the phenomenological roots of nonstandard mathematics

ROMANIAN JOURNAL OF INFORMATIONSCIENCE AND TECHNOLOGYVolume 8, Number 2, 2005, 115–136

The phenomenological rootsof nonstandard mathematics

Stathis LIVADAS

Department of Mathematics,University of Patras, Patras, 26500, Greece

E-mail: [email protected]

Abstract. In this paper we intend to interpret the axiomatical and formal

structure of nonstandard mathematics in terms of a phenomenological analysis

by means of two approaches: On the formal-deductive level by means of conser-

vative enlargements of relatively definite axiomatical systems to which Husserl

made reference in his 1901 Gottingen lectures ([9], Abhand. VI, VII) and on

a formal ontological level by means of the reduction of principles of analytical

logic to subjective evidences of experience. In the latter approach we attempt

a phenomenological interpretation to the notion of urelements and that of pro-

longation principles in nonstandard structures. Further, we demonstrate the

relevance of the shift of the horizon approach in Husserlian sense in the con-

struction of alternative models of nonstandard mathematics in the intensional

part of nonstandard analysis.

Keywords. Horizon of life-world, nonstandard theories, non-Cantorian

theories, relatively definite system, urelement, Continuum.

1. Introduction

It is well-known that a more systematic axiomatical approach to foundations wasmotivated out of the foundational crisis of mathematics in the beginning of last cen-tury with the consolidation of the Zermelo-Fraenkel plus the Axiom of Choice (ZFC)theory which is generally proposed as the axiomatical theory of standard mathemat-ics. This is essentially a theory born out of the Cantorian construction of the class ofordinal numbers with some additional axiomatical tools so as to avoid circular pitfallsor conceptual confusion brought upon e.g. by Frege’s cumbersome definition of setsas classes of objects generated by predicative formulas.

116 S. Livadas

It should be reminded here that the original Cantorian axioms at the time ofGrundlagen in 1883 involved a definition of sets as “anything that can be counted” (aset is the range of a one-to-one function with domain a proper initial segment of theordinal numbers) but it was by the use of a form of the commonly used today PowerSet Axiom that Cantor was able to prove for the first time that the real numbersform a set. He was also able to show that the power of the set of real numbers (or ofthe continuum) was that of the set of functions from the natural numbers to a pair,otherwise that c = 2ℵ0 ([14], IV, pp. 95–98). The use of the Power Set Axiom meantthat a conception of sets was introduced in such a way that they could not be countedin principle or well ordered in a definable way.

This has its own special meaning in the context of this article as the underlyingconceptual base of A. Robinson’s nonstandard approach is that of the “refutation” ofthe actual infinity concept on which the Cantorian and subsequently the ZFC systemis founded and the adoption of ideal elements in an enlarged domain of discoursecorresponding to a model theorist’s principles of elementary extension and saturationin an enlarged domain ([2], Ch. 3, p. 29). It is true that although, by the wordsof A. Robinson, non-standard analysis viewed syntactically introduces new deductiveprocedures rather than new mathematical entities, it attempts, in effect, to recali-brate the platonistic claims associated with the formal results of Zermelo FraenkelSet Theory which formalizes the concept of actual infinity in the form of Infinity andPower Set axioms.1

As a matter of fact though, the Axiom of Choice (AC) and Zorn’s lemma whichhave a significant use in the proof of the Compactness Theorem and the constructionof ultraproducts in [20] and [21] presuppose a concept of actual infinity and it is provedby Z. Szczepaniak that the well-ordering principle (which is logically equivalent to theAxiom of Choice) does not follow from ZFC minus Power Set (ZFC−) if ZFC− isconsistent ([31], p. 339).

Robinson’s analysis is generally considered as a key part of the extensional non-standard analysis in which nonstandard entities e.g. infinitely large or infinitely smallentities are thought of as possessing an objective “existence” in extensional termswhereas nonstandard and non-Cantorian theories considered as the intensional partof nonstandard analysis are generally interpreted inside an “observer’s” witnesseduniverse where nonstandard entities are linked to his observational modes and limi-tations.

In these non-Cantorian theories e.g. the Internal Set Theory (IST) of E. Nel-son properly interpreted, the Alternative Set Theory (AST) of the Prague School(Vopenka, Sochor et al) and their offsprings, nonstandard entities do not have anobjective “existence” and are introduced axiomatically by means of ad hoc axioms orof external formulas that involve new undefined predicates.

The claim I’ll try to support is twofold:

1 In A. Robinson’s view, actual infinity could be substituted by the introduction of in-finitely large or infinitely small numbers whose existence is “neither more nor less real than,for example, the standard irrational numbers. This is obvious if we introduce such numbersaxiomatically; while in the genetic approach both standard irrational numbers and nonstan-dard numbers are introduced by certain infinitary processes” ([20], Ch. X, p. 282).

The phenomenological roots of nonstandard mathematics 117

First, that the notion of consistent enlargements of standard systems and the questof ideal elements in saturated extensions of their domains was implicitly introducedby E. Husserl in the ideas of completeness and relative definiteness of axiomaticalsystems developed in his Gottingen lectures [9] . This is mainly done in subsections2.1 and 2.3.

Second, that the nonstandard “ontological” approach to infinities or infinitesi-malities has a common conceptual ground with the phenomenological approach toirreducible individualities in the process of constitution of phenomenological percep-tions (Wahrnehmungen) (mainly, subsections 2.2 and 2.4). Our account of this isbased on Husserl’s reduction in Transzendentale und reale Logik [10] of the laws ofanalytical logic to subjective evidences directed intentionally to distinct unities in thelowest degree of the intentionality of experience inside the impredicative (or ratherpre-predicative) continuous unity of experience as such.

It should be also stressed the influence that the phenomenology of time conscious-ness developed mainly in Vorlesungen zur Phanomenologie des inneren Zeitbewusst-seins (“Lessons for the Phenomenology of the inner time-conscience”) [11] held ona modeling of the idea of intuitive continuum in the works of H. Weyl and L.E.JBrouwer. Taking into account the Husserlian idea that perceptual intuitions are anal-ogous to mathematical intuitions, in the intuition, for instance, of natural numbers assequences of immanent unities in consciousness constituted out of phenomenologicalperceptions2, one proceeds to a modeling of the continuum of real numbers by meansof choice sequences of natural numbers where at any given stage one can generate areal number defined in terms of the generating terms in their (temporal) extensionand not as an outcome of a limiting procedure in the classical sense.

This is conceptually linked with the phenomenological constitution of the contin-uous unity of the flux of time consciousness as such out of the concrete and distinctmultiplicities of immanences of perceptual objects in it, in the original sensation plusthe retention and protention scheme ([28], pp. 206–212). I choose not to deal furtherin this paper with the phenomenological influence on the intuitionistic approach ofmathematics for this might have the breadth to be the subject-matter of anotherarticle.

2 Husserl claimed that the mathematical objects given in intuition are founded on per-ception but they are not perceptual objects. Referring to his views Richard Tieszen statesthat:

“The intuition of mathematical objects is said to be “founded” on perception because per-ceptual acts provide the concrete, immediate and non-reflective basis of all our experienceand any intuition of abstract objects must be “constituted” from this basis. [...] What I havetaken to be essential to perceptual intuition is of course the possibility of having sequences ofmental acts directed to individual perceptual objects, so this feature should be viewed as play-ing a crucial role in the founding of the intuition of mathematical objects. On the construalof the intuition of natural numbers I have given, this feature does play a crucial role since thestructure or form of sequences in perception (which one gets by absrtacting from qualitativedifferences in perceptual acts) may be viewed as a necessary condition for intuition of naturalnumbers” ([25], p. 414).

