Is There an Ontology of Infinity?

Is There an Ontology of Infinity? Dr. Stathis Livadas Independent Scholar e-mail:[email protected] Patras, Greece April 13, 2020 Abstract In this article I try to articulate a defensible argumentation against the idea of an ontology of infinity. My position is phenomenologically motivated and in this virtue strongly influenced by the Husserlian reduction of the ontological being to a process of subjective constitution within the immanence of consciousness. However taking into account the historical charge and the depth of the question of infinity over the centuries I also include a brief review of the platonic and aristotelian views and also those of Locke and Hume on the concept to the extent that they are relevant to my own discussion of infinity both in a purely philosophical and epistemological context. Concerning the latter context, I argue against Kanamori’s position, in The Infinite as Method in Set Theory and Mathematics, that the mathematical infinite can be accounted for solely in terms of epistemological articulation, that is, in the way it is approached, assimilated, and applied in the course of the construction of mathematical hierarchies. Instead I point to a subjectively constituted immanent ‘infinity’ in virtue of the a priori as well as factual characteristics of subjective constitution, underlying and conditioning any talk of infinity in an epistemological sense. From this viewpoint I also address some other positions on the question of a possible ontology of the mathematical infinite. My whole approach to the question of the infinite in an epistemological sense hinges on the assumption that the mathematical infinite subsumes the infinite of physical theories to the extent that physics and science in general deal with the infinite in terms of the corresponding mathematical language and the specific techniques involved. Keywords: A priori; continuous unity; entity; factual; immanent infinity; mathematical infinite; ontology; ordinal; set theory; temporal consciousness; universe of sets. 1 Introduction Is there indeed an answer to the question of the title? And if so can it be a sound and well-founded one in case it does not involve a multi-level discussion across pure philosophy, starting at least from Greek antiquity, through to modern epistemology in taking into account recent advances in the entire epistemological edifice? After all such an answer 1 would transcend the realm of absolute being in the platonic sense corresponding to the Greek term εἶναι and enter into a dialectic composition with an epistemological sense of being corresponding to the Greek term γίγνεσθαι. This is the primary intention of this article and its ultimate goal is to provide a convincing argumentation against the possibility of founding infinity in the ontological sense of εἶναι from the vantage point of phenomenological analysis, something that implies bringing into the fore the process and the a priori modes of subjective constitution. In this motivation the concept of infinity as it is basically understood (leaving aside metaphysical idealizations), starting from the Greek classical texts to our days, may in fact be divided into two radically different subspecies: the one taken to be the ‘closed’ infinity conceived as the immanent1 content of the flux of consciousness inexhaustibly extended over any conceivable boundary without any spatiotemporal and causal constraints, and the real world ‘infinity’ which is not really an infinity but an objective spatiotemporally founded and indefinitely extensible external reality which exists as such and such according to definite causal laws derived by human experience underlying a specific presence in the world. In short, the main argument to be defended is that it is impossible to provide a foundation of infinity in purely ontological terms insofar there exists a world, thought of in a most primitive sense, in which which human egos are included as subordinate realities and yet this world cannot be taken in an objective sense but as a secondary being posited by a constituting consciousness which ‘exists’ unconditionally in itself, i.e., one which “nulla ‘re’ indiget ad existendum”. This is done in the Sections 1 and 3 in which I mostly rely on the original Husserlian writings, namely in Ideas I, [19], Logical Investigations, [21], Lessons on the Phenomenology of Inner Time Consciousness, [14], Experience and Judgment, [12], Late Texts on Time Constitution, [23], On the Theory of Intersubjectivity, [17], etc. As a matter of fact Husserlian phenomenology in particular in its post-psychologistic, transcendental phase, is the backbone of my overall argumentation, though not restrictively, against an ontology of infinity and to a significant extent the leit motiv of the discussion of infinity as epistemologically articulated, e.g., in relation to Kanamori’s views in [25]. This is mostly based on the fact that a phenomenologically motivated approach toward the concept of infinity in general not only helps unfetter infinity from idealistic illusions in an ontological sense, but can also epistemologically account for a sound meaning of the concept in view of its status in formal-mathematical theories. In these terms I give a special attention to the fact that a subjectively founded conception of infinity in the sense of an objective fulfillment in actual presentation entails both a priori and factual characteristics thus offering grounds to refute ipso facto any ontologically based counterarguments (Section 2). Having in mind that the question of the infinite is a key philosophical preoccupation over centuries of rational thought, I thought it purposeful to briefly comment on some key ideas of Plato and Aristotle on the infinite in Timaeus, Parmenides (Plato), Physics, Metaphysics I-IX, and Metaphysics X- XIV (Aristotle). I have also given a special attention 1 The term immanent, widely used in phenomenological texts, can be roughly explained as referring to what is or has become correlative (or ‘co-substantial’) to the being of the flux of one’s consciousness in contrast to what is ‘external’ or transcendent to it. For instance, a tree is transcendent as such to the consciousness of an ‘observer’ while its appearance in the modes of its appearing within his consciousness is immanent to it. 2 to the aristotelian treatment of the infinite in Physics in relation to my own subjectively oriented position and the possibility of a reduction within immanence. This is done in Section 5. Finally taking into account the literature discussion involving the epistemological aspects of the infinite I engage in Section 6 in a thorough treatment of the infinite principally in logical-mathematical terms, in which case I point to an implicit assumption of a kind of impredicative substratum, possibly reduced to a subjectively constituted ‘closed’ infinity, which circularly reproduces itself in the ascension toward higher order infinities in the context of large cardinals theory. It should be said that this particular section involves some level of knowledge of standard and higher order set theory and consequently could be skipped, at least in its strictly technical details, by the more philosophically oriented reader without losing the larger philosophical picture. Also a special mention and account is made of the work of A. Badiou on a presumed ontology of mathematical infinity in his well-known treatise Being and Event [1]. One could possibly include the formal syntactical ways, along with the generated semantical content, in which various alternative mathematical theories treat the question of the infinite. But given the general intention of the paper, this would go a bit far away both in content and length and lean the balance too much toward the epistemological aspect. In any case I find nothing in the elaboration of my arguments that would not vindicate my general position also with regard to alternative mathematical theories. The reader will find no particular discussion of the infinite in physical theories as such, for instance in general relativity theory, to the extent that physical and positive science in general deal with the issue in terms of their mathematical metatheory, in which case infinity as treated in terms of the physicalistic language alone is actually a sort of ever shifting ‘horizon’ of the predominant each time physical model. 2 A phenomenologically motivated discussion of infinity Husserl had stated that what is called phenomenon has its origin a standing-flowing self-presentification, that is, the self-presenting, flowing absolute ego in its standing-flowing life, a life that is a constant living-experience, intentional and conscious apprehension, a standing-flowing validation of being (Seinsgeltung) in multiple modalities, a validation of being with content or meaning which inherently belong to the substance of the flowing ([23], p. 145). This position takes into account that any phenomenon in real world terms cannot be thought of as detached from the world conceived in the sense of life-world2 with an open ended horizon in a definite subjectively reducible sense and in relation to at least one phenomenological reduction performing subject. On these grounds, the infinity of the world, according to Husserl’s views in the Late Texts on Time-Constitution (Späte Texte über Zeitkonstitution), [23], is an open-endedness, the motivated possibility of new experiences and determinations through experience. Husserl considered that this infinity, 2 The phenomenological notion of life-world can be described in rough terms as an indefinitely extensible horizon of our special reduction-performing co-presence in the world, this latter meant as the primitive soil of our experience. A major work in which this notion is further elaborated is Husserl’s well-known Crisis of European Sciences and Transcendental Phenomenology [20]. 