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INFINITY

Gianfranco Basti

I. Infinity, from the Ionian Philosophers to Aristotle. 1. The Presocratic Philosophers: the Ionians, Pythagoras, Parmenides and Democritus. 2. Plato and Aristotle. – II. The Reflections of Thomas Aquinas and the Theological Significance of the Notion of Infinity – III. Infinity within Galilean and Newtonian Science – IV. The Infinity of Cantor. 1. Three Kinds of Infinity. 2. The Notion of Transfinite. 3. The Power of the Continuum. From the Antinomy of the Power Set to the Axiom of the Power Set. – V. Concluding Observations.

The term “infinity” has a vast range of significances and applications, having been employed within the common language, since antique times by philosophers, theologians, mathematicians and poets. The term is applied in an “objective” sense, to indicate an effective characteristic of “unboundness” of something (a straight line, the sequence of integer numbers, etc.), or also of “absolute perfection” of the Being (lat. Ens, essentially with reference to God). The notion has been also applied in a “subjective” sense, to indicate the perception had by the experiencing subject, who considers as infinite something objectively finite, but much greater than the self (the height of the sky, the breadth of the ocean, etc.). In this last case we have a relational type usage of the term, to indicate a sort of “ratio” or “scale” between the subject and the object, a relationship that can also be inverted with respect to the infinitely small (infinitesimal), rather than the infinitely large. Etymologically the significance of the term includes a negation (gr. à–peiron, lat. in–finitus) characterized by a privative alpha or by another negative preposition: non-finite, that is without boundaries (lat. fines), without limits, not only in an “extensive” sense, but also within the “intensive” sense of limitations and imperfections, and is therefore supplied with the fullness of the positivity, that is to say of the total “actuality”. Infinity, as extensively agreed upon, therefore presents itself to be not completely crossable, not traversable from side to side (a straight line cannot be completely run through); and intensively agreed upon, it consequently presents itself as not totally graspable, and as such is inexhaustible or incomprehensible (lat. in-comprehensus, non circumscribable), unavailable to a complete intellectualization by means of specific acts of human knowledge, which are by their nature necessarily limited and of a finite number. In its negative sense, wishing to indicate something that has no limits or boundaries, the term is also sometimes employed as synonymous with “indefinte”, and as such “indeterminate” and “amorphous”, without properties and face, thereby totally deprived of any “actuality”: not as though it were nothing, but rather as if it were a pure disposition to receive whatever determinations, that is, a pure potentiality. In the history of thought, the philosophical access to infinity has been marked, particularly from Aristotle onwards, by the notion of “potential” infinity, whilst in the theological sphere Thomas Aquinas underlined the property of “actual” infinity; in more recent times, George Cantor elucidated upon the access to infinity in mathematics with his enrichment of the notion through the concept of “transfinite”.

I. Infinity, from the Ionian Philosophers to Aristotle

1. The Presocratic philosophers: the Ionians, Pythagoras, Parmenides and Democritus. The first philosophical school, the so-called School of Miletus (Asia Minor) is characterized by the research of the principle (gr. arché) of all things, which Thales (640-560 B.C.) recognized to be water. For Anaximander (610-547 B.C.) all things are properly regarded as “definite”, while their principle is to be considered as “indefinite” (á-peiron). There thus appeared for the first time the term “infinite”, with the sense of “indefinite” and “indeterminate”. For Anaximenes, (585-528 B.C.), the materialization of the ápeiron of Anaximander brought a recognition of air as the origin of all things, through a mechanism of rarefaction-condensation.

In the school of Pythagoras, which developed in Croton in Greater Greece in the 6th century B.C., starting from the teaching of its founder Pythagoras of Samo, the theses of Anaximander and Anaximenes – for the first time in the history of western thought – were elaborated resorting to a mathematical foundation. For Pythagoras all things derived from the synthesis of the “definite-indefinite” and of the “limited-unlimited”. The essence of all things is that which pertains to geometrical figures. They are ultimately formed by points or undividable unities, therefore number-points. The things that are definite are so because they are “measurable” (since they are extensive entities or geometrical figures) and “denumerable” (since they are composed of undividable unities). The reality was born, then, from the harmony of the opposites: and first of all from the fundamental opposition of the “limited-unlimited”, or respectively of the “uneven” (one, limited, form) and of the “even” (two, unlimited, matter), since from one and two all the numbers and all the geometric shapes can be constructed. The other oppositions that derive from the preceding are those of “straight-curve”, “rest-motion”, etc. By this philosophical thought a glimpse can be caught about the role of the unlimited, of infinity understood as a “disposition” to receive determinations.