118 S. Livadas

Finally, taking into account that nonstandard objects are new, ideal elements inconsistently enlarged domains of Cantorian axiomatical systems we may view themas beyond the horizon of countable infinity objects in a sense related to the notion ofthe horizon of our intersubjective life-world described in [12]. This has to do mainlywith the AST approach to natural infinity as explicitly stated by P. Vopenka [29] andit is developed in section 3.

2. A phenomenological interpretation of nonstandardanalysis

2.1. Husserl’s idea of enlargement in logical-deductive systems

In his Gottingen lectures of 1901 to the German Mathematical Society, E. Husserlpresented his views on a matter that had preoccupied him for at least a decade, namelythe problem of imaginary entities in mathematics conceived as a logical-deductivesystem. His main preoccupations about them were three:

1) Under what conditions can one freely use within a formally defined deductivesystem concepts that make no sense according to the definition of the system and canthus be characterized as imaginary?

2) How can one be sure of the validity of the reasoning when one has also made ap-peal to imaginary entities in reaching a conclusion which otherwise uses the languageand the rules of deduction of the formal system? and

3) Under what conditions is it permissible to expand a well-defined deductivesystem and create a new one in relation to which the older one stands as its formallimitation?

We should remark that the matter of these lectures can be put in the context ofa then ongoing discussion regarding the place of imaginary entities in mathematicswith G. Frege banning in his logic any combination of signs and rules that did notdesignate an object, arguing that “unless an equation contained only positive num-bers, it no more had a meaning than the position of chess pieces expressed a truth andcondemning the theory by which one might but set down rules by which one passedfrom given equations to new ones in the way one moved chess pieces” ([18], p. 89).

In his critic of the Fregean approach Husserl maintained that constraints in formaltheories banning reference to imaginary or non-existent objects (Husserl regardedas imaginary numbers in arithmetic the negative, rational, irrational and complexnumbers) only restrict the scope of logical-deductive theories and have nothing todo with the nature of the logical-deductive structure as such. His view was that thetheory of complete manifolds (vollstandige Mannigfaltigkeiten) was the key to thesolution as to how non-existent concepts could be dealt with as real ones withinthe realm of non-imaginary numbers. In that sense each grammaticaly constructedproposition exclusively expressed in the language of the domain and coming in apurely analytical procedure out of a finite number of concepts and propositions drawnfrom the essential nature of the domain under consideration could operate freelywith imaginary concepts and be true or false in virtue of the axioms of the domain.


In such complete manifolds, he regarded the concepts true and formal implicationsof the axioms as equivalent ([18], p. 92), a statement akin to Godel’s completenessstheorem for first-order predicate calculus.

In fact, according to Husserl’s view any system of arithmetic is absolutely definite 3

and the reason for this is that “the formulas of the language of arithmetic can alwaysbe reduced to systems of equations and inequalities which are decidable” ([7], pp.433–434). In support of this, he held that any numerical equation is decidable sinceit can be reduced by means of operation forms (which are, in fact, quantifier-freeformulas) according to the axioms to an identity (true) or not (false) and any algebraicformula is decidable too because it is decidable for any numerical instance ([9], VI, p.443). The weak points of this argument are two:

First, it is problematical whether you can call an arithmetical system completejudging by its operation forms only and second, it is questionable whether you cancall a general statement (an algebraic formula) decidable whenever its instances aredecidable ([7], p. 434). Of course, Husserl’s argument about the completeness ofarithmetical systems stands in obvious contradiction with Godel’s incompletenesstheorem proved in subsequent years but this contradiction can be easily explained interms of the difference between them in the interpretation of the term decidability.

As it concerns Husserl’s conditions for the possibility of the introduction of imag-inary elements in enlargements of formal domains the key to it is his idea of relativedefiniteness of a manifold. With respect to this notion:

“If P is a proposition that says: P holds for the manifold whose existence is provedby (axiomatical system) A, then this proposition is, for this manifold, either true orfalse in virtue of the axioms.

The manifold contains all the objects proved as existent in virtue of A, which doesnot exclude, that the same axioms hold for a larger manifold, but in such a way thatthe extra objects are not defined to exist or proved to exist by means of the axioms(without the addition of new ones)” ([9], VII, p. 454).

In other words, an axiomatical system is relatively definite (relativ definit) when-ever a proposition that makes sense according to it can be decided within its ontologi-cal domain so that any theorem deducible in its enlargement must contain exclusivelyconcepts that are valid in the narrower one and thus not imaginary or must con-tain imaginary concepts (new axioms and new elements in the expanded ontologicaldomain defined by the new axioms).

Moreover, supposing that A and B are two consistent systems of axioms andB extends A so that the ontological domain of B includes properly that of A thuscontaining imaginary elements from the perspective of A, then:

If B proves any proposition P in the language L(A) of A, P must be necessarilyproved inside A on the condition that the variables of P are restricted to the domainD of A. In that case, imaginary elements of the domain of B are not necessary inproving assertions involving exclusively objects of the domain of A ([7], p. 429).

Absolute or even relative definiteness of an axiomatical system ensures that its

3 Roughly speaking, a system of axioms is defined by Husserl as absolutely definite (absolutdefinit) in case any proposition that has a sense according to it, is decided in general.

120 S. Livadas

domain is complete in Hilbert’s sense, namely that no object can be adjoined to itand the enlarged structure be still a model of this axiomatical system. In an “upward”directed approach Husserl’s imaginary numbers (or elements) are adjoined in consis-tent extensions of relatively definite systems and cannot by themselves introduce newelements into the extended formal domain to which they belong nor describe in anyway this domain inside the structure of which they are essentially imaginary or “ex-ternal”. We should note, in addition, that Husserl’s view of imaginary numbers is thatof pure symbols rather inaccessible to intuition much more so than “huge” naturalnumbers generated by abstraction from given collections of objects ([7], p. 434).

It seems remarkable, though, that later in Formale und Transzendentale Logik(1929) he reduced the laws of analytical logic to subjective laws of evidences of ex-perience putting consequently the concepts of definiteness and saturation inside aformal-deductive theory under a new purely phenomenological perspective. In rela-tion to the meaning of the (absolute or relative) definiteness of a manifold developedpretty earlier in the double lecture at Gottingen in the winter of 1901–1902 but notfurther pursued in subsequent Logische Untersuchungen, he asked:

“How can one know a priori that a domain is a nomological (absolutely definite)domain and if one takes as an example of such a domain the space in its spatial formsthe set of immediately evident axioms that is posed “grasps” completely the essenceof the space, that is, it suffices to determine a nomology?

And then, a fortiori, in pure formalisation or in the free construction of forms ofmanifolds: how can one know, how can one prove that a system of axioms is a “defi-nite” system, a “saturated” system?” ([10], Ch. III, pp. 131–132, transl. of the author).

The critical turn in Husserl’s view of a universal manifold in the sense of a Leib-nizian mathesis universalis is its reduction to a problem of phenomenological interpre-tation of a global analytical logic. A doctrine of a universal manifold has to define,in virtue of axiomatical forms, every other manifold which has to include its ownfundamental propositional forms with their relevant logical categories that are impli-cated systematically in the morphology of judgements. In other terms, it has to beconstructed upon a prior discipline, that of the morphology of judgements (as catego-rial significations in apophantic attitude) which will be ultimately reduced to primeevidences of experience in phenomenological terms ([10], Ch. III, p. 136).