3 thought of as ‘external’, has its counterpart in the inner infinity which ‘belongs’ to each particular real being. It is only bounded by the scope of my own ‘I can’, this latter thought of course in eidetic terms.3 Space is constituted as a reality through my respective all-around possibility of its unfolding and this possibility has its own subjective horizon essentially associated with inner temporality. Space is not bounded in an exact sense yet it has a finite limit (in accordance with a subject’s own finiteness).4 For Husserl, in his final and predominant phase of transcendental phenomenology, infinity as subjectively constituted and further objectified as an immanence in consciousness is rooted in inner temporality in contradistinction to the ‘external’ infinity of the material world, the infinity of physical theories, which is in fact an ever receding horizon of the finiteness of the life-world, the latter being convergent but not identical to its forms of mathematization. In this temporally founded perspective a presumed ontology of infinity would be de-constructed in favor of the subjective character of inner time constitution conceived in the level of constituting and not that of constituted to which belongs the objective, scientific time. As a matter of fact this reduction of infinity, as temporally founded, to an absolute subjective source would entail in turn other questions pertaining to the impossibility of an ‘ontology’ of the absolute origin of temporal consciousness, But this is a deep enough question which is purely philosophical in character and, except for the original Husserlian texts, has been treated to a certain extent elsewhere.5 In any case even though Husserl did not write much on the concept of infinity proper,6 he had made clear this kind of subjective reduction of infinity throughout his texts even alluding to the possibility of ‘men’ of other planets and other galaxies, on the presupposition of sharing the same or analogous eidetic attributes, of ‘co-founding’ an infinite, homogenous spatiotemporal world (ibid. p. 373). At another point in the Theory of Intersubjectivity, Third Part, Husserl referred to the possibility of a subjective reduction of infinity in the following implicit terms: “Meine endliche Zeit ist doch unüberschreitbar auch in dem, was ich früher hätte aktualisieren können, wie in dem, was ich künftich verwicklichen kann. Aber ich tue so, als hätte ich Zeitflügel, als hätte ich ein Vermögen der Bewegung durch alle Zeiten, als könnte ich eine Einheit der Weltanschauung, einer möglichen Welterfahrung konstruieren als mir eigene, in der ich in unendlicher Immanenz 3 By eidetic laws or eidetic attributes in the world of phenomena one can roughly communicate to a non-phenomenologist what relates to the existence of objects or states-of-affairs as regularities by essential necessity and not by mere facticity. One may also consult E. Husserl’s Ideas I, ([19], Engl. transl., pp. 12-15). 4 “ Die Unendlichkeit der Welt ist diese Offenheit, diese motivierte Möglichkeit neuer Erfahrungen und erfahrender Ausweisungen. Diese Unendlichkeit, hier als äussere gedacht, hat ihr Gegenstück in der inneren Unendlichkeit, die zu jedem einzelnen Realen gehört. Sie ist begrenzt durch den Umfang meines ‘Ich kann’. Der Raum ist als realer konstituiert mit meinem jeweiligen Allseitig-ihn-erschliessen-können, und dieses Können hat seinen eigenen subjectiven Horizont, der zwar nicht voll exakt umgrenzt ist, aber doch einen endlichen Limes hat.” ([23], p. 164). Transl. of the author: “The infinity of the world is this openness, this motivated possibility of new experiences and determinations through experience. This infinity, here thought of as external, has its counterpart in the inner infinity which belongs to each particular real being. It is bounded by the scope of my ‘ I can’. The space is constituted as a reality through my respective all-around possibility of its unfolding and this possibility has its own subjective horizon which is not bounded in an exact sense yet it has a finite limit.” 5 See, e.g., [33]. 6 An explicit reference is found in the first paragraph of Section 3. 4 alle Unendlichkeiten der Erfahrung durchlaufen könnte.”; [18], pp. 239-240. (Transl. of the auth.: My finite time is thus insurmountable also in what I had prior been able to actualize, as also in what I will be able to realize in the future. Yet I do so, like I had time-wings, like I had a property of motion through all times, like I could have a unity of the intuition of the world, (like) I could construct a possible world experience as my own, in which I could run through all infinities of my experience in the infinite immanence). In Husserl’s view every science of being can be transformed into a ‘metaphysics’ insofar as it is associated with the phenomenological knowledge of essences and on this ground acquires ultimate meaning clarification and also ultimate determination of its truth-content. This is a metaphysics that owes its origin to the essence of knowledge and correlatively with it to a twofold knowledge attitude: the one purely oriented to the being itself as appearing, perceived and thought of by consciousness, the other oriented to the enigmatic essential relations between being and consciousness (Phänomenologische Psychologie, [15], Einleitung des Herausgebers, p. xx). In the sense of being as a phenomenon Husserl associated, as said in the first paragraph, the notion of a non-reducible primordial phenomenon whose origin is the presentifying, absolute ego in its standing-flowing life. The purely subjective (factor) as reflection of a ‘higher degree’, as consciousness of what stands in virtue of ontological essence, is characterized as the temporalized living consciousness in the universal temporality, the temporalized infinite flux of consciousness in which are included re-presentations and appearances of what is ontological in their flowing forms and re-identifications ([23], p. 362). Consequently questions of pure ontology are reduced to the question of the modes of an individual actual and possible experience, as part of the universal experience, appearing as ‘living experience’ in the stream of my inner temporality (ibid., p. 357). Relevant to the phenomenology of inner time-consciousness, in the well-known description of the unity of the flux of consciousness in terms of the transversal and longitudinal intentionality, Husserl posed the question of whether there is an ultimate now which has no past in advance of the enactment of the a priori conjunction just passed by - original now - not yet. In case of a positive answer this would lead, on the ground of evidence, to the possibility of a de facto ‘empty’ time, a time which would be totally incompatible with the presence of a subject with a priori constitutional-eidetic capacities ([22], p. 64). More than this, a subjective presence implies that there is no conceivable notion of time but timeas-fulfilled in a way that each givenness to phenomenological perception (Wahrnehmung) is necessarily extended in time and not simply a temporal punctuality. It is due to the temporal essence of phenomenological perception that to each necessarily prevalent now is associated a gradual ‘descent’ (of retentions) into a haziness which does not appear as such essentially (ibid., pp. 34-35). The allusion, on the one hand, to this de facto character of the gradual vagueness of the retentional tails to the past and, on the other, the appeal (among others) to rememoration (Wiedererinnerung) as an essential mode of the temporal flux of consciousness are going to be accounted for in the next, insofar as they provide strong clues to the belief that an ontology of infinity can be refuted solely on phenomenological, even simply subjectively founded grounds. For now and in view of my intentions, I point out also from Husserl’s Lessons on the Phenomenology of Inner Time-Consciousness, [14], that the continuity of the modes of procession of the duration of an object is juxtaposed to the continuity of the modes of 5 procession of each ‘point’ of duration itself in the sense that the former are determined by the continuity of the modes of procession of the latter. In this respect and in what may be thought a critical association of infinity as generated by inner temporality with infinity-in-spatiality, every temporal point of an object has its fulfillment, its temporal ‘thisness’ of content and thus as a matter of fact a spatial extension ([23], p. 64). Husserl attributed the co-existence form of spatiality to the inner form of temporality in the sense that inner temporality makes possible that various world objects in any interval of the same temporality co-exist and in this way the same temporal points and time-intervals with the same corresponding objective contents are fulfilled. It is remarkable that in the Late Texts on Time Constitution Husserl gave a founding priority, concerning the selfconstituting character of inner temporality, to the essential form of constituting present (or living present) which he indirectly linked to the essence of the transcendental ego (ibid., p. 4). Talking at the level of constituted, each new now in the temporal flux is transformed into a continuous tail of retentions fading away into a hazy past and yet a durating object as immanent unity can be recalled by rememoration as an actual now which is not authentically its original presence (Gegenwärtigung) but its re-presented presence (Vergegenwärtigung) generating anew a continuous sequence of retentions and so on. Underlying the