An initial connection between the metaphysical and the anthropological contexts arises with Parmenides of Elea (520-440 B.C.), who brings to light for the first time the notion of “being” in Western thought. Parmenides affirms the identity between thought and being: he conceives being univocally, like a “very general genus”, as a unique notion, more universal than the others, but one which is not yet recognized «according to different ways of significance», as Plato and Aristotle will discover and set forth later. We must turn to Parmenides for the formulation of that fundamental law of logic that is the “principle of non contradiction”, which for him is also the fundamental “metaphysical” law. But he does not yet know the distinction between a metalogical and a metaphysical use of that principle, which is indispensable to avoid confusions and errors. With this premise, Parmenides affirmed the purely apparent character of the quantitative multiplicity, of the qualitative diversity, and of becoming in its different forms. Melissus of Samo, his disciple, drove out the contradiction of this vision: if being is a unique entity, it cannot be unlimited, because the “unlimited” indicates a non-being, a negativity (here “infinite” as lacking of delimitations, is meant as “indeterminate” and imperfect). It will be therefore limited, as a “sphere” is. But if it is limited, and all being is by definition in the sphere, who limits the sphere? Such limits must be “outside” of the sphere. But if all the being exists within the sphere, who or what could limit it except the non-being? Yet, non-being does not exist and, whatever being may be considered, limited or unlimited, it will have to do with non-being: herein lies the antinomy. As can be seen, Parmenidean metaphysics, like all other rationalistic metaphysics — but also, today, like any formal metalogic  — intimately tends towards antinomy.

A first reply to Parmenides was through Democritus of Abdera (460-370 B.C.) who demonstrated the non-contradiction of multiplicity. In fact, if Zeno, a disciple of Parmenides, had demonstrated the self-contradiction of the infinite divisibility of the extensive material (cf. Koyré, 1971, pp. 9-35;  MECHANICS, II.3), it would be furthermore necessary to admit the existence of the utmost undividable parts (atoms), and thereby the extensive reality needed to overcome the accusation of the self-contradiction of the notion of numerical multiplicity pointed out by Parmenides and Zeno. Therefore, in order to justify the concept of multiplicity we do not need to invoke the existence of “nothing” (absolute non-being), because it is enough to employ the concept of “vacuum” (i.e. empty space, or absence of matter: a relative non-being). The entity of vacuum is not an absolute non-being, but the pure and simple “deprivation” or “absence of matter” ( MATTER, VI). The self-contradiction arises from the “absolute” opposition between being and non-being, but nothing forbids that something be such when compared to some particular thing, and not with something else. The vacuum is the “absence of matter”, nor the absurd existence of the non-being. The vacuum is not nothing, but rather the “non-being of something”. Analogously, when related to the number “1”, by which we enumerate a certain discreet or atomic entity, the number “0” is not completely representative of “nothing” but rather denotes the absence of that entity or, more exactly the emptiness of it (the “empty set” as referred to by the mathematicians within the framework of modern set theory). But this last refinement will reach mathematical thought later on, in the Middle Ages, with Arabic mathematics. All of this opens up the door a bit more to the possibility of considering an entity which is “infinite” in a positive sense, i.e. without relative negation of being, and no longer in an indeterminate and negative sense.

2. Plato and Aristotle. It will nevertheless be with Plato, and afterwards and above all with Aristotle, when the explanation of the concept of being will be developed, no longer univocally, but taking into account that «it can be said according to different ways of significance» ( ANALOGY, II.4): In so doing, an important change is performed from the purely negative notion of infinity (such as an indeterminate one) to one that is positive (infinity “in act”). Plato realized the first step in this direction with his dualistic conception. There exist in the cosmos two worlds: one material, composed of entities in a continuous state of becoming, and the other “immaterial”, composed of immobile entities in their fixed state, not becoming. This is nothing but the definition of the “diversity”, that even within the limits of a dualistic scheme, already implies the surmounting of the presumed contradiction affirmed by Parmenides, so introducing a possibility for multiplicity. The “diversity” between A and B, for example, although implying that A means non-B and that B means non-A, does no longer imply the notion of an absolute non-being. That is to say it denies only something determinate of A or of B, not “all of A” or “all of B”, but only the “form a” of A and the “form b” of B. Diversity, therefore, implies the relative non-being, not the absolute non-being. The “different” entities are not opposing each other because of an opposition of contradiction (A/non-A), but they simply oppose for the opposition of contrariety (A/non-a). It refers to an opposition regarding the form, not an opposition with respect to whole being, because affirming B does not deny all of A, but rather only its a form. Evidently, then, the entity A or the entity B is not only composed of the “form” but also of a “matter”, physical or intelligible, corresponding to the reference of a physical or logical entity. The development of this very valuable Platonic intuition, will be due to Aristotle, with his doctrine of act and potency.

Through the distinction of two irreducible principles, “matter” and “form”, constitutive of the essence of every physical entity, regardless if it is substance or accident, Aristotle furnishes us with a reply to the problem of Parmenides about the presumed contradiction within the concept of becoming. In order to realize this, he considered matter as a kind of “potentiality to exist”, or “potential being”, and form as “actuality of being”, which actualizes matter determining it to exist as a specific actual being. “Becoming” is not a transition from being to non-being or vice-versa, but rather from the state of potential being to the state of actual being and vice-versa ( METAPHYSICS, I.1; MECHANICS, II.2). The distinction between potential and actual being, allowed Aristotle also to introduce a diversification of the entities that accounted for the multiplicity of all the “nuances” encountered in the experience of real beings, and also to consider infinity in terms of potency and act. In this way was born the distinction, that later became classical, between “potential infinity”, always subject to “enlargement” and never considered within its totality, and “actual infinity” which is a concept of infinity with respect to fullness, considered simultaneously and in its totality.