In this context, a phenomenological reduction of apophantic sentences in formal-deductive systems will be linked, on a formal ontological level, to the quest of idealelements as irreducible individualities beyond “the horizon” of standard elements ofnatural intuition though any ontological concerns about the status of the concepts ofinfinity would leave, by the words of A. Robinson, a logical positivist indifferent.

2.2. Reduction of analytical-logical principles of deductive systemsto prime evidences of experience

Husserl stated in Formale und Transzendentale Logik that a pure analytics hasto lead to a phenomenological analysis of a vast amplitude and depth if it is to bereally a theory of science and indeed found the possibility of an authentic science thatmakes available the principles of the justification of its authenticity.


As a concrete example of his position he gave the fundamental form (which heconsidered as one of the idealisations that play a universal role in analytical logic) ofthis way in infinitum that produces an infinity by iteration and has as its subjectivecorrelate the form one can always do. It is evident that this constitutes an idealisationsince nobody can always perform de facto something anew. All the same though, thisidealisation is evidently instrumental in the “morphology” of analytical principlesconcerning infinity axioms or the construction of number systems e. g. given any set,one can always have a new set to which the former is disjoint and, in addition, adjointhis new set to the first one. Or, given any number a one can always form a newnumber a + 1 and in this way starting from 1 form the infinite sequence of (natural)numbers.

Following this analysis in which he sought to lay bare the subjective, constitu-tive origins of analytical principles lying “hidden” behind infinite constructions byiteration or ideal existences in formal mathematics, Husserl referred to the analyticalprinciples of contradiction, of the excluded middle and the laws of modus ponens andmodus tollens too:

“In the purely objective perspective the analytical principle of contradiction is aprinciple on the mathematically ideal “existence” (and co-existence) that is, on theco-possibility of judgements at the stage of distinction. But it is on the subjective sidethat it is found the a priori structure of evidence and the effectuations that usuallycome out of this structure, structure whose uncovering puts in evidence the essentialsubjective situations that correspond to its objective sense. [...] All judgements mustbe put in contact with “things themselves” to which they refer and they have toconform to them whether in a positive or negative completion” ([10]: §75, §77, pp.257, 261, transl. of the author).

Positive completion is to be meant in the sense that the judgement in questionis evidently true in the verification and sufficiency that comes out of the coincidenceof the categorial objectivity which is “viewed” in the presumed judgement with thatsame objectivity given as a givenness (Gegebenheit) “in-itself” ; whereas in the nega-tive case the judgement is evidently false to the extent that in the partial completionof the judgement there is manifested as a givenness “in-itself” a categorial objectivityopposed to that one in the intentionality of the original judgement and thus is bynecessity “annuled”.4

4 (i) Cf. in relation to the notions of positive or negative completion with C. Popper’sprinciple of Falsifizierbarkeit in his Logik der Forschung (Springer, Wien 1934) regarding thescientific “status” of universal theories in contrast to the Verifizierbarkeit principle of logicalpositivists.

(ii) Referring further to the analytical principle of the excluded middle Husserl noted thatsince it decrees that “all judgements can be led, in principle, to completion” it implies anideality to which there is no corresponding evidence, which also holds for many other non-evident judgements, whereas referring to the universal character of analytical principles (orof universal-quantifier logical formulas) he stressed their intersubjective character in termsof the formation of ideal unities out of a multiplicity of subjective experiences that hold forevery being.

122 S. Livadas

Going deeper in the reduction of the principles of analytical logic (and thus offormal mathematics) to subjective evidences Husserl held the view that on a purelyanalytical level every judgement and thus every sentence in apophantic logic can bereduced by syntactical “deconstruction” to its ultimate object-substrates so that thepropositions reached in the final stage can be no longer held to be “analytical”. These“final nuclei” have to be objects of intuition in the sense that they are irreducible indi-vidual substrates of analytical propositions corresponding to absolute and individual“some-things” for the possibility and essential structure of which nothing can be saidin analytical terms even that they by necessity appropriate a temporal form ([10],§82, p. 276).

These “individual objects” reached in the most fundamental level of every syntac-tical and correspondingly semantical reduction have no further syntactical structureand their existence can be only “grasped” by experience prior to any analytical formof judgement. Every most original judgement being the subjective form of the effec-tuation of the most original and direct evidence must be a priori directed to individ-ualities donated by first-degree experience in its prime and strongest sense definedprecisely as direct reference to individualities.

Taking into account his doctrine of the universal genesis of consciousness in gen-eral, Husserl thought of judgements5 not only as complete products of a “constitution”or “genesis” bearing in them a kind of historicity with respect to their original sensebut also in their most original and primitive forms as leading to a genetical reductionof predicative evidences to the non-predicative evidence which is experience itself.

In this way, the givenness of “things in-themselves” and also of any modalitiesrelating to them (proprieties, relations etc.) and the subsequent construction of ana-lytical forms of judgements of a higher level do not exclusively belong to the predica-tive “universe” but also to the unity of possible experience which intentionally refersto these irreducible syntactical elements (whether material or formal individuals).This makes possible the cohesion of the content of any original judgement becauseit is based on the synthetic unity of experience transposed (in a phenomenologicalsense) in the flux of consciousness of a subject who forms judgements of any degreeof evidence ([10], §89 (b), pp. 295–296).

This Husserlian reduction of the principles of analytical logic to subjective primeevidences of experience in the construction of apophantic sentences (or mathematicalpropositions) leads to three important conclusions in relation to the aim of this article:

First, the reduction of the laws of analytical logic and consequently of those ofclassical mathematics to subjective evidences points to the underlying conceptualframe of nonstandard mathematics. This stands essentially in the discarding of theplatonic nature of the existing ZFC system which adopts the idea of actual infin-ity in its axiomatical forms and their logical implications and its substitution by a

5 It should be noted that the term judgement in Formale und Transzendentale Logikis used in the sense of formal logic which in Husserl’s view bears the double character ofapophantic logic and formal ontology, disciplines which were regarded by him as equivalenttaking into account that they are in universal correlation up to the very last details and thusvalue as one and the same science ([10], §42, p.151).


witnessed universe correlated to the presence of a potential “observer” in which in-finities or infinitesimalities are defined axiomatically but nonetheless refer to his sub-jective “observations”.6

Second, the most fundamental reduction to constituent individualities as primeevidences of experience bearing no further inner structure points to a hierarchy ofinfinitesimals of various orders in which the infinitesimals of a given order appear tobe points without structure to the immediately lower order until we unravel their ownstructure in a kind of a Russian doll game and reveal a class of infinitesimals of a stillhigher order playing provisionally the role of points.

In nonstandard analysis this relates, for instance, to the definition of points in thenonstandard model R∗ of the standard set of real numbers R which have an innerstructure as equivalence classes of infinite sequences of standard reals modulo anultrafilter F over the set of natural numbers, (see [21]). In this context the standardreal numbers in the classical sense are the irreducible individuals of R∗.