capability of presentifying a durating object as a noematic7 one, which encloses also the continuity of its appearances over duration, is what Husserl termed the double intentionality of consciousness, namely the form of transversal intentionality in the a priori scheme retention-original presence-protention and that of longitudinal intentionality. The first one makes that each original impression is a priori tied with primary memory (i.e., retention) and a-thematic expectedness (protention) while the second one makes that the sequence of retentions, of retentions of retentions and so on as well as ultimately the temporal flux of consciousness itself are constituted as a whole in continuous unity. It turns out that longitudinal intentionality in this a priori sense invalidates eo ipso any talk of a possible traversability of infinity in objective terms as immanent fulfillment within consciousness. To the extent that the double intentionality, as an a priori intentional form making possible the unity of the absolute flux of consciousness as inexhaustibly fulfilled by ever new objects as immanent appearances, may found the possibility of immanent continuous unity independently of any spatio-temporal constraints raises in turn the question of its proper objectivity in real world terms. K. Michalski refers in [34] to the double intentionality of temporal consciousness in these terms: It is thanks to the fact that retention is characterized by a ‘double’ non-objective intentionality that the consciousness of temporal succession is at all possible ([34], p. 138). In fact as intentionality cannot be thought of independently of what is intended, and this latter even if it is simply an object of imagination cannot be but a being-in-objectivity, it is really an issue whether intentionality in general and a a fortiori 7 A noematic object, a phenomenological term, is an object as meant constituted by certain modes as a well-defined object immanent to the temporal flux of a subject’s consciousness. It can then be said to be given apodictically in experience inasmuch as: (1) it can be recognized by a perceiver directly as a manifested essence in any perceptual judgement (2) it can be predicated as existing according to the descriptive norms of a language and (3) it can be verified as such (as a reidentifying object) in multiple acts more or less at will. More in Husserl’s Ideas I: [19], pp. 229-232. 6 the double intentionality of consciousness can be regarded as a priori, non-objective acts themselves or just clues on the level of constituted of an underlying absolute, subjective origin of inner temporality. In Experience and Judgment Husserl stated that the back and forth passing into states of consciousness (e.g., passing to the state of rememoration spontaneously and in instantaneity) presupposes in a certain way the conception of an ‘infinite’ time. On this apparently ‘naive’ assumption he went on to conclude that the difficult problems of the apprehension of absolute temporal determinations of objects, the constitution of their location in objective time, and in general the manifestation of the continuity of absolute objective time by means of the subjective time of lived experiences, all this constitutes the great theme of a more worked out phenomenology of time-consciousness ([12], Engl. transl., p. 167). At this point I take advantage of the preceding discussion to point to the impasse reached in trying to found an ontology of infinity in the absolute sense of being (εἶναι), especially if one clings to the general position that “[..] the spatio-temporal infinity of the world requires the endlessness of the absolute consciousness”, (transl. of the author of: “[..] die räumlich-zeitliche Unendlichkeit der Welt fordert Endlosigkeit des absoluten Bewusstseins”) ([17], p. 17). As absolute consciousness is for Husserl the residue left over from the annihilation of the world of experience, it is therefore considered ‘in advance of’ objective time and may be consequently thought of as the non-objective temporal ground for the constitution of ‘infinite’ time and for an extended in (‘infinite’) time ‘infinite’ world. 3 Actual infinity as subjectively founded Husserl described immanent infinity, in the following terms as the unconstrained extension of any conceivable spatial and temporal stretches in imagination “The fact that we freely extend spatial and temporal stretches in imagination, that we can put ourselves in imagination at each fancied boundary of space or time while ever new spaces and times emerge before our inward gaze - all this does not prove the relative foundedness (Fundierung) of bits of space and time, and so does not prove space and time to be really infinite, or even that they could be really infinite, nor even that they really can be so. This can only be proved by a law of causation, which presupposes, and so requires, the possibility of being extended beyond any given boundary” ([21], Engl. transl., p. 45). In fact this kind of immanent infinity was recognized in a certain sense as such from Husserl’s earlier so-called psychologistic stage at the time of the Philosophy of Arithmetic. In that work, preceding his later espousing of transcendental phenomenology, he referred to inner experience as the evident factor for the possibility of representing a multiplicity of objects as an instantaneous lived experience ([16], p. 24). He claimed also that temporal succession8 is what precisely characterizes multiplicity as multiplicity although this succession is attainable only to the extent that contents are found in whatever simple or complex relationship and thus can be put together as a whole in mental re-presentation (ibid., p. 25). 8 Temporal succession is thought of in the Philosophy of Arithmetic as the psychological pre-condition for the built up of number meanings and generally of the meaning of multiplicities ([16], p. 28). 7 In his later works Husserl was increasingly intent on establishing the foundation of continuous unity, as thematically presented in reflection, on the modes of the self-constituting temporal consciousness. In Experience and Judgment, in particular, he explicitly referred to a special kind of constitution of unity that provides the basis for special relations, having specifically in mind formal relations ([12], pp. 153-154 & pp. 188-189).9 In an apparent connection to his preceding ideas on the concept of formal ontological objects in Formal and Transcendental Logic, he called this kind of unity a formal-ontological one and it was described as a unity extending to all possible individuals or non-individuals being originally given as objects independently of a concrete material content. The unified whole of a collection of objects as re-presented becomes thematized and therefore objective in the continuous apprehension of these objects one by one and in their totality. In this sense, one may provide a subjective foundation to universal-existential predicate formulas of an indefinite range inasmuch as a proposition of the kind ‘each and every thing (everything possible and hence everything actual) such that....’ is possible to be intuited by consciousness as equivalent to the metatheoretical proposition: ‘each and every thing such that... is, in principle, capable of being colligated’. As a further clue that this kind of unity might be reducible to an original form of the self-constituting temporality of the transcendental ego, Husserl characterized this collective unity as essentially founded not on material elements nor on the essence of things as this latter is taken into consideration only insofar as it makes differentiation possible ([12], engl. transl., pp. 188-189). Rather to make a collection of objects (e.g. a set of objects or a class of sets) a thematic objectivity itself in actual presence within immanence, an act of a higher level is required not one of passive receptivity but instead one of productive spontaneity. This was termed a retrospective apprehension (rückgreifendes Erfassen), an act whose content is that of the thematization by the constituting ego of a collectivity, pre-constituted by the polythetic act of colligation, into an identifiable and re-identifiable object-meaning possibly posited as a substrate of judgments (ibid., pp. 246-247). On this account Husserl characterized time-consciousness as the original seat of the constitution of the unity of identity in general, making further clear that the outcome of temporal constitution is a universal form of order, of succession and coexistence of all immanent data. Talking about content, we talk about syntheses which produce the unity of a field of sense and this is meant to be a higher level of constituting activity. But this again presupposes the temporal structure of the passive field itself which precedes all acts and further the origin, in temporal terms, of the passive unity of the pregivenness of a plurality of perceived or even imagined things, that is, the absolute subjectivity thought to be the original source of self-constituting temporality. Husserl put it this way: “[..] the unity of the intuition of time is the condition of the possibility of all unity of the intuition of a plurality of objects connected in any way, for all are temporal objects; accordingly, every other connection of such objects presupposes the unity of time.” (ibid., p. 182). It is important to note that the unity originating in the being of consciousness as such not only underlies the possibility of reflecting on things in general as identical, durating 9 This obviously refers to relations between formal-ontological objects, including such mathematical objects as sets, classes of sets, functions, domains of functions in the form of Euclidean