The potential infinity of the “matter” introduced by Aristotle is said to be an essential, irreducible indetermination of the finite material substratum of the physical entity. This essential indetermination of matter is defined by Aristotle as «a delimited being, even always different» (Physics, III, 206a, 34), similar to that which will later be the introduced by Cantor (see below, IV). The potential infinity of matter, then, is what the matter without form would be, an «always different being», the pure casuality of becoming. For example, a casual sequence of numbers is a sequence in which no repetition can be found, no periodicity, no function or “law” capable to predict successive values on the basis of the preceding data. The matter, as a constitutive principle of each physical entity, refers essentially to the intrinsic instability of the motions of a substratum of elements. Therefore, according to Aristotle, matter deals with both potential being (matter itself) and actual being (matter that receives a form). The potential infinity of the physical matter is, in short, an absolutely unpredictable becoming, without any stability or periodicity, and without any law or mutual order of the parts, neither in space nor time. Then, for Aristotle, physics cannot be reduced to geometry, contrary to the majority of Greek and Modern thought ( RELATIVITY, THEORY OF, IV.5). It is worth underlining how the Aristotelian philosophy of nature actually does give us a better account of the state of the art of contemporary physics, following the discovery of the prevailing role of the dynamic instability in the study of real physical systems ( DETERMINISM/INDETERMINISM, II.4).

II. The Reflections of Thomas Aquinas and the Theological Significance of the Notion of Infinity

With respect to Aristotle,  Thomas Aquinas (1224-1274) pointed out some important aspects of the problem of infinity that found in more recent times also a confirmation by the mathematical approach to infinity that is today considered to be classical, as it has been developed by George Cantor. Aquinas demonstrated that while the notion of “infinity in act” is, as such, contradictory, it is not contradictory, instead, to refer to “actually infinite objects” (“absolutely” or “relatively”). Regarding this matter, it is of fundamental importance to underline that, different from what is often believed, Thomas did not refuse actual infinity as such, but he specified the conditions that are necessary to correctly refer to it. In particular, it is important to understand in which sense Aquinas considers it contradictory to speak of «collection-objects infinite in act» (infiniti in actu), whilst it is not completely contradictory to speak of «actually infinite objects» (actu infiniti), both relatively (secundum quid), and absolutely (simpliciter). Bearing in mind that within the context of actually infinite objects (in an absolute sense) there can exist only one Absolute or Subsistent Being — they are philosophical attributes that Aquinas recognizes to be proper only to  God —, the medieval Master proposes a distinction regarding three kinds (lat. genera) of infinity: a) «potentially infinite» objects; b) «actually infinite objects in a relative sense»; c) «actually infinite objects in an absolute sense». They all result in perfect consistency, and in opposition to the contradictory notion of objects “infinite in act”. The Thomistic conception corrects and enlarges that of Aristotle, who instead recognized only two types of infinity within science, namely that regarding potency (consistent) and that with respect to act (inconsistent). It would therefore be erroneous to believe that the adjustment carried out by Cantor — using the methods of set theory to rigorously confront the different types of infinity that exist within mathematics — denies the Scholastic philosophical doctrine that considered the notion of infinity in act tout court contradictory, except when applied to divinity.

A careful study of the work of Aquinas demonstrates that the notion of “relative actual infinity” (infinitum actu secundum quid) was perfectly recognized by him to be non-contradictory, and therefore admissible as a logical entity, also if it is not a “constructive” notion (as infinitum “in actu”), because it would be contradictory to conceive of an infinity as potential and, at the same time, as completely actualized. It therefore makes sense, for Thomas, to speak of the actuality of a “relative infinite” only as the negatively conceived infinite of a certain totality, according to one of its specific modalities of being. For example, Aquinas says that it is perfectly consistent to affirm that the totality of the natural numbers is infinite, because there does not exist nor can there exist a maximum natural number of each infinite sequence of natural numbers. In other words, if one wishes to speak of an “upperbound” of a set such as the natural numbers, one must consider it as “exterior” to the set of the naturals; it must be thought of as something belonging to another kind (lat. genus; a transfinite number in the sense of Cantor) and not belonging to the sequence itself as its “maximum” number.

Regarding the infinity considered as an attribute of God, one should remember that he dedicated a whole “question” to the subject in the Prima Pars of the Summa Theologiae (I, q. 7: De infinitate Dei) immediately after the Goodness of God and immediately prior to his exposition about the Presence of God in all things (gathering there some results of his previous reflection on infinity: cf. q. 8, a. 4). In that Quaestio God is recognized to be the «infinite and perfect Subsistent Being». The order of the four articles is descending, from the philosophical-theological level to the physical, and finally to the mathematical level. He goes through the question «if God is infinite» (a. 1), the ones «if there is anything other than God which is essentially infinite» (a. 2), «if anything can be actually infinite in size» (a. 3), and finally asking «if there can exist an infinite number of things» (a. 4).

