, J.: Infinities, infinitesimals, and indivisibles: the leibnizianlabyrinth. Studia Leibnitiana pp. 236–251 (1975)[14] Edwards, H.: Eulers definition of the derivative. Bulletin of the AmericanMathematical Society (4), 575–580 (2007)[15] Fletcher, P., Hrbacek, K., Kanovei, V., Katz, M.G., Lobry, C., Sanders, S.:Approaches to analysis with infinitesimals following robinson, nelson, andothers. Real Analysis Exchange (2), 193–252 (2017)[16] French, S.: Action at a distance. In: The Routledge Encyclopedia of Phi-losophy. Taylor and Francis (2005). . doi:10.4324/9780415249126-N113-1[17] G¨odel, K.: Consistency of the Continuum Hypothesis.(AM-3), vol. 3.Princeton University Press (2016)[18] Goldblatt, R.: Lectures on The Hyperreals. Springer (1998)[19] Gray, J.: The real and the complex: a history of analysis in the 19thcentury. Springer (2015)[20] Hardy, G.H.: Orders of infinity. Cambridge University Press (2015)[21] Henle, M.: Which Numbers are Real? The Mathematical Association ofAmerica (2012)[22] Kanovei, V., Reeken, M.: Nonstandard analysis, axiomatically. SpringerScience & Business Media (2013)[23] Lolli, G.: Metamathematical investigations on the theory of grossone. Ap-plied Mathematics and Computation , 3–14 (2015)[24] Mares, E.: Relevance logic. In: E.N. Zalta (ed.) The Stanford Encyclopediaof Philosophy, spring 2014 edn. (2014). Available online at http://plato.stanford.edu/archives/spr2014/entries/logic-relevance/ [25] Mayberry, J.P.: The foundations of mathematics in the theory of sets,vol. 82. Cambridge University Press (2000)[26] McKubre-Jordens, M., Weber, Z.: Real analysis in paraconsistent logic.Journal of Philosophical Logic (5), 901–922 (2012)[27] Mortensen, C.E.: Models for inconsistent and incomplete differential cal-culus. Notre Dame Journal of Formal Logic (2), 274–285 (1990)4628] Mortensen, C.E.: Inconsistent Mathematics, vol. 312. Springer (1995)[29] Mortensen, C.E.: Inconsistent Geometry. College Publications (2010)[30] Musser, G.: Spooky Action at a Distance: The Phenomenon that Reimag-ines Space and Time–and what it Means for Black Holes, the Big Bang,and Theories of Everything. Macmillan (2015)[31] Normann, D., Sanders, S.: Computability theory, nonstandard analysis,and their connections. The Journal of Symbolic Logic (4), 1422–1465(2019)[32] Nugraha, A.: Na¨ıve infinitesimal analysis. Ph.D. thesis, School of Math-ematics and Statistics University of Canterbury, New Zealand (2018). https://ir.canterbury.ac.nz/handle/10092/16627 [33] Popkin, G.: Einstein’s ‘spooky action at a distance’ spot-ted in objects almost big enough to see (2018). Avail-able online at .doi:10.1126/science.aat9920[34] Priest, G.: The logic of paradox. Journal of Philosophical Logic (1),219–241 (1979)[35] Priest, G.: Minimally inconsistent lp. Studia Logica (2), 321–331 (1991)[36] Priest, G.: Paraconsistent logic. In: Handbook of Philosophical Logic, pp.287–393. Springer (2002)[37] Robinson, A.: Non-standard analysis. Princeton University Press (1974)[38] Rosinger, E.E.: On the safe use of inconsistent mathematics. arXiv preprintarXiv:0811.2405 (2008)[39] Rosinger, E.E.: Basic Structure of Inconsistent Mathematics (2011).URL https://hal.archives-ouvertes.fr/hal-00552058 . MSC00A05,00A69,00A71,00A99,03Bxx,03B60,03B99[40] Sanders, S.: On the connection between nonstandard analysis and con-structive analysis. Logique et Analyse pp. 183–210 (2013)[41] Sanders, S.: The effective content of reverse nonstandard mathematics andthe nonstandard content of effective reverse mathematics. arXiv preprintarXiv:1511.04679 (2015)[42] Sanders, S.: From nonstandard analysis to various flavours of computabilitytheory. In: International Conference on Theory and Applications of Modelsof Computation, pp. 556–570. Springer (2017)4743] Sanders, S.: The gandy–hyland functional and a computational aspect ofnonstandard analysis. Computability (1), 7–43 (2018)[44] Sergeyev, Y.D.: Arithmetic of infinity. Edizioni Orizzonti Meridionali(2003)[45] Sergeyev, Y.D.: The olympic medals ranks, lexicographic ordering, andnumerical infinities. The Mathematical Intelligencer (2), 4–8 (2015)[46] Shalm, L.K., Meyer-Scott, E., Christensen, B.G., Bierhorst, P., Wayne,M.A., Stevens, M.J., Gerrits, T., Glancy, S., Hamel, D.R., Allman, M.S.,et al.: Strong loophole-free test of local realism. Physical review letters (25), 250402 (2015)[47] Spalt, D.: Cauchy’s continuum-a historiographic approach via cauchy’ssum theorem. Archive for History of Exact Sciences (4), 285–338 (2002)[48] Susskind, L., Hrabovsky, G.: The Theoretical Minimum: What You NeedTo Know To Start Doing Physics. Basic Books (2014)[49] Tao, T.: A cheap version of nonstandard analy-sis. https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/ (2012)[50] Weber, Z.: Inconsistent mathematics. In: J. Fieser & B. Dowden (Eds.).The Internet Encyclopedia of Philosophy (2009). Available online at [51] Weber, Z.: Extensionality and restriction in naive set theory. Studia Logica (1), 87–104 (2010)[52] Weber, Z., McKubre-Jordens, M.: Paraconsistent measurement of the cir-cle. Australasian Journal of Logic (1) (2017)[53] Weston, M.M., Slussarenko, S., Chrzanowski, H.M., Wollmann, S., Shalm,L.K., Verma, V.B., Allman, M.S., Nam, S.W., Pryde, G.J.: Heralded quan-tum steering over a high-loss channel. Science advances4