or non-Euclidean spaces, etc. 8 unities over time but also grounds the possibility of reducing the concept of an ideal infinity in progression to an act of fulfillment and completion in the present now of consciousness. In this sense the unity of a physical thing “stands over against an ideally infinite multiplicity of noetic mental processes of a wholly determined essential content and which can be surveyed despite the infinity, all of them united by being consciousness of the ‘same thing’. This unification becomes given in the sphere of consciousness itself, in mental processes which, on their side, also belong again to the group which we have delimited here.” ([19], Engl. transl, p. 323). To the extent that physical objects are reducible to noematic immanent ones under certain eidetic norms, something that happens also with their unity founded in the unity of consciousness, the statement above also holds good for categorial objects corresponding to pure content-free object-forms with a priori associated, empty-of-content, specific relational forms. This latter class that includes all objects of a formal-mathematical theory, i.e., all categorial forms in general, numbers or number-theoretical forms, sets, functions of pure analysis and their Euclidean or non-Euclidean domains (ibid., p. 33). Husserl offered a kind of metaphorical explication to account for the apparent contradiction between the all-sided infinity of continuum meant, e.g., in the sense of an ideal infinity of multiplicities of appearances of one and the same thing and the ‘closed’ unity of the completion of the ‘running through’ of appearances. Thus the idea of continuum as such and the idea of the perfect givenness prefigured by the idea of continuum are presented in an intellectual seeing (Einsicht) in the way an ‘idea’ can be intellectually seen by designating by virtue of its essence the peculiar type of intellectual content. In these terms the idea of an infinity motivated according to its essence is not itself an infinity. The fact that this infinity cannot be given in principle and in rem, does not preclude but rather requires the intellectually seen givenness of the idea of this infinity (ibid., p. 331). This seems to be a way out of the apparent contradiction of a ‘finite infinity’ even though, as with Husserl’s obsessive search for a foundation of the immanent unity of consciousness in temporal terms, one can possibly raise the issue of the circular character of an endless regression of reflecting-reflected. In this view one may have a notion of actual infinity independent, as it stands also with formal individuals, of spatio-temporal and consequently causal constraints. This kind of immanent infinity appearing in the form of a continuous whole in the actual now of reflection and possibly extended indefinitely in all ‘directions’ while preserving the identity and relations of any well-defined objects within it as temporal fulfillments is, in fact, what makes mathematics such an effective and inexhaustible tool to describe real world processes while by the same token conditioning (partly at least) the non-decidability of key foundational questions in axiomatic set theory.10 This kind of infinity is not to be confused with real world causality-constrained infinity conceivable only as spatiotemporally unbounded and, phenomenologically considered, invariably extending its eidetic state-of-affairs horizon ideally in infinitum. It is an infinity meant in radical phenomenological reduction as a correlate of inner-time consciousness and therefore prone to the circularities and inherent impredicativity induced by its non-objective origin, namely the absolute subjectivity 10 This claim is largely associated with a subjectively based approach to the foundations of mathematics, see: [30] & [31]. Also for a more detailed description of mathematical objects in phenomenologically motivated terms the reader may consult [31] & [37]. 9 (i.e., the transcendental ego) of consciousness as being-in-constituting (and not constituted) prima causa. In consequence infinity in this sense may open up an extra-theoretical discussion relevant to the conception of infinite sets as completed totalities especially in the case where there is no possibility to carry out a recursive enumeration on these sets or generate them by firstorder predicate formulas that exclude power-set operations and certain ‘external’ predicates to a standard first-order system. 4 4.1 Some prompts on infinity in classical tradition A brief review of infinity in the platonic-aristotelian metaphysics In the platonic Parmenides a characteristic example of the contradictions that may result from ontological definitions that do not take into account a notion of subjective constitution is the person’s Parmenides inference that the part (τό μόριον) “is a part, not of many nor of all, but of a single form and a single concept which we call a whole, a perfect unity created out of all; this it is of which the part is a part” ([6]: 157 E). In this case we do not come upon a relation between primary elements and their aggregate by a progression of iterative acts and their active synthesis but instead between a part and a containing whole which is specified as a single concept (ἰδέα) and a single form, so as to underline the different genus to which the whole belongs with respect to its parts or generally to others. It turns out that ambiguities or contradictions found in platonic Parmenides in this and other places, e.g. in [6]: 159 B, may be due to a certain vagueness of ontological notions mainly engendered by the transformation of the objects of experience, to which we bear a specific and at least not analytically describable kind of relation, into formal objects of apophantic logic in ignoring the context established by the evidence of a constituting temporal consciousness. If for Plato in Timaeus the impossibility of thinking of an all-encompassing unity in terms of plurality, even of an infinite one, is reflected in the description of the Universe as the single visible living being containing all living beings of the same natural order, Aristotle, in his part, had gone into lengths in Metaphysics to show that the infinite cannot exist actually nor can be an essence unto itself. On the assumption of actuality any part we might take of it would be infinite, being a substance and not an attribute of a subject, and consequently either indivisible or infinitely divisible if it is divisible. But the same thing cannot be many infinities at once just as air is part of air by being of the same genus. Consequently infinity is indivisible and impartible (ἀδιαίρετος καί ἀμέριστος). Yet what is actually infinite must be attribute of something spatio-temporally existing, therefore it must be a quantity (which is partible) and for this reason it must be an accidental attribute of that of which it is accidental which eventually means that actual infinity cannot be a self-standing essence in an ontological sense. A still more interesting point is Aristotle’s subsequent aporia as to how and in virtue of what can mathematical magnitudes be taken as a unity, more concretely, in which way a collection of mathematical unities can be regarded as a unity in its own right. Further and most important, he seems to suggest that the underlying unifying factor for objects in the 10 world can be reasonably supposed to be the soul, or part of the soul or some other influence apart from which things are a plurality and may disintegrate. Yet even though he alluded to a subjectively founded unifying factor of an indefinite plurality of, e.g., mathematical objects, a reference which oddly enough is not substantially commented in aristotelian secondary literature, he virtually left open the question as to the cause in virtue of which mathematical magnitudes, being divisible and quantitative, are formed in unity and cohesion ([38], p. 184). In any case he did not delve further into a subjectively founded interpretation for the underlying reason of unity in terms of plurality.11 Concerning the apprehension of infinity in physical terms Aristotle stated in the first book of Metaphysics (Metaphysics I-IX) that even though infinity as well as void can be said to exist potentially, they cannot exist as separate states-of-being in actuality but only in knowledge in other words only as mental states-of-affairs. Views on infinity and void are more systematically taken up in Aristotle’s Physics (resp., Book III, iv-viii and Book IV, vi-ix) where Aristotle admitted to the difficulties posed by the problem of the infinite in which many contradictions result whether we suppose it to exist or not. Therefore he wondered: “If it exists, we have still to ask how it exists; as a substance or as the essential attribute of some entity? Or in neither way, yet none the less is there something which is infinite or some things which are infinitely many?” ([35], Engl. transl., p. 43). In the course of ensuing argumentations Aristotle reached the conclusion that since ‘being’ has more than one sense, the sense of being of the infinite is not that of an ontological being rendered unto a definite substance but consists in a process of coming to be and passing away. In particular talking about numbers, if numbers are bounded below by the least indivisible one, it is always possible in the direction of largeness to think of a larger number consequently this infinite is a potential, not an actual one. Moreover since each larger number is associated with a process of bisection with regard to the immediately smaller number, its infinity as inseparable from the process of bisection which is not a permanent actuality but consists in a process of coming to be like time and