Within the logic of the Quaestio, the infinity of God derives from his reality of being a Pure Act, that is from the convergence between his fullness of Being and his unconditional capacity to be causa omnium formarum, without having within Himself any composition of matter. When attributed to “matter”, the infinity always contains some restriction; instead the “form” is itself unlimited, and only knows limitations if it is joined to the matter it actualizes, something that cannot happen in God. The infinity that is attributed to God is not the infinity that refers to quantity, but rather that which refers to the Pure Act. This Act determines, and therefore transcends, every quantity in the material order. The infinity of all the forms which are destined to join to the matter (but also the infinity of the matter destined to join to many possible forms, such as those which can assume the matter of a determined trunk of wood worked on by an artist) is always limited (and therefore only infinite in a relative sense), because it is conditioned by the “relative” infinity of that which it is composed. Regarding the  angels, in which no composition with matter is present, the non absolute infinitude of their forms is due to the fact that they possess, even in their spiritual multiplicity, a “determinate” nature, something that cannot be said of God. If something is to be considered as infinite other than God, it can be considered only as a «relative infinity and not an absolute, full infinity» (q. 7, a. 2, resp.). The consequence is that absolute actual infinities cannot be given in the dimensional order (bodies), neither physical nor geometric, because bodies are always considered to be entities possessing a determinate form.

The theme of the attribute of God as “Infinite” remains dear to mediaeval theology, which had perhaps known the first speculatively interesting formulation in the reflections of Anselm of Canterbury (1033-1109), when at the beginning of the Monologium he refers to God as «what nothing “greater than” can be thought» (id quod maius cogitari nequit). Later on, in the itinerary of the Proslogium, Anselm speaks of God as «something always greater than whatever one could think» (quiddam maius quam cogitari possit). In the first case it could be considered a notion that any human being is by nature predisposed to recognize as meaningful, while in the second case it is rather a conclusion reached by a believer who compares his knowledge of God with everything that is not God. Close by to other attributes such as Omnipotence, Truth, Goodness or Omniscience of God, the philosophical attribute of God’s “Infinity” has found a place in some professions of faith of the Magisterium of the Roman Catholic Church, as that reported by Vatican Council I (1870): «Creator and Lord of heaven and earth, omnipotent, eternal, immense, incomprehensible, infinite in intellect and will, and in every perfection» (DH 3001). On the other hand, it must be remembered that the term is predominantly philosophical, and it is hardly present, as such, in the Scripture. In fact, Sacred Scripture considers the theme of the divine “infinity” under other lights, which are mainly existential and historical-salvific in character. So are God’s omnipotence and lordship over history and his incomparable love for humankind. This has been revealed to us in Christ, as beautifully expressed in the known Paoline citation: «And that Christ may dwell in your hearts through faith; that you, rooted and grounded in love, may have strength to comprehend with all the holy ones what is the breadth and length and height and depth, and to know the love of Christ that surpasses knowledge, so that you may be filled with all the fullness of God» (Eph 3, 17-19).

III. Infinity within Galilean and Newtonian Science

It is without doubt correct to identify the birth of modern science and mathematics, in their useful cross-fertilization, with  Galileo Galilei (1564-1642); particularly the proof he offered for the law of fall of bodies and the consequent necessity to admit “actual infinity” as an indispensable component of the theoretical framework of the “new” science (cf. Koyré, 1957). In fact the Galilean supposition that each moving body must actually cover “all” the infinite «momenta of smaller “fastness” (or greater “lateness”)», contrary to the Medieval (non-Aristotilean) theory of the impetus, necessarily implied the acceptance of the actuality of infinity. Torricelli (1608-1647) and Cavalieri (1598-1647), who followed the Pisan scientist applying his “geometry of infinitesimals” and deepening the message contained therein, even without a knowledge of Archimedes’ Metodo (discovered in 1906), employed the idea of the divisibility of a continuum within an actually infinite set of undividable parts. It is also known as Cavalieri thought, yet different from Torricelli who had a more pragmatic attitude, that one could ascribe to the method of the indivisibile infinitesimals (Principle of Cavalieri) the dignity of a formal demonstration, not considering it a simply heuristic procedure, as supposed by Archimedes. And this, regardless of the fact that Cavalieri never affirmed explicitly that the continuum was composed of an infinity of indivisible infinitesimals, an assertion which is not rationally demonstrable (cf. Koyré, 1973, pp. 334-361).

Galileo, instead, introduced in all his scientific discussions a principle that I could call “Parmenidean”. He linked the existence of infinitesimal elements, on one side, to the necessity to conceive them, and on the other side to the non-contradictory nature of their existence, following in this respect a very particular way of reasoning. He affirmed that the continuum is composed of the infinite indivisibile «non quanta» it possesses (i.e. unextended elements). Thus he intended to overturn the criticism of the Aristotelians affirming that, if an extended quantity can be divided into an infinite number of parts, then it is supposed that the parts themselves are infinite, otherwise the «division would stop» at some step. On the other hand, if these infinite parts were «quanta» (extended), their infinity would imply that the resultant extension of their composition was infinite. Therefore every continuum is composed of infinite undividable parts that are non-quanta (cf. Lombardo-Radice, 1981, p. 33).