In this respect we should take into account the role of urelements which are consid-ered elements of a higher order structure with no further inner structure themselvesin the definition of internal and external sets in nonstandard structures e.g. by meansof an injective mapping ∗ : S −→ S∗ where to every element s ∈ S in classical(standard) sense there corresponds an inner set (urelement) (s)∗ in S∗ but not con-versely since the copy ∗s ∈ S∗ may also contain nonstandard elements ([16], §2. 5,pp. 44–45). It is to be noted too their ad hoc introduction in certain nonstandardtheories (e.g. the ZFBC theory) that deny the Regularity (Foundation) Axiom andthus allow for the existence of infinite ∈ -chains of sets x1 3 x2 3 ... 3 xn 3 ... . 7

Third, it gives in phenomenological perspective the dialectical opposition betweendiscreteness and continuity and moreover motivates a nonstandard approach to thenotion of continuity as a non-predicative unity of naturally intuited individualitiesformalized in parts of nonstandard mathematics by means of ad hoc prolongationaxioms in ad hoc saturated enlargements of domains of classical axiomatical systems.

6 This is rather evident in the semantical interpretation of the Alternative Set Theoryand the Internal Set Theory of E. Nelson as will be seen in next subsections.

7 The Foundation (well-foundedness) Axiom stands in conformity with the natural in-tuition that if sets are conceived of as collections of objects which in their turn could becollections of other objects and so on in infinitum there must be ultimately a collection of“things” which are not collections any more (urelements).

In another nonstandard approach the Foundation Axiom is denied in the construction ofhypersets and substituted by the Antifoundation Axiom (AFA), proposed by P. Aczel, basedon the assumption of a forgetful functor metaphor in which sets arise by means of directedgraphs (accessible pointed graphs or apgs) in a process of abstraction that “forgets” theexact nature of the constituent nodes of the graphs, the “distinguished” node at the topof the graph being a well-founded or not set b. The hypersets are defined to be preciselythe non-well founded sets that is, a set b is a hyperset if there exists an infinite descendingsequence of sets a1, a2, ..., an, ... such that ... ∈ an+1 ∈ an ∈ ... ∈ a1 ∈ b. Otherwise b iswell-founded. See ([5], pp. 36–37).

124 S. Livadas

2.3. Nonstandard models as conservative extensions of modelsof classical mathematics

Generally, extension in mathematical terms means that an object occupies a defi-nite place in “space” and is related with concepts like set, class, element, etc. Thus ifa set-theoretical formula P (x) expresses a property p then A = {x; x ∈ V and P (x)}is the extension of property p inside a set V. In the case of a mathematical object e.g.an abstract set A then any function f : T −→ A represents an extensional aspect ofA (a list of elements {f(t) : t ∈ T} of A) which is an approach used in A. Robinson’sdefinition of a higher order structure M .

In that sense the extensional part of nonstandard analysis whose significant partscan be considered Robinson’s nonstandard analysis ([20]) and Zakon’s nonstandardultrapower constructs ([21]) is thought to be fundamentally based on extensions ofthe classical Cantorian objects of mathematics whereas the intensional part of non-standard analysis on the rather subjective observations of a potential “observer” re-alized in a local and non-Cantorian way inside an intersubjective “Universe”. Theselast terms are basically meant, at least by P. Vopenka [29] in the sense of Husserl’sLebenswelt as it was developed mainly in Krisis der Europaischen Wissenschaften unddie transzendentale Phanomenologie [12].

Robinson’s quest of ideal elements, in a model theorist’s saturation approach, isimplemented inside the domain of consistent enlargements of (standard) axiomaticalstructures in a way that is basically in accordance with both Leibniz’s idea of ex-tended mathematical universe in the sense of preservation of “standard” qualities inthe extended one and Husserl’s idea of consistent enlargements of (relatively) defi-nite formal-deductive systems developed in subsection 2.1. With respect to Husserl’sGottingen approach the nonstandard elements are “imaginary” with respect to anysentence inside the domain of a standard system and cannot by themselves decideany assertion inside the standard domain. Further, if A and B are two consistentsystems of axioms and B extends A then B cannot prove any proposition P in thelanguage of A that, when restricted to the domain of A, A itself cannot decide (proveor disprove).

In Robinson’s nonstandard language this means that no proposition can be provedinside a B-model of the B-enlargement HB = K

⋃KB of a stratified set of sentences

K which when restricted in all its variables to the domain of K cannot be decided inthe model M of K ([20], Ch. II, pp. 33–34). Generally, the Extension and Transferprinciples with respect to a standard set S and its nonstandard extension S∗ cannow be seen from a certain point of view to be proper formalizations in nonstandardcontext of the Husserlian views on the extendibility of relatively definite manifolds.8

8 If a standard infinite set S is given and also the superstructure V (S) on S, then anonstandard model for V (S) consists of:

• a superstructure V (S∗) on a nonstandard extension S∗ of S and

• an embedding ∗ : V (S) ↪→ V (S∗),

that satisfy the following axioms:

• Extension Principle: S is a proper subset of S∗ and (∀s ∈ S) [s∗ = s]


It is to be noted though, that while Husserl was regarding imaginary elements asrather inaccessible to intuition, Robinson’s nonstandard numbers are, in fact, “by-products” of theoretical constructions involving universal quantifier formulas insidean essentially subjective in infinitum horizon of finite sets of constants occuring in astratified set of sentences K ([20], Ch. II, §2.9.) and are correspondingly intuited asexceeding any standard entity of common intuition.

And, naturally, even if A. Robinson stated that on an ontological level nonstan-dard numbers are no less real than standard irrationals, the new deductive proceduresintroduced stand in contrast with the platonism of the actual infinity principle em-bodied in Cantorian mathematics and further introduce implicitly a kind of shift ofthe “horizon” of natural intuition. There exists, though, a critical detail in mostnonstandard constructions that we should not overlook:

Concerning Robinson’s introduction of nonstandard elements by the constructionof B-enlargements of standard models or Zakon’s set-theoretical, non-constructiveversion of equivalence classes of infinite sequences modulo an ultrafilter over the set ofnatural numbers we note a use of the Axiom of Choice and its logical equivalent Zorn’slemma9. In the latter case Zorn’s lemma is used to ensure the existence of an ultrafilterextending the Frechet filters of all cofinal subsets of natural numbers whereas in theformer case an explicit use is made of the Axiom of Choice and Zorn’s lemma inthe proof of the Finiteness Principle by means of an ultraproduct constructed asa model for a set K of sentences ([20], theor. 2.5.1, pp. 13–19). The FinitenessPrinciple (or Compactness Theorem) for the first-order Predicate Calculus to whichthe corresponding principle for higher order theories is reduced is, in fact, fundamentalin proving the consistency of B-enlargements ([20], Ch. II, pp. 13–34)10.

• Transfer Principle: A proposition φ in the language L(V, S) holds in V (S) iff its ∗-transfer holds in V (S∗), that is V (S) |= φ iff V (S∗) |= φ∗.

See [24].9 An alternative approach in the construction of a nonstandard extension of the system

T of a standard type theory is offered by P. Martin-Lof in ([15], pp. 164–168) by adjoiningthe axioms for a single choice sequence α = f0(f1(f2(...))), that is,

• αi ∈ Ai

• αi = fi(αi+1) ∈ Ai

for i = 0, 1, ..., where each Ai is a non-empty set containing at least an element αi for each i.But then, one deals with the intuitionistic notion of a choice sequence where strong con-

tinuity principles must be assumed in case these objects are implicated in certain logicaloperations, (see [26], §2.6, pp. 14–19).