the number of time (ibid., Engl. transl., p. 50). It is noteworthy that in the last paragraph of the part viii of Book III Aristotle seems to imply that there is a possibility of an indefinite magnification of a quantity in thought12 which however does not correspond to an infinite in real world terms adding by the same token that time, movement and also thinking are infinite in the sense that each part taken in the actual now passes in succession out of existence (ibid., Engl. transl., p. 51). I take the last statement, even if it is marginal in Aristotle’s main argumentation on the nature of infinity, as an indication that the reduction of infinity to an immanence in 11 Hellman and Shapiro have developed, in [9] and [10], a formal axiomatical system to construct a ‘regions based’ one-dimensional continuum which follows the Aristotelian credo that continua are not composed of points (as spatially postulated). However they do not follow Aristotle in a very important aspect, namely in that they make essential use of the actual infinite rejecting the notion of potential infinity espoused, at least on epistemic grounds, by Aristotle and many other mathematicians and philosophers since then. 12 The possibility of reaching an infinity in thought in the sense of immanence which does not correspond to a spatiotemporally founded one is also implied by the statement that “not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought” (Greek orig. of the underlined: διὰ γὰρ τὸ ἐν τῇ νοήσει μὴ ὑπολείπειν) ([35], Engl. transl., p. 43). 11 consciousness associated in an essential way with the nature of time may point to certain aristotelian insights about the nature of infinity that run contrary to a foundation of the infinite in terms of ontological being. 4.2 Infinity in the perspective of Locke’s and Hume’s philosophies Even though Husserl had fundamental disagreements with Locke’s and Hume’s views on ontology and metaphysics he had great esteem for both philosophers especially for Hume whom he regarded (along with Descartes) as a precursor of transcendental phenomenology. His primary critique concerning Locke was that he could not see that the problems of knowledge in their purity and in principle are incompatible with the objectivism of his method, that these problems require to put radically into question the whole universe of objectivity and hold it exclusively in the sphere of pure (absolute) consciousness ([13], p. 76). In the same viewpoint, while praising Hume’s Treatise as being the first outline of a pure phenomenology, he nevertheless remarked that it was actually in the sense of a pure sensualistic and empirical phenomenology. In this regard Hume’s nominalistic reduction could not found the identity of ego, making each one the same person, out of an aggregation of constantly interconnected perceptions. Moreover Hume’s inductive-empirical objectivism would be at odds with the necessity of the eidetic character of the Husserlian description of consciousness. Consequently the idea of infinity that both Locke and Husserl held cannot in any way be reducible to an immanental form correlative to the intentional modes of a self-constituting temporal consciousness. More specifically, Locke upheld the negativity of the ideas of infinity and infinite divisibility on the grounds of adding to the sense impressions of extension and divisibility (upon extension) the ‘negativity’ of unbounded or unlimited continuation in the understanding that the idea of an end is, on intuitive grounds, rather positive than negative. Yet the notion of infinity in these terms and a fortiori that of infinite divisibility, even in the Lockean negative sense,13 presuppose a notion of infinite time as an in rem condition for the kind of conception of the negativity of an unlimited continuation,14 thereby making the concept of infinity, as it is the case also with Hume, ultimately dependent on and derivable from a progressing series of impressions of perceptions and reflections. It follows that Locke reduced infinity to the sense Husserl precisely denied, namely to an ever extending sensual ‘substratum’ founded (in the Husserlian sense of Fundierung) on real world spatiotemporality and the accompanying causality. Evidently this would be an infinity of virtually no conceptual relevance with the formal-mathematical notion of infinity, at least as this notion has been established from the time of Cantor’s introduction of the theory of transfinite numbers and has evolved since then in the development of axiomatic set theory and the theory of 13 In An Essay Concerning Human Understanding, Locke argued about the ‘negativity’ of the idea of infinity on the grounds of the finiteness of human mind for which an infinite extension (over time) cannot but produce a negative idea of infinity, that is, a confusing and indeterminate one out of insufficient understanding (see [24], p. 66). 14 Even though Locke denied that we might have any positive idea of infinite number, space, or time he nonetheless claimed that the ideas of ‘causal power’ and ‘unity’ are found to be ubiquitous within experience ([8], p. 18). As argued in the present article, unity (but not causality) can be integrated in the phenomenological sense of immanent infinity. 12 large cardinals, taking also into account the intuitionistic idea of infinity and the intuitive continuum. In short, the Lockean semi-empiricist position on infinity, in contrast to the Husserlian immanentization of infinity as eidetically ‘constrained’, cannot epistemologically account for a mathematically proper notion of infinity, for instance, of the kind generated in a non-nominalist sense by logical-mathematical formulas quantifying over an indefinite domain. In a phenomenological perspective Hume’s position can also prove faulty even if Hume discarded any argumentation based on the ‘negativity’ of an unlimited continuation adjoined to finitely many impressions, and espoused instead only the negativity of the vacuum, as the latter idea was entertained by the natural philosophy of the time. Yet this would leave someone clueless as to the possibility of a conception of infinity to the extent that memory, imagination and reason would be only left to elaborate on the ‘materials’ provided by the senses and consequently could not make more in conceptual complexity than these ‘materials’ would allow. This would evidently imply that the finitistic and limiting character of impressions would by necessity constrain the accessibility to any conception of infinity in an absolute sense ([24], p. 69). To sum up both Locke’s and Hume’s account of infinity left no room for a complete riddance of the concept of infinity of the spatiotemporal and causal constraints Husserlian phenomenology sought to abolish by treating infinity and infinite objects as categorial objects in their specific eidetic characteristics and ultimately associating infinity, in terms of transcendental phenomenology, with inner temporality and its constituting subjective origin. 5 Is there infinity as an entity? In view of the discussion of the previous sections one may reasonably claim that the possibility of an ontological foundation of infinity can be correlated with a possible ontology of intuitive continuum insofar as actual infinity, at least in phenomenological terms, can be re-presented as a complete, immanent objectivity in the form of the continuous unity of consciousness. It should be noted here that while the intuitionistic view of intuitive continuum as an impredicative, immanental substrate devoid of any quality or specification is close to the phenomenological description of internal time,15 yet the (mathematical) infinite in Brouwer’s conception of freely evolving choice sequences is associated with indefinitely processing mental processes. The fact certainly is that Brouwerian intuitionism undertook an interpretation of mathematics in terms of the notion of time which, among others, leads to the rejection of the principle of excluded middle. However it is on the grounds of the phenomenologically motivated principle of two-ity with regard to the progression of choice sequences, an intuition reducible to the intentional modes of protention-retention16 in tem15 In what Brouwer described as the primordial intuition of mathematics intuitive continuum is the “substratum, divested of all quality, of any perception of change, a unity of continuity and discreteness, a possibility of thinking together several entities, connected by a ‘between’, which is never exhausted by the insertion of new entities” [3], p. 17 in: [39], p. 205. 