To understand a similar, rather original way of reasoning, we must know that for Galileo, in compliance with Greek Euclidean mathematics, the arithmetical “unities” were not “numbers” but rather the expression of proportionality ratios existing between continuum quantities. This principle had an immediate experimental significance for him, since he carried out his calculations through the usage of only integer numbers and fractions; these were ratios derived from the measurement of continuum quantities, which he subsequently varied in order to render such ratios all “rational”. As historians have shown, he worked with two parallel tables, one which reported the variations (double) observed for the length l of his pendulum, and the other that reported the times t necessary for the pendulum to complete a half period of oscillation, measured in “grains of water” by his very ingenious water clock (by which he could regulate the intensity of the water flow). Employing such parallel lists, he discovered the law of fall of bodies in the form of the square of the times (t²). In fact he realized that, assuming each time t was the geometric mean between the number 2 and the length of the pendulum, the two tables become correlated row for row. Properly varying the two magnitudes in question, he further verified, in short, that the relation was bi-univocal only if this relationship was satisfied (cf. Drake, 1990).

When considered within its original experimental context, one can better understand Galileo’s idea of the continuum, as composed of indivisible non-quanta. With an all Italian geniality he attempted to lead the new experimental science back into the riverbed of great Greek tradition, that did not consider unity (a fundamental of discreet quantification) a number, but rather as a generator of reciprocally irreducible numerical sets, obtained through the sequence operation. Vice-versa, the beginning of modern mathematics corresponds to the renunciation apropos this reciprocal irreducibility between numerical sets founded on the non-numerical character of unity; modern mathematics considered as numbers also those irrational limiting of ratios that the Greeks instead considered as non-numbers. A consideration that, through the invention of infinitesimal calculus and the progressive definition of the concept of limit, brought us to the theory of real numbers as conceived by Dedekind, and therefore to the Cantorian theory of sets, that was the first attempt to give a coherent systematics to the whole problem ( CANTOR, IV). Correlatively, the history of modern science brings us to the renunciation of the original formulation of infinitesimal calculus based on “indivisible infinitesimals”, defended by  G. Leibniz (1646-1716), thus resolving the paradox of Zeno, implicit in the affirmation that the sum of the infinite quanta parts brings us always to an infinite result, thanks to the discovery of convergent series and their application to the physics created by  I. Newton (1642-1727). A discovery, which was strictly related to that algebraic formulation of geometry to which  R. Descartes (1596-1650) brought a relevant contribution. To found the infinitesimal calculus, we no longer need the concept of actual infinity, or alternatively to consider “all” the points, “all” the lines or “all” the planes included between whichever two points are on a straight line, between two sides of a polygon or between two surfaces in a volume. Starting instead from a “finite” subdivision, we can make an approximate calculus to reach a limiting result. The approximated calculus is then transformed into an exact calculus when the number of the parts tends toward infinity, becoming every part “evanescent”, and in this sense infinitely small, that is to say “infinitesimal”. Thus the notion of “limit” was implicitly introduced by Newton, a notion which through successive stages, became formally rigorous in the 18th century. So we are led finally to the days of  George Cantor (1845-1918).

IV. The Infinity of Cantor

1. Three Kinds of Infinity. It is by now historically certain that Cantor considered three kinds of infinity (cf. Hallett, 1984, pp. 8-10), with a modality of distinction that has many points in common with Thomist conceptions (see above, II): a) the “potential” infinity, indeterminate and incrementable, b) the “transfinite” infinity, determinate and incrementable, c) the “absolute” infinity, determinate and non-incrementable. Also for Cantor, the latter infinity can be said only of God, the Absolute Being. Such distinction was used by the mathematician of St. Petersburg to oppose two kinds of ideologies typical of the Enlightenment and regarding modern science and mathematics: one of a Spinozian nature, the other Kantian, disproving their two main fundamental principles.

The former is based on the Spinozian equivalence between God and Nature (Deus sive Natura), which is a fundamental principle of modern theoretical atheism ( PANTHEISM, II.2). Such equivalence is based upon the supposed reducibility of both the notions (God and Nature) to the same notion of actual infinity, completely determined, which is taken as the immanent foundation both of the order of nature itself, and of the necessary and universal character of the explanations of the «new geometrically conceived Galilean-Cartesian science» of nature. The latter is the Kantian conception that considers actual infinity as a limit towards which potential infinity tends, and founded upon the four Kantian anti-metaphysical antinomies, regarding the idea of world: finiteness/infinity, discretion/continuity, indeterminism/determinism, caused/uncaused. All those antinomies were based on the very same conception of the absolute actual infinity understood as the limit towards which the finite tends.

Against both of these bastions of rationalist 18th century philosophy and its anti-metaphysical program, Cantor poses the distinction between relative actual infinity, or “transfinite”, as a mathematical notion ( CANTOR, III), and the “absolute” actual infinity, as a metaphysical and theological notion, typically attributed to the divine nature, and absolutely unreachable by pure mathematical knowledge. Unfortunately, Cantor thought his view on infinity as opposed to the Thomist conception, because of his insufficient knowledge of Aquinas’ thought, together with the insufficient scholarship of some of his interlocutors. Therefore he was led to believe that he had to systematically oppose the Scholastic philosophical doctrine with his conception of actual infinity in mathematics. The necessity of actual infinity here re-appears in a sense that joins the “Parmenidean” instance with the “Platonic” instance. The necessity for the existence of actual infinity is so linked by Cantor with the necessity for its conceivability (Parmenidean instance), properly in relation to the rigorous definition of the notion of limits within the analytical calculus and regarding the definition of Dedekind of “real number” as the limit of a sequence of “rational numbers” not belonging to the sequence itself. These two notions in fact imply that, in order to let mathematics be founded on them in a really auto-consistent way (Parmenidean instance), the indefinite variation of the finite (potential infinity) requested by the notion of limit has to suppose the a priori “complete determination” of the domain of variation (Platonic instance). «There is no doubt that we cannot do without the variable quantities within the sense of potential infinity; and that from this can be demonstrated the necessity for actual infinity. In order that there is a variable quantity in a mathematical theory, the “domain” of its variability must be, strictly speaking, known ahead of time through a definition. Thus, the said domain must not be itself something variable, otherwise every base founded for the study of mathematics would vanish. Consequently, this domain is a definite set of values, and is thereby actually infinite» (Cantor, 1886, p. 9).