10 These classical infinity axioms (Axiom of Choice or Zorn’s lemma) are also used in theconstruction of proper ultraproducts in the proof of a strong extension principle that leadsto a κ-saturated enlargement of a universe U in nonstandard ZFBC theory where again asin the theory of hypersets the well-foundedness axiom is invalid and substituted by a globalchoice and a superuniversality axiom.

A universe W is κ-saturated, if for every system {Ai|i ∈ I}, Ai ∈ W , |I| < κ, such that⋂i∈J Ai 6= ∅ for every finite J ⊆ I, one has

⋂i∈I Ai 6= ∅ ([3], pp. 747–748). Loosely speak-

ing, Ballard and Hrbacek have proved the existence of an extended nonstandard universeinside ZFBC theory which preserves a large enough supply of urelements even in the absenceof a well-foundedness axiom.

126 S. Livadas

What does the use of the Axiom of Choice and its equivalents specifically meanin the context of nonstandard theories?

On a formal-deductive level almost nothing, for A. Robinson already admits thatnonstandard models are constructed within the framework of contemporary (classical)mathematics and “thus affirm the existence of all sorts of infinitary entities” ([20],Ch. X, p. 282). To my knowledge, to a larger or lesser degree this is true for allnonstandard extensional models and generally conforms on a formal-deductive levelwith Husserl’s Gottingen view of consistent extensions of relatively definite manifolds.

But on a formal ontological level it leaves an open question:No matter how explicitly professed the scope of at least some nonstandard theories

of extensional type to circumvent or to outright reject the platonic nature of Cantorianmathematics embodied in the actual infinity axioms, they cannot avoid introducing itin the extended axiomatical system by adjoining new infinity axioms as it is the casee.g. with the introduction of the stronger than AC Global Axiom of Choice (GAC) inZermelo-Fraenkel-Boffa Set Theory with Choice (ZFBC) or “backdoor” in the detailsof proofs of fundamental theorems mostly in the application of the Axiom of Choice(or its logical equivalents) in the construction of ultraproducts, or ultrapowers ingeneral, since these very axioms conceptually presuppose a notion of actual infinity.11

This question in another context concerns also those theories of the so-calledintensional part of nonstandard analysis characterized as non-Cantorian (AST, IST,THS, etc) to the extent that they reduce the notion of uncountable infinity to ad hocprolongation principles from countable classes to the vagueness of continuum or thead hoc axiomatization of the effect of a new undefined, non-logical predicate in thelanguage of ZFC theory.

We think that this question goes deep enough to ultimately reduce to the dialec-tical opposition between discreteness and continuity of infinity which we’ll discuss toa very limited extent in a phenomenological and nonstandard context in section 3.

2.4. The phenomenological interpretation of urelementsand prolongation principles in nonstandard theories

In subsection 2.2 we dealt to some extent with Husserl’s reduction of the laws ofanalytical logic (and consequently of mathematical propositions) to prime and directevidences of experience. This led to an ultimate reduction of analytical sentencesin their most fundamental form to their constituents – “substrates” apprehended bythe intentionality of experience as irreducible individualities bearing no further innerstructure. Further, these individualities as objects of intentionality are apprehendedinside the synthetic unity of pre-predicative experience genetically constituted in theconstituting flux of consciousness of the subject.

In the following I’ll try to demonstrate that urelements in the axiomatical structure

11 An attempt has been made to introduce pseudo-ultrapowers in models of ZF theory with-out the Axiom of Choice where, nevertheless, the fundamental theorem of ultrapowers holds.This is done, though, by means of constraints that considerably weaken the mathematicalstrength of the construction as it concerns countable models of ZF and other weakeningconditions ([23], pp. 1209–1212).


of nonstandard theories can be phenomenologically interpreted by these no furtherreducible individuals – “substrates” and the prolongation principles in nonstandardextensions of classical domains as retaining their essentially individual and relationalcharacter in the induced vagueness of continuum.

Intuitively, the urelements are defined to be those mathematical objects which arenot sets so in formal sense we could define a set x to be an urelement if x = {x}.

If a theory ZFCσ is constructed as an extension of ZFC by adding a new constantU to ZFC (the Universe of ZFC) together with the axiom (∃x) [x ∈ U ], where wechoose U to be the set of urelements, then there are two kinds of sets; the sets of ZFCand the sets of the extension ZFCσ. The elements of U (the urelements) are calledthen inner sets and anything else (the sets of a set theory with urelements) outer sets.If x is an inner set then the outer set x∗ is defined:

x∗ = {y ∈ U : y ∈ x}An outer set of this form is called an internal set and any subset of U which is not ofthis form is called external set ([16], §2.3, §2.5, pp. 39–45).

If we consider, for example, a natural number n ∈ N in classical sense then byinduction there is a unique inner set x ∈ N∗ where the corresponding outer set x∗

has cardinality n (N∗ is the nonstandard copy of ). This is implemented by meansof an injective mapping i : N −→ N∗ which is not a surjection (ibid. p. 45). Thatmeans for each (standard) natural n a unique inner set (urelement) i(n) is defined inN∗ by means of i but the inverse does not hold. There are infinite numbers x∗ ∈ N∗

which are not images by i of any x ∈ N .Generally, if S and S∗ are respectively a standard and a nonstandard superstruc-

ture it has been proved (by Los theorem and Mostowski’s collapsing function) thatthere is an elementary isomorphic embedding of superstructures ∗ : S −→ S∗ whichis not a surjection (see [24]) in the sense that there are s∗ ∈ S∗ for which there arenot s ∈ S such that (s)∗ = s∗.

If s is a finite number of S then (s)∗ = {y∗; y ∈ s} = s∗ but if s is infinite then(s)∗ = {y∗; y ∈ s} ⊂ s∗ since s∗ can also contain nonstandard elements.

Trying to give a phenomenological interpretation to the urelements in nonstandardtheories we can claim at this point:

The urelements are irreducible individuals (atoms) of classical systems which inthe context of nonstandard theories preserve in isomorphic embeddings into nonstan-dard structures their essential invariance as to their individuality in-itself (ExtensionPrinciple) and their relational properties (Transfer Principle). In that sense theycan be interpreted phenomenologically as the irreducible individualities - objects ofthe intentionality of most prime experience in Husserlian sense which are objects -“substrates” of analytical propositions of most fundamental level.

It should be stressed here the fundamental importance of the existence of a suffi-cient number of urelements in the domain of classical axiomatical theories on the wayto their nonstandard extensions. This is evident in Robinson’s nonstandard analysisin the definition of an extended B-model M of a (stratified) set of sentences K, inthe sense that a sufficient number of constants (urelements) α is presupposed in thelanguage of the theory so that:

128 S. Livadas

For every element b ∈ B (roughly understood as a binary relation in usual inter-pretation) there is a constant α of the language of the theory not occurring in K suchthat the set of sentences

Xbg = Φτ (b, g, α)

holds in M , where g belongs to the set of constants which occur in the set of sentencesK ([20], §2.9, p. 34).

The same could be said about nonstandard theories that deny the existence ofFoundation Axiom in their axiomatical structure and thus allow for the existence ofinfinite ∈ -chains of sets.

In ZFBC theory, for instance, the existence of a sufficient number of urelements asa proper class Ur = {x; x = {x}} is secured by Superuniversality Axiom (BA) ([3],pp. 742–743) and it is preserved in an extended nonstandard universe W by means ofa strong Extension Principle to which we already made reference in footnote [10]. InHyperset Theory which also rejects Foundation Axiom, a structural representation ofany set is made possible by AFA that essentially ignores the cumulative hierarchy ofset formation of standard theories and reduces to the fundamental notions of points -nodes as atoms with no inner structure and to arrows between pairs of nodes (edges)(footnote [7]).