16 These specifically phenomenological terms can be roughly described as a priori forms of intentionality toward an original impression just passed-by to the past (retention) and an a-thematic ‘expectation’ to the 13 poral consciousness, that Husserl’s student Oskar Becker criticized Brouwer’s intuitionism. In more concrete terms Becker claimed that Brouwer’s thinking about time was insufficient to make sense of free-choice sequences, namely of ‘lawless’, non-repetitive constructions which in intuitionistic analysis were needed to introduce the real numbers.17 Even as Becker ‘deviated’ from orthodox (Husserlian) phenomenology to attribute an interpretational role to the historicity of Heideggerian Dasein, he still endorsed the phenomenological idea of the possibility of reflecting each time upon a potentially infinite sequence of mentally implemented iterations in virtue of a whole, thus allowing finite human consciousness to grasp transfinite structures as totalities in actuality. Therefore by taking recourse to a constituting subjectivity attitude he hoped, at odds with Brouwerian intuitionism, to vindicate large parts of the Cantorian theory of transfinite numbers ([36], pp. 577-578). These given, an interesting and widely debated question is whether the possibility of existence of entities (in virtue of objectivities within the world), as infinity and continuum may be taken to be, can be answered empirically or categorically. The author clearly rejects the former position simply by pointing to the fact that infinity (at least as an epistemic term) is never attained empirically either in empirical physical sciences or formal logical-mathematical ones but only as an ideal limit generated by abstractions made in thought. And yet in choosing the second option and stating that the question of existence of the continuum (and infinity for that matter) is a categorial question, we face the circular inconvenience that categories are themselves categorial terms insofar as categories may not signify ontological differences but rather the ways the mind puts experience into some sort of order so that successful consequences may follow. This means that categorial distinctions are not set up by the nature of things but depend on the activity of the mind (irrespectively of the content one gives to the term mind) in terms of which they are ‘detached’ from reality through conscious experience which is temporal and therefore continuous. It is interesting and clearly supportive of my argumentation against an ontology of the infinite, H. Lee’s view in Are There Any entities? that the question of whether there exist entities in a self-standing ontological sense is closely bound up with the question of whether reality is a substantial unity composed of discrete, ontologically self-standing parts, objects and classes of objects, or whether it is a process composed of continuous events. This is claimed by Lee to be perhaps the most fundamental metaphysical dispute in contemporary philosophy as it summarizes the underlying question of continuity and the relation between the continuous and the discrete ([29], p. 125). It turns out that entities as categorial terms are meaningless without some sort of constituting activity of the mind which is moreover a discursive activity if entities as such and in relation to others are going to be carriers of a meaning. Most importantly the whole approach, which in places reminds of Whitehead’s ‘ontology’ of events in Process and Reality, presupposes a temporal conscious experience with respect to a reality which is a concrete continuum. The way to delineate entities as mental objects in terms of beginnings, ends, limits and boundaries is to break the continuum only in abstraction since mind “cannot actually stop process” (ibid,. p. 127). Entities, therefore, are not to be considered as existing in ontological future (protention). For more the reader may look at [14]. 17 For a review of the fundamentals of the intuitionistic version of mathematical foundations, in particular the ad hoc continuity principles, the reader may consult [11]. 14 self-sufficiency independently of the mind, yet the concept of entity becomes a fundamental categorial tool for the purpose of understanding and reaching the level of need for social communication. In short, what I find relevant here to my own approach are the following claims: 1) entities in general do not correspond to ontologically self-standing beings but are reduced to some kind of constituting-transforming activity of the mind in a way that they ultimately become hypostatizations 2) any constituting and further transforming activity of the mind is performed in terms of human experience which, conscious or unconscious, is temporal and as such it is continuous. On these purely intuitive grounds one may have a refutation of the ontological claims for the infinite without necessarily espousing phenomenological analysis at least in a strict and constraining sense. Of course the debate of what may stand as ontological entities either in metaphysical or epistemic terms is a far-reaching and evolving one especially in the light of the progressing epistemological edifice including the content one may give to the term ontology itself. A major question, for instance, is the way infinity is mediated through logico-linguistic formulation to obtain a semantic content within a formal mathematical environment. However this latter discussion would probably take us out of the scope and the limits of the present article.18 6 Is there an epistemic foundation for an ontology of infinity? As already stated my argumentation is based on the supposition that the mathematical infinite subsumes the infinite of physical theories to the extent that physics and science in general deals with the infinite in terms of the corresponding mathematical metatheory and consequently through the expressional means of formal mathematical language and the specific techniques involved. Modern philosophical views of infinity have, if anything else, clarified our understanding of the difficulties and questions encountered in the view of infinity as a potentiality with regard to the finiteness of the natural world, going as far as to be regarded as an ‘externally’ imposed disruption to the finite order. A. Badiou’s epistemological approach to the question of infinity in Being and Event, [1], is in fact its reconceptualization in formal-mathematical terms, more specifically in terms of set and set-forcing theories. More concretely, axiomatic set theory is thought to revolutionize our understanding of infinity in two ways: (1) it makes possible a view of infinity as actually existing in the form of a complete, even though non-‘traversable’, whole and (2) it abolishes the ‘naive’ homogeneity of infinity by articulating a sequence of infinities of various orders. As a result in Badiou’s view infinity, as actual and susceptible to articulation in thought, is made part of the world without a need to refer to an external metaphysical or ‘divine’ reason. In fact actual infinity in the sense of an impredicative ‘substrate’ upon which logical-mathematical formulas bounded by universal-existential quantifiers may lay ontological claims by generating infinite mathematical objects is implicitly present in most of foundational mathematics and logic. Moreover as I have argued earlier actual infinity as founded in each one’s temporal 18 The interested reader may look at [32] for a detailed presentation of the issue. 15 consciousness is indeed the only one immanently existing whose evidence is any infinite objectivity as a completed totality in reflection. Consequently, on this supposition, the known canonical scale of infinities generated by the circular notion of sets as collections of objects, the indefinite repetition of the power-set operation within infinite cardinalities or the list of large cardinals, e.g., inaccessible, supercompact and even larger ones generated by strong infinity assumptions, are in fact formal-mathematical intricacies conditioned and presupposing a temporally founded continuous unity whose ‘derivative’ form is actual infinity in presentational immediacy. In a parallel situation in forcing theory19 Badiou proposes to consider the generation of a generic set G as an indiscernible multiple out of finitistic conditions which themselves belong to a particular situation. In other words an indiscernible whole, the generic set G, is generated by means of subjective acts performed over a set of finitistic conditions. In view of this apparent incompatibility reducible to the sphere of subjectivity, Badiou wonders how “an ontological concept of the pure indiscernible multiple exists” ([1], p. 358). In accordance and concerning the nature of a generic set as a pure indiscernible multiple, he concludes that “at base its sole property is that of consisting as pure multiple, or being. Subtracted from language, it makes do with its being.” (ibid., p. 371). This is a state of affairs which is naturally the outcome of Badiou’s ontology in which the subjectivity factor is standing ‘opposite’ to the transcendence of the event and is deprived of the essential property of being co-constitutive with the event (be it a physical or mental one) in the modes in which it can be co-constitutive within-the-world. Yet what are thought to be essential properties of genericity, i.e., the filter and generic properties proper, are not mere subtracts of the formal language for they can be seen as abstractions presupposing subjectively founded meaning-acts as those of set formation (in terms of passive and retrospective apprehension; see Sec. 3, par. 4), of ordering, of the conservative extension of a set-theoretical structure over an indefinite horizon, etc. These can be considered as mathematical acts performed independently of any spatio-temporal and causal constraints, being only subjected to certain eidetic necessities of an intentional-constitutive consciousness and also mediated by previous mathematical experience in reference to the world as a ground of experience. The fact that a generic set G retains the property of being indiscernible (in Badiou’s sense), even though it does not introduce any new ordinals in the extended forcing model M[G] of ZFC (i.e., the standard Zermelo-Fraenkel Set Theory with the Axiom of Choice), can be seen as a concrete case in the mathematics of the infinite where one can think of an infinite set in excess of constructivist steps, as having a host of desired properties even if they are not generated by definability formulas associated with Gödel’s constructive universe L. Badiou sets a dividing line between the subject on the one side and the event (which belongs to ‘that-which-is-not-being-qua-being’) on the other, in the sense that “situated in being, subjective emergence forces the event to decide the true of the situation” ([1], pp. 429-430). Infinities meant in Badiou’s general ontology are associated to one or the other degree, as 19 Forcing theory is, roughly speaking, an ingenious mathematical technique initiated by the American mathematician P. Cohen in 1963, and further developed to a full-fledged mathematical theory, mostly motivated by the interest to resolve certain key mathematical questions such as the Continuum Hypothesis. More can be found in Cohen’s own work, [4], and Kunen’s [28]. 