The three notions of infinity are strictly related to the three fundamental principles that give unity to all Cantorian thought: reductionism, finitism and the instance (or the postulation) of an Absolute. The necessity that mathematics, at its more fundamental level, must deal with “complete domains of variation”, considered as its proper objects, brought Cantor to affirm his thesis that, at such a foundational level the mathematics must concern itself with collection-objects, to be more precise, with “sets” . The operation of reducing to sets all the objects which are conceivable without contradictions, can be defined as the “reductionism” of the set theory of Cantor. He decided afterwards to regard the same mathematical infinity as “finite objects”, that is to say as “closed totalities”, a concept that brought him, as we know, to develop his theory of “transfinite”. He hoped to be able to reduce within the notion of transfinite also the concept of the continuum, in particular that which is linked to the re-elaboration made by Dedekind. Finally, and probably as a result of a rooted Platonic prejudice, he perceived the necessity that all conceivable mathematical entities, and therefore all the sets, were parts of some “absolute collection”, in which they would result all definite, in order that they could enjoy their utmost consistency. At the same time, it was necessary that this absolute collection was not mathematically determinable, and in particular was not itself a set, in order to escape from the corresponding antinomies ( LOGIC, III.3).

The geniality of Cantor, universally recognized in mathematics only after David Hilbert (1862-1943), consisted in the fact that he had already understood the nucleus of every antinomy. Though he was close to various aspects of the thought of Von Neumann (1903-1957), Cantor distanced himself from the Hungarian mathematician, because the foundation he gave for the absolute collections was not axiomatic, but rather metaphysical. However, the Cantorian demonstration, through the antinomy of the “power set” (the set of all subsets of a given set), of the inconsistency and contradictory nature of the “universal set”, or “set of all sets”, is not a sort of accident within the course of the development of his research. It is something profoundly linked to the Platonic roots of his metaphysics regarding the ultimate non-predicability of the One that, at least in this context, is profoundly anti-Parmenidean. The Absolute cannot be conceived, without contradiction, as a univocally definable specific set, or more precisely as the “totality of being” or a “semantic whole”. At the same time, the other root of the Cantorian metaphysic, that is the Parmenidean one, linked to the assumption that the conceivability formally implies existence, helps us to see that, indeed, one cannot be Platonic without being Parmenidean. By Cantorian terms: «everything which is not logically inconsistent exists» and further exists within the «absolute collection»; it follows that each entity exists insofar as it belongs to the said Absolute collection, and vice-versa each entity belonging to the Absolute collection is implicitly [unpredicatively] considered as actually existing. From this point arises the authentic “original structure” of every contradiction.

The ruinous “indecision” between the Platonic instance and the Parmenidean instance is in my opinion the ultimate root of the theoretical weakness of the Cantorian theory of sets. The first implies the non-definability of the Absolute as a set or a “closed” infinite totality, and therefore its “transcendence” with respect to predicative mathematics, and its simultaneous necessity with respect to the mathematics as Absolute collection where all the non-inconsistent entities already exist and remain for ever. The second instance implies that existence depends only on the logical non-inconsistency, and therefore on belonging to the universal set. The question about the “unity” of the concept of set and about the foundation for this unity, is the central question and like the culmination of the Cantorian reflections, particularly after the discovery of the so-called antinomy of Burali-Forti (1861-1931), which points out the inconsistency of the idea of “absolute ordering”. This consideration, risen from within Cantorian theory, prompted Cantor himself to reflect on the real consistency of his idea of absolute collection, though, without ever making him arrive at a doubt with respect to the existence of a set that ultimately must not be founded on its state of belonging to the absolute collection.

After solving the philosophical problem of a correct understanding of the Cantorian concept of “actual infinity”, we arrive to consider the more constructive aspect of his set theory, i.e. his positive consideration of the actual infinity, the “transfinite”. This notion introduces us immediately to the consideration of the antinomy of the “power set”. According to Cantor, this antinomy is a very precious verification that the Absolute cannot be conceived of as an ordinary set.