We’ll try to demonstrate also that nonstandard entities are defined in nonstandardextensions as irreducible individualities too, retaining by ad hoc means an essentialindividuality in-themselves and in their relational properties.

If we take, for example, Robinson’s introduction of the elementary nonstandardmodel of analysis R∗ then the nonstandard number α ∈ R∗ which is greater than allnumbers of R exists by the axiomatical construction of a B- model R∗ of a set ofsentences K where B = {q}. That is, the sentence

(∃q ∈ B) (∀g ∈ ∆b) (∃α)Φ(0,0)(q, g, α)

holds in R∗ where the (concurrent) constant q is interpreted as the well-known orderof the reals (the formula Φ(0,0)(q, x, y) means x < y). The nonstandard number αbelongs to the set of constants of the language but does not belong to the set Γ ofconstants which occur in the set of sentences K that hold in the standard model R.

Consequently, we can assume the number α to be an irreducible individualityin-itself “viewed” from the standard model R which preserves in the nonstandarddomain of R∗ the same relational properties as “viewed” from the “optical” field ofR. In a formal sense, for any a, b ∈ R the formula [Φ(0,0)(q, a, b)] holds in R∗ iff itholds in R.

Generally, the irreducible character of the atoms of a standard structure andtheir relational properties are preserved in elementary isomorphic embeddings intononstandard superstructures by the axiomatically postulated transfer and extensionproperties (footnote [8]).

In certain alternative nonstandard theories (e.g. AST) the preservation of theindividuality of the standard elements of the theory as atoms and of their relationalproperties is ensured by prolongation axioms which deal essentially with the behaviorof internal entities in the “horizon” and in the transfer between two kinds of indices;


the finite naturals and the infinitely large “supernaturals”. The denotation of thisprinciple as a prolongation one reflects exactly the fact that properties concerning thefirst category of indices are extended or prolonged to the second one.

In next subsection 3.1 we’ll specifically refer to the role of Prolongation Principle inthe frame of Alternative Set Theory as an axiomatical means to preserve the essentialnature and properties of elements of countable classes in the path to an ever shifting“horizon” and beyond it to the vagueness of continuum. Whereas in subsection 3.2we’ll give a similar interpretation to certain axiomatical principles adjoined to thenew nonlogical predicate standard added to classical ZFC theory.

Summarizing, we can conclude that urelements in the conceptual frame of non-standard structures can be captured by the meaning that Husserl gave in Formale undTranszendentale Logik to the objects – “substrates” of analytical sentences in the mostfundamental level of experience as no further reducible individualities which preservetheir essential individual and relational character inside the impredicative syntheticunity of experience. The invariant essential character of the atoms – objects of math-ematical propositions in the extension from standard to nonstandard structures ispreserved by ad hoc prolongation axioms or elementary isomorphic embeddings. Thenonstandard elements can only be “viewed” as individualities in-themselves by meansof the “observational” frame of the standard system and can be thought of as the newindividuals for a still higher-order nonstandard system.

We should emphasize here Husserl’s analysis in Ding und Raum ([13], §48, p. 166)of the phenomenological relevance of the concept of “points” as “visual atoms” in aprocess of fragmentation which ultimately leads to minima visibilia. He noted there“the essential similarity to itself of the visual field, on a large and small scale” andexplained that “it is obviously this immanent similarity which, as evident generic sim-ilarity, justifies the transposition of the eidetic relationships discovered, so to speak,in the macroscopic universe, to the microscopic “atoms” situated beyond divisibility”(see [19], §6.2, p. 351).

3. The phenomenological relevance of the intensional partof nonstandard analysis

3.1. Husserl’s idea of the horizon and its axiomatizationin the language of AST

In previous subsections we made reference to the fact that in the intensional casethe existence of infinitesimals or infinities has a subjective “observational” characterlinked to the modes of intentionality of an “observer’s” consciousness who “sees” theabsolute ZFC framework with a local and non-Cantorian way. In virtue of this, Alter-native Set Theory (AST) and Internal Set Theory (IST), properly interpreted, alongwith some ultrafinitist ramifications (e.g. J. Hjelmslev, S. Lavine, A. S. Yessenin-Volpin) and lately Nonstandard Class Theory (NCT) and Theory of Hyperfinite Sets(THS) [1] are generally considered as the main pillars of this alternative nonstandardapproach.

130 S. Livadas

It must be made clear that by local and non-Cantorian way of “observing” itis meant that the classical idea of actual infinity incorporated in Cantor’s systemand consequently in ZFC theory is refuted and substituted in AST by a notion ofnatural infinity which manifests itself in the “observation” of phenomena present invery large sets e.g. in the intuition of the topological continuum in the “observation”of a concrete physical surface. As a “very large set” is considered the set which fromthe classical point of view is finite but which, nevertheless, “is not located as a wholeand along with each of its elements, in front of the horizon currently limiting the viewof the given set. Here we understand “view” in much more than a visual sense, thatis, essentially as a grasping and holding of some field of phenomena.” ([30], p. 116).

On a formal-syntactical level, AST works with sets and classes as objects. Sets aredefinite, (may be very large, but) sharply defined and finite from the classical pointof view in the sense that in its universe of sets, AST accepts the axioms of Zermelo-Fraenkel theory with the exception of the axiom of infinity – it is proved in fact, byA. Sochor, to be a conservative extension of ZFfin ([22], p. 145). Classes representindefinite clusters of objects such as the class N of natural numbers in the classicalsense. Thus the Extended Universe of Sets of this theory includes some extra axiomsin addition to those of the Universe of Sets which are not set-theoretical formulas:

• The axiom of existence of classes:

For any property φ(x) of sets from the Universe of Sets the Extended Universecontains the class {x; φ(x)}.

• The axiom of existence of proper semisets:

There is a proper semiset. In the AST language,

(∃X) (Sms(X) ¬Set(X)).

In AST an important role is played by proper semisets which are classes insidesets and represent, roughly, blurriness and non-surveyability in the observationinside very large sets.

• The Prolongation principle:

For each countable function F there is a set function f such that F ⊆ f .

It is important here to have in mind that countability of a function is, in fact,countability of a class of ordered pairs of elements and that a set function canbe an uncountable set of ordered pairs of elements.

See ([29], Ch. I).As we have stated, infinity in AST sense is not beyond but already present in very

large sets e.g. in the application to certain macro or micro-scale phenomena whichshow themselves in large sets and in this sense precisely is characterized as naturalinfinity in contrast to classical infinity of Cantorian set theory; the latter becomesrelevant in relation to real world phenomena when it applies its classical infinity resultsto natural infinity. Natural infinity in its most basic form should be understood ascountable natural infinity in the sense of an initial grasp of a local finiteness and the


subsequent constant shift of the horizon of “sharpness” and “discernibility” until itstabilizes to something constantly unchangeable and definite in the sense of classicalcountability ([30], pp. 116–118).

P. Vopenka states explicitly that the term horizon is understood in the sense ofE. Husserl’s Lebenswelt linking, in effect, this notion and the AST definition of acountable class of elements: “If a large set x is observed then the class of all elementsof x that lie before the horizon need not be infinite but may converge toward thehorizon. The phenomenon of infinity associated with the observation of such a classis called countability” ([29], Ch. I, p. 39).