16 it happens with the extrinsic indiscernibility of a generic set G, with the transcendental character of the event which is generally thought of to be a major prerequisite of truth. A phenomenologically motivated view could dismiss the artificial distinction between subject and event insofar as the event in the sense of implementation of a mathematical act carried out in objective time cannot be conceived independently of the self-evident presence of a consciousness endowed with specific intentional-constitutional properties within the world of experience. In this sense, referring again to the method of forcing, the subject cannot force the event to ‘decide’ the truthfulness of a situation existing on the ground level (by its presumed veracity in the extended model) on the supposition of existence of a generic indiscernible part of the ground model M , a position that seems to me a rather superficial interpretation of the subjective influence in shaping mathematical truths. One should rather assume that the existence of the indiscernible part G of the ground model with the presumed generic properties represents the capacity of the (mathematical) consciousness to apprehend infinite mathematical objects with specific combinatorial properties and extend or restrict them indefinitely in a consistent way eventually granting them globally desirable properties, in the fond of a continuous unity which is not as such part of the predicative universe of the formal system in question. In the On Ontology and Realism in Mathematics, [7], H. Gaifman has claimed that the problem of the ‘natural questions’ of a given area in mathematics has to do with the relations between ontological and epistemic factors. A a matter of fact major advances in the history of mathematics have been made by reorganizations of the epistemic framework which made it more transparent and thus more efficient. However, in his view, the translation of a mathematical theory into set theory is an ontological reduction that destroys the epistemic organization of the area since it is indifferent, if not counterproductive, to the mathematician who works in that area.20 Natural questions arising in a particular mathematical area depend on the epistemic organization, and the undecidability of such a question shows to the mathematician that his grasp of the area was defective. What appeared as a well stated question that had a unique solution turns out to have more than one answer depending on the corresponding version of set theory. Speaking metaphorically, it is an undermining of the epistemic by the ontological, the latter appearing here in the form of set theory to which the given mathematical theory is reduced. An easy example to consider is any theorem or corollary in the mathematical metatheory of general relativity conditioned on the continuity of the corresponding topological structures and consequently on the solid foundation of tensor equations across space-time. These problems can be reduced to the set-theoretical question of establishing the power (cardinality) of the continuum which, as well-known, is a problem with two alternative, conflicting answers depending on the settheoretical model one works with. 20 It is questionable, though, whether the translation of a mathematical theory, be it a pure or applied mathematics theory, to the axiomatic set theory is indeed an ontological reduction properly meant or just a reduction to the meanings and methods associated with the semantics and the syntactical structure of set-theoretical models. Of course a supplementary reduction on the level of primitive logical-mathematical ideas such as an indefinite collection of abstract objects, the non-logical predicate of belonging, ∈, etc., may lead to questions pertaining to the nature of primitive mathematical objects themselves, but then whether one chooses an ontological interpretation in a platonic context or a subjectively based one is a matter of choice and sound argumentation. 17 Hilbert believed that the two, epistemic and ontological, can go together. The success of Hilbert’s program would guarantee that we can safely work in a given area, pretending that the actual infinities that figure in our reasoning are real and thus pretended realism is safe. And the slogan ‘in mathematics there are no ignorabimus’ would mean that our epistemic apparatus is able to solve any mathematical problem that arises in our research. After all the problem, e.g. that of determining the power of the continuum, was not forced on us by the physical world. It arose in the domain of pure thinking and therefore pure thought can possibly handle it. However, in Gaifman’s view, Gödel’s undecidability results introduce a wedge between the epistemic and the ontological which makes us conclude that pure thought is not enough ([7], p. 410). As a matter of fact immanent infinity, that is, ‘infinity’ as constituted in temporal consciousness in the form of an objective fulfilment exhibiting a priori eidetic characteristics, is implicitly presupposed in the treatment of the mathematical infinite even in case one discards an ontological commitment to infinity in favor of its comprehension through epistemic articulation. This is Kanamori’s position in The Infinite as Method in Set Theory and Mathematics where “the commitment to the infinite is to what is communicable about it, to the procedures and methods in articulated contexts, to language and argument. Infinite sets are what they do, and their sense is carried in the methods we collectively employ on their behalf.” ([25], p. 39). Yet in spite of his view that assimilation of methods along hierarchies can be viewed as commitments to the infinite in the sense that different assumptions and techniques are employed respectively in the introduction of the countably infinite of natural numbers, the continuum of real analysis, and the (empyrean) infinite of higher set theory, Kanamori urges a kind of ecumenical approach to the infinite leaving, without further specification, some clues to an irreducible semantic content of ‘prior’ proofs relating to the resources of the underlying elementary system of a statement (ibid., p. 40). This is a claim that leaves by itself some room for doubt concerning the claim that mathematical infinite can be accounted for only in terms of epistemic articulation, that is, in the way it is approached, assimilated, and applied in the construction of mathematical hierarchies. On the assumption, in Kanamori’s sense, that the infinite in epistemic terms can be a way out of the impasse of a purported ontology of infinity yet it seems one may still stumble upon theoretical obstacles and circularities whose origin may be indeed the lack of an ontology of infinity.21 I refer in particular to two logical paradoxes, the Russell paradox generated by Frege’s attempt of the reduction of arithmetic to logic and the Burali-Forti paradox generated by the conception of the class of all ordinals as a completed mathematical object. In both cases M. Dummett has suggested that the underlying reason is our failure to recognize and properly reason with what he calls ‘indefinitely extensible concepts’, a primary instance of which is the concept of ordinal numbers ([5], p. 316). Given that for any given ordinal the position the particular ordinal occupies in the natural ordering is given by its very existence, the answer about the place occupied by a new ordinal put 21 Of course mathematicians treat infinite in everyday mathematical practice as an ideally existing yet concrete and complete mathematical object (or rather such a collection of objects), part of the predicative universe of a formal theory. This was acknowledged as far back as Aristotle’s Physics ([35], Engl. transl., p. 50). Yet this conventional attitude, dear to logical positivism, in no way affects the question of a possible ontological treatment of the infinite. 