2. The Notion of Transfinite. In his study of infinity within the fields of mathematics and philosophy, when introducing the theory of Cantor’s concept of transfinite, Lucio Lombardo Radice (1981) affirms that the constructive center of the idea of Cantor is the criticism of the notion of limit that Kant had introduced into modern philosophy, along with his antinomies of the continuum. That is to say, the criticism of the belief that «the limiting entity of the finite is the Absolute». This would be true if there existed one mode only of considering actual infinity, but for Cantor it is not true: there also exists, as we know, the transfinite infinity, or to be more precise, the “determinate” infinity, specified within a limit, yet nevertheless “incrementable”, where not all the terms of the series are “actually defined” (i.e. enumerated). Cantor showed that every infinite “countable” set of elements, a set, he means, that can be put in a bi-univocal correspondence with the infinite denumerable set N of the naturals (N = {1,2,3,4, …, n, …}) «has the same infinite cardinal number of elements». It is easy to show that, for example, the set Z of the relative integer numbers (Z = {… –n … -3, -2, -1, 0, 1, 2, 3, …, n, …}) is countable. With an analogous procedure it can be shown that also the set Q of all the rational numbers is countable, and that even the set U of all the countable sets is again countable. This very powerful theorem shows us that an infinite set can have as the same “power” (cardinality) as one of its proper parts (subsets). But to affirm that an infinite set has the same power as one of its parts, does not signify an absolute negation of the classical principle according to which . As a matter of fact, (or to be “equipotential”) does not mean at all (cf. Lombardo Radice, 1981, pp. 52f): a “part”, by definition, lacks something that the “whole” has, and therefore can never be identical to the whole. For the same reason much attention must be given to the attribute “actual infinity” applied by Cantor to his concept of the transfinite. In the language of Thomas Aquinas, the infinities “determined but incrementable” of Cantor’s theory, that is the infinite totalities that he teaches us to treat actually, would be simply “actual” infinities which are not “in act”, since they are incrementable. That is, they are infinities only under some “special respect” (secundum quid), while they are essentially “finites”. And it is so because all that is determined and/or specified, is for this same reason “delimited” within an essence, even if it is “infinite” with respect to some modality or property. In the case of Cantor, the element of indetermination of the transfinite is really the fact that it is incrementable, the fact that an infinite totality can be conceived as specified in itself, even if its elements cannot be caught all simultaneously in detail. «The terms of an infinite series are all given by the constructive law provided to define the series, which makes useless to enumerate them individually: it is the law that characterizes the series, and no longer “all” the terms actually enumerated» (Zellini, 1980, p. 195).

Starting from the specific difference of the “countability”, Cantor has defined a criterion of uniformity which allowed him to extend rigorously the concept of denumerability to numerical sets other than N. Having discovered how to correlate the elements of an infinite set in order to show if it is countable (to put their elements into a bi-univocal correspondence with those of N by means of a particular order), Cantor has also been able to discover that Z and Q have the same power as N. Having a specific difference in common (the countability), they belong all to a same type (lat. genus), which he defined the «transfinite of minimal order»: the fundamental transfinite order of the infinite with the power of a countable set. But since he did not know the metaphysical Thomist principle regarding the real difference between essence and existence, and persisting in his Parmenidean prejudice with respect to the existence of any logically non inconsistent entity (essence = existence), he did not recognize that if he had re-read his discovery in the light of the essence-existence difference, he would have found the way to obtain the generalized and universal «principle of limitation of the extension of a set», that he sought in vain throughout his life and would have solved every antinomy. Thus, he continued to interpret the incrementability of a transfinite set as a pure logical contingency. And he continued to search “elsewhere” for those elements that are not actually defined inside the transfinite continuum, and to consider them as formally existing within the interior of the Absolute collection, so failing in the attempt to found on the transfinite the reductio ad unum that he had so long and unsuccessfully looked for. From here arose the final inconsistency of his justification of the notion of set and, in particular, of the notion of number based on set theory, that has opened up the road to future axiomatizations.

3. The Power of the Continuum. From the Antinomy of the Power Set to the Axiom of the Power Set. Another very important consequence of the Cantorian theory is the fact that the real numbers, placed by Dedekind into a bi-univocal correspondence with the points of the “continuum”, and therefore the continuum itself, exceed the power of a countable set. In fact the real numbers are defined by Dedekind as limits of a sequence of rational numbers, a limit that lies “outside” each of these sequences. The problem that Cantor then posed was the following. The power of the continuum is greater than the power of a countable set (called aleph-0): but is the power of the continuum the one immediately greater than that of a countable set? Since Cantor did not succeed to construct transfinite cardinals that were included between the power of the countable and the power of the continuum, he formulated the conjecture of the so-called continuum hypothesis (Continuum Hypothesis, CH), i.e. «the continuum has the power immediately greater than that of the countable».

Immediately related to the notion of power set understood as «the set of all the subsets of a given set», there is the «antinomy of the power set»; it is precisely a contradiction arising when considering a set of maximum power, or «universal set». If we observe the construction of the power set of a set A, we see that it includes also A itself as one of its subsets. It follows that «any set is a [proper] subset of its power set; therefore the set of all sets, or universal set U, may not exist», just because it should be a proper subset of its power set, and it would not therefore be any more the universal set, in contradiction with its same definition.

As we have seen, the demonstration of the antinomy of the power set was functional for the Neoplatonic notion of «Absolute collection» and therefore for the notion of transcendence as pure “indescribability”. Within Cantor’s thought, this notion was strongly conditioned by the Augustinian theology relating to number, which placed the actually infinite totality of the numbers as fully known within the mind of God (cf. Augustine, De civitate Dei, XII, 18; cf. Hallett 1984, pp. 35-37).