In a purely phenomenological sense a notion of the world as exceeding the percep-tion of things is already found in the first book of Ideen under the term of “halo” (Hof)or, more frequently, “horizon”. In that context horizon takes its meaning from whatHusserl called Horizontintentionalitat (horizontal intentionality) which originally, atleast, was characterized by the preoccupations of a psychology of “attention” in themost broad sense12. Much later the term horizon is put in a more general frame andincorporated in the notion of life-world (Lebenswelt)13 as an ever shifting “bound” ofthe noematic field of a subject existing in terms of correlation with Lebenswelt.

Thus, the countable class FN of natural numbers is considered in AST as a repre-sentative of the path towards the horizon in the formal sense of hereditary finitenessof each of its segments. The formal definition of a countable class in AST is thefollowing:

A pair < A,≤> of classes is called an ordering of type ω iff

1. ≤ linearly orders A,

2. A is infinite and

3. for each x ∈ A the segment {y ∈ A; y ≤ x} is finite.

A class X is countable iff there is a relation R such that < X, R > is an ordering oftype ω. A class is uncountable iff it is neither countable nor finite.

It is a theorem of AST that each countable class is a proper semiset thus givingan intuitive characterization of semisets as classes with hereditarily finite segments([29], Ch. I, pp. 39–42).

A drastic and qualitative shift of the horizon leading to the vagueness of naturalor intuitive continuum is induced axiomatically by the Prolongation Principle which

12 “This intentionality has multiple and diverse modes: retentional and protentional hori-zon that frame the “grasp” of an object, active and passive motivation in the accomplishmentof various acts, in the realization of the possibility of an experience over another, in the en-largement or retraction of my field of view. The horizontal intentionality is thus responsiblefor the continuity of the life of the subject [...] and the principle that integrates all acts inthe continuous unity of life, in the flux of consciousness of the subject” ([6], Ch. I, p. 99,transl. of the author).

13 “To be sure, everyday induction grew into induction according to scientific method,but that changes nothing of the essential meaning of the pregiven world as the horizon of allmeaningful induction. It is this world that we find to be the world of all known and unknownrealities. To it, the world of actually experiencing intuition, belongs the form of space-timetogether with all the bodily shapes incorporated in it; it is in this world that we ourselveslive, in accord with our bodily, personal way of being.” ([12], part II, p. 50).

132 S. Livadas

is, in fact, a saturation principle on a nonstandard model of Peano arithmetic. In asemantical-phenomenological interpretation of AST a remarkable consequence of theProlongation Principle is:

If a perceived (i.e. definable inside a intersubjective universe including an “ob-server”) state of affairs φ holds of every element of a sequence (xn), n ∈ ω, (where ωis the cardinality of countable infinity in AST) progressing towards the horizon, then(xn), n ∈ ω, is extendible to a sequence (xβ), β ≤ α, that crosses the horizon and itsmembers also satisfy φ.

Put in a somewhat less formal language, any totality of elements as atoms per-ceived in a phenomenological sense in front of the horizon of AST countability canextend beyond the horizon preserving the prior characteristics of its elements ([27],p. 394). In that sense any segment of a countable class can be perceived as a totalityof urelements which can be extended beyond the horizon of countability retaining theessential characteristics of its elements (e.g. individuality, ordering).

This is a saturated nonstandard enlargement in the sense of nonstandard theo-ries already discussed, only that P. Vopenka puts the emphasis on the original andstraightforward way that countable infinity is interpreted inside AST as hereditaryfiniteness towards a phenomenological horizon that incorporates axiomatically non-standardness.

3.2. The interpretation of IST nonstandard approach

In relation to Internal Set Theory, which is described by E. Nelson as a variantof A. Robinson’s nonstandard analysis on the syntactical level “without enlarging the

world of mathematical objects in any way” ([16], Ch. I, p. 1), there is a tendency tosee it as part of intensional nonstandard mathematics. In reaching this conclusionit helps to refer to Nelson’s introduction of a new nonlogical predicate standard inthe language of ZFC theory which along its axiomatical tools can be interpreted asindirectly inducing the presence of an “observer” in an intersubjective universe.

The axiomatical equipment of the new undefined predicate standard which playsa rather syntactical role consists of the following three axioms where by the terminternal formula is meant a formula of ordinary mathematics which does not involvethe predicate standard even indirectly otherwise it is called external:

• The Transfer Principle (T):

∀stt1 . .∀sttn[∀stx A ←→ ∀x A] where A is an internal formula whose only freevariables are x, t1, . . ., tn.

The intuition behind (T) is that if something is true for a fixed, but arbitrary,x then it is true for all x.

• The Idealization Principle (I):

∀stfin x′∃y∀x ∈ x

′A ←→ ∃y∀stx A where A is again an internal formula.

To say that there is a y such that for all fixed x we have A is the same as sayingthat for any fixed finite set of x’s there is a y such that A holds for all of them.Put more naively, we can only fix a finite number of objects at a time.


• The Standardization Principle (S):

∀stX∃stY ∀stz [z ∈ Y ←→ z ∈ X and A] where A is any formula external orinternal.

Intuitively, we can say that if we have a fixed set, then we can specify a fixedsubset of it by giving a membership criterion for each fixed elementSee ([16], pp. 3–11).

In line with the phenomenological interpretation of urelements and prolongationaxioms in nonstandard theories proposed in subsection 2.4 we could interprete the Ide-alization Principle as prolonging the “horizon” of finiteness of a collection of standardelements by preserving their standard (“fixed”) character whereas Transfer Princi-ple could be interpreted as prolonging indefinitely the “horizon” of standard elementswith respect to their essentially individual and relational character expressed in termsof the classical (standard) system ZFC.

Evidently, in spite of their syntactical role in the the theory these three axiomsinduce a nonstandard extension in the domain of “fixed” elements where the term“fixed” in informal mathematical discourse can be used as an intuition of the newpredicate standard.

By this axiomatical means, specifically by the idealization principle (I), it is easilyproved that there exists a nonstandard number in any infinite set including the setof natural numbers N ([16], Ch. I, p. 5). As an outcome, continuity and topolog-ical properties involving new nonstandard definitions of infinitesimal and unlimitednumbers are not related to any particular mathematical model but, instead, they areintroduced by the enriched language of IST.

This theory is proved to be a conservative extension of ZFC theory in the sensethat every internal statement (a statement expressed in the language of ZFC) whichcan be proved in IST can be also proved in ZFC ([17], pp. 1192–1197).

Though it is not in the scope of the present article to deal in depth with thetheoretical problem of the dialectical or complementary opposition of discreteness andcontinuity on any interpretational level it happened, as a matter of fact, to remarkthat there is an underlying notion of actual infinity as a “background frame” in anyapproach introducing new nonstandard entities by consistent enlargements of classicalaxiomatical systems.

This was manifest in nonstandard theories – retaining as valid the FoundationAxiom or not – in the use of the Axiom of Choice and Zorn’s lemma in the constructionof proofs of fundamental enlargement theorems. In the intensional case it was by anad hoc axiomatical principle (Prolongation Principle in AST) or by the introductionof a new nonlogical predicate with ad hoc axiomatical tools (the predicate standard inIST) that the countable infinity horizon was shifted to the vagueness of uncountableinfinity.