18 in the end of this ordering must presuppose the existence of a pre-existing class of all ordinals as a determinate whole which is precisely what generates the Burali-Forti paradox. Dummett sees an obvious analogy between the notion of an indefinitely extensible concept and the more philosophically familiar notion of potential infinity applied in a sense that may also include the choice sequences of intuitionistic analysis, as ever progressing ones either by a recurring formula or freely expanding, something that requires the application of intuitionistic rather than classical logic to reason about these concepts (ibid, p. 319). Yet one knows that Cantor’s well-ordering principle which corresponds to a defining characteristic of ordinals was reduced in 1904 by Zermelo to the Axiom of Choice, AC, namely the abstract existence assertion that for every set X there is a choice function, i.e., a function f such that for every non-empty subset Y ∈ X, f (Y ) ∈ Y . In this way Zermelo showed that a set is well-orderable (in other words it can be ordered by the wellordering of ordinals) if and only if its power set has a choice function, that is, the axiom AC takes hold. It turns out that the extremely important in foundational mathematics Axiom of Choice has a three-fold implication: (a) it is based on a meta-mathematical concept of actual infinity meant as a kind of abstract, completed infinity on which to define the choice function as indefinitely extensible (b) it has been proved to be independent from the rest of the axioms of the standard Zermelo-Fraenkel Set Theory, ZF, a result that attests, if we accept Gödel’s position (about the Continuum Hypothesis) in the earlier version of What is Cantor’s Continuum, to a problematic mathematical state-of-affairs and (c) it is, nevertheless, regarded a necessary principle “for infusing the contextualized transfinite with the order already inherent in the finite.” ([25], p. 38). The fact is, being in principle an indefinitely extensible infusion of order, AC is conditioned as such on the prior existence of an abstract (completed) infinity in actuality which in turn cannot be grounded otherwise but as immanently induced in consciousness with no real space-time foundedness. By this measure it cannot be a potential infinity in Dummett’s sense above in association with the concept of ordinals. Given that the mathematical infinite is layered in degrees of complexity, starting from the countably infinite of natural numbers, it is worthwhile to show the way a metatheoretical, in fact a completed, subjectively founded infinity conditions other assumptions and circularly reproduces itself in the ascension through hierarchical levels.22 In these terms a question that can be raised in the study of large cardinals theory is the extent to which the assumption of a standard level of reality, more concretely the conception of the intuitive and, in abstraction, of the formal-mathematical continuum (the latter associated with the uncountably infinite of real analysis), influences in one way or another the formal-axiomatical development of the theory. We may reasonably ask whether this level of reality must be implicitly assumed, for example, in extending ad infinitum the domain of application of 22 An interesting, though no further discussed on a subjective level, view of the mathematical continuum is E. Belaga’s in Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design. Belaga suggests that any phenomenologically and ontologically faithful axiomatization of the continuum should include a non-locality postulate (in the sense this term is understood in quantum information processing) to formally account for the following property of the continuum: All ‘points’, or ‘elements’ of the continuum are non-locally, i.e., simultaneously and at any moment, accessible. This non-local accessibility extends to all well-defined ‘slices’ (subsets in a set-theoretical terminology) of the continuum ([2], pp. 34-35). 19 quantifiers of first-order or second-order formulas in the various stages of construction of large cardinals; e.g., in the definition of κ-complete filters and ultrafilters for κ > ω. And of course we may raise the same question over the application, in the same context, of the Axiom of Choice or of its logical equivalents. Over the last decades the main tool for introducing new and ever more powerful infinite cardinals is by means of ultrapowers and elementary embeddings,23 that is, through truth-preserving injective mappings from the set-theoretical universe V into inner models M 24 equipped with structural properties associated each time with an ascending scale of infinity and reflected in the increasing resemblance between V and M . It turns out that by elementary embeddings a sense of connection with Cantor’s Absolute can be obtained by increasingly enhancing the ‘simulation’ of M toward V thus vindicating Gödel’s later position that every axiom of infinity should be derived from the extremely plausible principle that V is undefinable, where definability is to be taken in a more and more generalized and idealized sense.25 Along the way Scott’s result on measurable cardinals, obtained by the elementary embeddings approach, succeeded in transcending Gödel’s delimitative universe of constructibility by contradicting the constructibility axiom V = L through the postulation of existence of a measurable cardinal. Later, Vopěnka and Hrbaček generalized Scott’s result by proving its generalization on the assumption of existence of a strongly compact cardinal. Both results are, characteristically, conditioned on the existential postulation of a κ-complete ultrafilter over κ > ω which is by itself associated with set-theoretical operations and universal quantifications on scales above ℵ0 (i.e., the first infinite countable cardinal or, correspondingly, the first limit ordinal ω). One may also argue for the implicit assumption of a standard level of reality, in the sense of an immanent continuous unity formally corresponding to the continuum of real analysis, in Woodin’s research in higher set theory with the aim of acceding to Cantor’s Absolute. More concretely, without entering into the tricky technical details, Woodin has tried to elaborate an inner model program which from a strategy of incremental understanding of large cardinals, with universe V forever hopelessly out of reach because of Scott’s result (mentioned before) and its ‘descendants’, could evolve into a program for perhaps understanding V itself by approaching it closely enough. This in turn has generated an axiom for an ultimate version of Gödel’s constructible universe L, namely the axiom V = Ultimate-L and its accompanying conjecture, the Ultimate-L Conjecture, which would supposedly prove, on the assumption of sufficiently large cardinals, that the Ultimate-L exists in close proximity to V and moreover would also settle some inherent problems in the development of set 23 For structures M0 =< M0 , .. > and M1 =< M1 , .. > of a language L, an injective function j : M0 −→ M1 is an elementary embedding of M0 into M1 (j : M0 ≺ M1 ) iff for any formula ϕ(u1 , ...., un ) of L and x1 , .., xn ∈ M0 M0 |= ϕ[x1 , ...., xn ] iff M1 |= ϕ[j(x1 ), ...., j(xn )] . 24 A class M is an inner model iff M is a transitive model of ZF under the ∈ predicate and contains the class of all ordinals, i.e., ON ⊆ M . 25 In fact the elementary embeddings (into inner models) approach has its inherent limitations by virtue of Kunen’s fundamental result in [27]. Kunen has proved, capitalizing on the application of AC and the Erdös-Hajnal theorem, that if j is an elementary embedding from V into V then j must be the identity mapping. See: [27], pp. 407-408. 20 theory.26 Yet Woodin’s axiom V = Ultimate-L is based on the construction of a structural generalization of L, HODL(A,R) , which has been verified as such for many (yet only conjectured for all) universally Baire sets A ⊂ R.27 This given, one can reasonably raise the following counterarguments: First, it is at least questionable from a foundational viewpoint whether one can apply on the level of purely set-theoretical constructions topological notions inherently associated with the structure of the set of reals which is doubted, in the first place, to be a set at least in terms of the standard definition of a set and the application of the power-set operation. Second, even in taking Woodin’s use of topological notions in the construction of the structural generalization HODL(A,R) of L as ‘lawful’, this can only give credit to my argument about Woodin’s progression to an ultimate level of proximity to V by being conditioned on the implicit assumption of a standard prior level of reality. In other words one must rely on a formal level of reality associated with the continuum of real analysis, reducible on the subjective level to an ‘infinite’ continuous unity which circularly reproduces itself in the level of higher set theory. There are other concrete instances in the axiomatization and further in the construction of mathematical theories and corresponding models in which a notion of actual infinity is applied which is not, as I have tried to show, an infinity conceivable in ontological terms. This can be a fortiori true for physical theories as such, independently of their mathematical metatheory, in which any conception of objective spatio-temporality may only be, at best, an unbounded ‘finiteness’ indefinitely extensible. Looking back, and this could serve as a conclusion to this article, my intention was to refute an ontological conception of the infinite both on the purely philosophical level and the epistemic one, in the latter case on the assumption that the mathematical infinite underlies and conditions physical and positive science in general in terms of the predicative linguistic universe through which every empirical-‘observational’ science is mediated. In this viewpoint I argued, by referring to some concrete situations, against an ontology of infinity in the terms the concept of infinity is epistemologically articulated by virtue of the axiomatic and proof-theoretical machinery of mathematics. Unfettered by any platonic or generally idealist preoccupations and focusing my argumentation toward a subjective constitution on the part of a constituting temporal consciousness within-the-world, I was strongly motivated by the Husserlian writings on the matter, to a significant extent applied and commented in this paper, so as to proceed to a ‘deconstruction’ of the ontological foundation of the infinite at least in the sense generally espoused in philosophy and science over centuries of platonic-metaphysical tradition. Next I tried to show the ways by which a phenomenologically founded infinite can epistemologically account for the actual infinity free from spatio-temporal and causal constraints that underlies formal-mathematical theory in its own right and also as a metatheory of physical ones. 26 To date Woodin has proved the Ultimate-L Conjecture only under certain restrictive conditions. See: [41], p. 3. 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