Later on, then, was discovered the antinomy of Burali-Forti, who showed that not only the idea of a universal set founded on cardinal numbers is inconsistent, but also the idea of “absolute ordering” leads to a contradiction. The latter depends on the original properties of the transfinite ordinals. In fact, while the properties of the finite ordinal numbers coincided with those of the finite cardinals, it is not so for the infinite ordinals. This antinomy placed into crisis the same Cantorian idea of “absolute”, as a maximal set of all the transfinite orderings. In fact, if this maximal ordinal set was to exist, its limiting element should belong (as it is the maximal set) and at the same time should not belong (as it is the limiting element) to the ordinal set that it orders.

V. Concluding Observations

The concept of infinity, in conclusion, is a notion with more than just a passing significance. Departing from the first thinkers who generally identified it with the “indeterminate” (Anaximander, Parmenides), or the “limitless”, picking prevailingly the negative value of imperfection, the concept then developed thanks to the comprehension of the different modes of significance of the transcendental notion of “entity” (analogia entis), until it assumed a positive value of full and perfect actuality, deprived of every limitation (Plato and Aristotle). This course of elaboration of the notion with its diverse significations was possible thanks to the concurrent contributions and co-operations of mathematics, metaphysics and theology. On the mathematical side came the need to explain the multiplicity from the numerical point of view: not only the concept of “one” exists, but also the concept of “two”, as observed the Pythagoreans; from this dyad the construction of all the sequence of the natural numbers became possible, and then the sequence of the integers and of the rational numbers, understood as successors of “one” (as, in modern times, can be found in the axiomatization of Peano who based his primitive concepts on the ideas of “one” and of “successor”); finally the set of the irrational numbers, that are defined as limits to which certain sequences of rational numbers tend (Dedekind). On the side of metaphysics came the need to explain the multiplicity of being (and therefore the nature of motion and becoming), that would result as a pure appearance in a Parmenidean perspective, but also the need to resolve the contradiction of a totally undifferentiated entity (Melissus).

This multiplicity, once founded for the entity as such, in physics-metaphysics (already with Democritus), and for the number (and later for the sets) in mathematics, could present itself, at least conceptually, as finite or as infinite, as incrementable or as non-incrementable. Aristotle, with his doctrine of the potency and of the act, succeeded to explain the differentiated multiplicity of the entity and to conceive the same infinity as potential, therefore as actually finite but incrementable to infinity or actually infinite and not ulteriorly incrementable. Thomas Aquinas and Cantor distinguished actual infinity as two types: relative (secundum quid for Aquinas and «transfinite» for Cantor) and absolute. An infinite object, in fact, when actually considered as a unique “thing”, can be “relatively” infinite when it does not include into its interior something that delimits it, as happens for the sequence of the natural numbers; it can be “absolutely” infinite when an infinity does not contain some limitations regarding either its interior or exterior, being therefore perfectly actualized, as can only occur for God. And it is only thanks to the analogy of being (lat. analogia entis) that we can conceive this absolutely in act Infinite actuality — avoiding the Parmenidean contradiction — as placed at a level of being that is “beyond every other type” of entity, beyond every distinction of the genera and of the differences (transcendence). On this level metaphysics meets up with theology: the absolute actual infinity of the metaphysical God recognizes itself in the revealed God of theology; the former is a glimpse caught from natural reason, that shows the existence of God and his main attributes, among which the infinity, the former is received by us through Revelation, the initiative of God himself that discloses to human reason both what we can know of Him, even with difficulty and the risk of mistakes, and what we could never even imagine.

Thus what is relevant from a logical point of view, both for mathematics and for metaphysics or theology, is the following theoretical result, which is part of the common foundations of these three disciplines: to consider and speak of actual infinity, and even of many kinds of actual infinities, some of which are relative and only one Absolute, is not contradictory.

I think it can be said, in conclusion, that after Cantor and Russell we have learnt today that, for a particular science, the rigor of an advanced formalism, which only a constructive approach can guarantee, can be used only with a subset of the proper objects of that science. To be precise, we have learnt that we have to limit the extension of the sets we want to construct, that is of those sets the existence of which can be demonstrated, and not only supposed by means of some specific axiom. The challenge that opens itself up today then is: must this subset of all the objects we construct be fixed or can it vary, thereby varying with it the axioms of the related formal system? In other words, need we suppose that all the objects inside the universal collection are actually existing, or may we consider as actually existing, time after time, only different subsets of them, and all the others to be only virtually existing? Here, by “virtually existing” we mean “called to actually exist” by means of a suitable demonstrative procedure — therefore constructively — every time that arises a logical necessity for it. This is the suggestion that the Thomist vision seems able to give to the search that challenges those scholars who are dealing today with the theory of foundations; by fact, they have arrived at the thresholds of metaphysics, and are confronting the same problems that occupied the greatest minds of antique times. And from this meeting can reasonably mature useful fruits for mathematics, metaphysics and theology; such fruits concern the level of a theory of common foundations for the diverse disciplines, and constitute the indispensable premises for a synthesis that restores to knowledge its equilibrium and unity.

Gianfranco Basti
(translated by Ruan Harding)

See also: ANALOGY; LOGIC; MATHEMATICS, SAPIENTIAL VALUE OF; MATTER; MECHANICS; METAPHYSICS;

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