It is our view that there exists a factor of impredicativity implicated in the no-tion of intuitive and consequently mathematical continuum which is reflected in itsindescribability in any first-order language. This impredicativity might be intepretedphenomenologically in terms of the synthetic unity of pre-predicative experience ge-netically constituted in the flux of consciousness of the subject. If phenomenological

134 S. Livadas

analysis is to prove of any further significance relative to the core of the matter weshould inquire into how one is led to the self-constituting continuous unity of the fluxof time consciousness out of discrete multiplicities of appearances (Erscheinungen) orimmanences of “objects” in it (see [11]).14

4. Conclusion

As the scope of this paper was to interpret the formal-axiomatical structure ofnonstandard mathematical theories in terms of a phenomenological analysis we at-tempted to do this on two levels: The formal-deductive and the ontological one.

In relation to the first one we remarked the convergence on a fundamental levelbetween the Husserlian view of imaginary elements as new elements in the domain ofconsistent enlargements of relatively definite manifolds and the introduction of non-standard elements in consistent (or conservative) enlargements of classical axiomaticalsystems.

Our analysis was oriented next into how analytical principles are reduced to directevidences in the lowest degree of pre-predicative experience in a phenomenologicalperspective. In so doing, we put under this perspective nonstandard methods thatintroduce new objects which stand beyond the horizon “viewed” from the scope ofthe domains of classical systems to which they are imaginary in the formal sense ofHusserl’s Gottingen views.

On a formal ontological level these new objects can be interpreted as irreducibleindividualities in a new level of reality which is an intentional correlate of the mostprime experience in the sense described in subsection 2.2. They are elements of ex-tended domains of standard systems retaining their former essential characteristics bymeans of ad hoc axiomatical principles but existing in a beyond common mathematical(and perceptual) intuition horizon. In Alternative Set Theory this was defined to bethe horizon of natural infinity put in formal terms as the horizon of a countable classinfinity and motivated by the phenomenological notion of the horizon of the life-worldin. In a structuralist approach based largely on category-theoretical notions talk ismade about the introduction of even new levels of reality by nonstandard extensions(see [8]).

Whether it can be asserted that phenomenological analysis is a deeper theoreticalunderpinning of nonstandard analysis of either an extensional or intensional specifi-cation it can be certainly argued that it provides an important theoretical tool to anew more natural, more intuitively motivated and thus non-Cantorian approach ofmathematics.

14 On the circularities of continuity involved in a mathematical modelization of the in-tentionalities of the flux of consiousness see J. Petitot’s Morphological Eidetics for a Phe-nomenology of Perception [19].


References

[1] ANDREEV, P., GORDON, E., A Theory of Hyperfinite Sets, arXiv:math.LO/0502393v1, full text available at: http://arxiv.org/abs/math/0502393, 2005.

[2] BALLARD, D., Foundational aspects of “Non”standard Mathematics, Amer. Mathe-matical Society, Providence, 1994.

[3] BALLARD, D., HRBACEK, K., Standard foundations for nonstandard analysis, TheJournal of Symbolic Logic, vol. 57, no. 2, 1992.

[4] BARWISE, J., Admissible sets and structures, Springer, Berlin, 1975

[5] BARWISE, J., MOSS, L., Hypersets, The Mathematical Intelligencer, vol. 13, no. 4,Springer-Verlag New York, 1991.

[6] BERNET, R., La vie du sujet, ed. PUF, Paris, 1994.

[7] DA SILVA, J. J., Husserl’s two notions of completeness, Synthese, 125, pp. 417–438,2000.

[8] DROSSOS, C., Structure, Points and Levels of Reality, Essays on the Foundations ofMathematics and Logic, Polimetrica Inter. Scient. Pub., ed. G. Sica, pp. 83–114, 2005.

[9] HUSSERL, E., Philosophie der Arithmetik, Husserliana XII, ed. Lothar Eley, M. Nijhoff,Den Haag, 1970.

[10] HUSSERL, E., Logique formelle et logique transcendantale, transl. S. Bachelard, PUFParis, 1984.

[11] HUSSERL, E., Lecons pour une phenomenologie de la conscience intime du temps,transl. H. Dussort, PUF Paris, 1996.

[12] HUSSERL, E., The crisis of European sciences and transcendental phenomenology,transl. D. Carr, Northwestern Univ. Press, 1970.

[13] HUSSERL, E., Ding und Raum: Vorlesungen, Husserliana 16, Ed. U. Claesges, TheHague: M. Nijhoff, 1973.

[14] LAVINE, S., Understanding the infinite, Harvard Univ. Press, 1994.

[15] MARTIN-LOF, P., Mathematics of infinity, Lecture notes in Computer Science,Springer, vol. 417, pp. 146–197, 1990.

[16] NELSON, E., Predicative arithmetic. Mathematical notes., Princeton Univ. Press,Princeton, 1986.

[17] NELSON, E., Internal Set Theory: A new approach to nonstandard analysis, Bulletinof the American Mathematical Society, vol. 83, 6, 1977.

[18] ORTIZ HILL, C., Tackling three of Frege’s problems: Edmund Husserl on sets andmanifolds, Axiomathes, 13, pp. 79–104, 2002.

[19] PETITOT, J., Morphological Eidetics for a Phenomenology of Perception, NaturalizingPhenomenology, Ch. 11, pp. 330–372, Stanford University Press, 1999.

[20] ROBINSON, A., Non-standard Analysis, North-Holland Pub. Company, 1966.

[21] ROBINSON, A., ZAKON, E., A set-theoretical characterisation of enlargements, Ap-plications of model theory to algebra, analysis and probability, pp. 109–122, (W. A. J.Luxembourg, editor), Holt, Rinehart and Winston, New York, 1969.

[22] SOCHOR, A., Metamathematics of the Alternative Set Theory III, CommentationesMathematicae Universitatis Carolinae, vol. 24, pp. 137–154, 1983.

136 S. Livadas

[23] SPECTOR, M., Ultrapowers without the axiom of choice, The Journal of SymbolicLogic, vol. 53, 4, 1988.

[24] STROYAN, K.D., LUXEMBOURG, W.A.J., Introduction to the Theory of Infinitesi-mals, Academic Press, New York, 1976.

[25] TIESZEN, R., Mathematical Intuition and Husserl’s Phenomenology, Nous, 18, no. 3,pp. 395–421, 1984.

[26] TROELSTRA, A., Choice sequences. A chapter of Intuitionistic Mathematics, Claren-don Press, Oxford, 1977.

[27] TZOUVARAS, A., Modeling vagueness by nonstandardness, Fuzzy sets and systems,vol. 94, pp. 385–396, 1998.

[28] VAN ATTEN, M., VAN DALEN, D., TIESZEN, R., BROUWER, L.E.J., WEYL, H.,The Phenomenology and Mathematics of the Intuitive Continuum, Philosophia Mathe-matica, (3) vol. 10, pp. 203–226, 2002.

[29] VOPENKA, P., Mathematics in the Alternative Set Theory, Teubner, Leipzig, 1979.

[30] VOPENKA, P., The philosophical foundations of Alternative Set theory, Int. J. GeneralSystems, vol. 20, pp. 115–126, 1991.

[31] ZARACH, A., Unions of ZF− models which are themselves ZF−, Logic Colloquium ’80,108, Studies in Logic and the foundations of Mathematics, pp. 315–342, North-HollandPub. Company, 1982.

the phenomenological roots of nonstandard mathematics

Documents