Synopsis of Georg Cantor, ‘Fondements d’une théorie générale des ensembles’
[‘Foundations of a general theory of sets’]
Georg Cantor (1845-1918) was the founder of modern set theory, which over the course of the early twentieth century came to serve as the most widely accepted foundational system for mathematics as a whole. The article translated here foregrounds several of the key principles animating the Cercle d’épistémologie, and deserves to be considered at some length.
Cantor is best known for his demonstration of variably sized infinities, or distinct infinite numbers. This demonstration overturned Aristotle’s longstanding judgment that, though conceivable as an unending potentiality (1, 2, 3...?), the infinite as actuality or completed ‘number’ was an incoherent idea: before Cantor, it was generally presumed that infinity, by definition, could only be thought as having a single supra-numerical or indeterminable ‘size’. But in the early 1870s Cantor proved that the infinite set of real numbers used in mathematical analysis or calculus was infinitely larger than the infinite set of all whole or ‘counting’ numbers. He then came to accept, ‘almost against my will’, the ‘point of view which considers the infinitely great not merely in the form of something growing without limit’ but as something that can be ‘fixed mathematically by [distinct] numbers in the determinate form of the completed infinite’ (CpA 10.3:42). Cantor demonstrates, in short, that ‘following the finite there is a transfinite (transfinitum), which might also be called supra-finite (suprafinitum); that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers’ (43).
Cantor’s demonstration of differently sized infinite number ‘powers’ or ‘classes’ hinged on his distinction of ordinal from cardinal conceptions of number. An ordinal number is determined by counting up from zero, and its place in the numerical order or sequence is what determines its value, i.e. as ‘first’, ‘second’, ‘third’, etc. By contrast, a cardinal number is a matter of its magnitude or ‘power’ [Machtigkeit]; the cardinality of a set is measured by the quantity of elements belonging to that set, without concern for the way in which they might (or might not) be ordered. To draw on an example invoked by Bertrand Russell: if we have say five teacups and five saucers on a tray, we can determine that each set contains five members by counting their elements up from zero; at the same time, without bothering to count out the number of cups or saucers, we can also establish that our two sets are of equal cardinality by simply comparing one against the other.
In the finite realm, ordinal value and cardinal value are coextensive as a matter of course, so this distinction might appear superfluous. In the transfinite realm, however, the ability to establish magnitudes or cardinalities without actually counting out the number of elements involved becomes crucially important. (To pursue our illustration: even if we have an uncountably large number of teacups and saucers, by comparing them one-to-one we can still establish whether we might have more of one or the other, and thus decide whether we are dealing with just one uncountable number, or two different ones). So long as they can be suitably ‘ordered’ in principle, Cantor believed, different transfinite numbers can be distinguished by their size or cardinality, without actually counting their elements. As Michael Hallett notes, however, what is striking and peculiar about Cantor’s conception of infinite numbers – and the basis for its ultimate ‘finitism’ – is precisely his ‘ordinal theory of cardinality’, i.e. his belief that all the transfinite cardinals ‘are capable of being ordinally numbered’, or ‘counted’ up from zero (1). Cantor effectively tries to treat the new domain of the transfinite more or less like an extension of the finite, i.e. as countable or orderable in principle. ‘For Cantor’, Hallett continues, ‘much more important than our ability to conceive of a collection as ‘one’ was God’s [infinite] ability to do so [...]. It is really God who has put elements together to form sets, not us’2
Cantor’s main concern in his ‘Foundations’, translated by Jean-Claude Milner specifically for the Cahiers, is with the general philosophical implications of his work. The piece was first published in French in the Acta Mathematica in 1882. A longer version was published the following year as part of his Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Manifolds]. Milner notes that he used the German as the basis for his translation (52), since the French article from 1882 was more of a reworking of Cantor’s ideas undertaken by his French colleagues than a proper translation. (It should be noted at the outset that this article was written before Cantor’s distinction between ordinal and cardinal numbers was formalised: what goes here by the name numéral – Milner’s translation for the untranslatable term Anzahl [as distinct from Zahl, the standard German term for number in the abstract sense of the word] – will evolve into the concept of ordinal number, whereas what Cantor calls ‘power’ will become effectively synonymous with cardinality).
Cantor begins his article with a distinction between what he terms the ‘infinite improperly understood’ [das Uneigentlich-Unendliche] and the ‘infinite properly understood’. The ‘improper’ (though not ‘bad’) infinite is conceived as an infinite indetermination. Understood in this way, the infinite remains derivative of the finite; the infinite is only ever understood as an extension of the finite, as an unending sequence of addition. This concept of the infinite is intrinsically indeterminate, and it inheres in the concept itself that it remain forever open-ended. By contrast, for Cantor the infinite proper is defined by the fact that it is always presented under a determinate form (36. The infinite is no less infinite for being determined, and one of the aims of the article translated in the Cahiers is to liberate the concept of determination from its traditional enclosure within the domain of the finite alone.
Cantor then distinguishes between distinct ‘classes’ or ‘powers’ of infinite numbers. ‘The first number class (I) is the set of finite integers or whole numbers [nombres entiers] 1, 2, 3 ..., ν ...; it is followed by the second number-class (II), consisting of certain infinite integers following each other in determinate succession; as soon as the second number-class has been defined, one comes to the third, then to the fourth, and so on’ (37).
In order to differentiate the magnitudes of these various classes, Cantor introduces two important new concepts. The first innovation involves what he calls the ‘concept of power’ [Mächtigkeitsbegriff, or concept de puissance]. ‘According to this concept, for any well defined system there is a determined power, and that same power is attributed to two systems when one can establish between them, element by element, a one-to-one correspondence’ (37). The second innovation involves the determination of a number by ordering the elements it contains in a determinate succession; such ordering determines the Anzahl [numéral] of a set. The ‘Anzahl of the elements of a well-ordered infinite manifold’ or set is ‘always expressed by a completely determinate number of our widened number-domain’, so long as the ‘ordering of the elements of the set is determinate’. Cantor explains that ‘a well-ordered set is a well-defined [i.e. “determinate”, “completed”] set in which the elements are bound to one another by a determinate given succession such that (i) there is a first element of the set; (ii) every single element (provided it is not the last in the succession) is followed by another determinate element; and (iii) for any desired finite or infinite set of elements there exists a determinate element which is their immediate successor in the succession (unless there is absolutely nothing in the succession following all of them). Two “well-ordered” sets are now said to be of the same Anzahl (with respect to their given successions) when a reciprocal one-to-one correlation of them is possible’ (38).
Now as Cantor observes, ‘for finite sets, power coincides with the Anzahl of elements, because, as everybody knows, such sets have the same Anzahl of elements in every ordering. For infinite sets, on the other hand, until now nobody at all has talked of a precisely defined Anzahl of their elements – even though a determinate power, entirely independent of their ordering, can be ascribed to them’ (37). Cantor then shows that the smallest infinite power corresponds to his first number class, and claims to show (on the basis of what he had anticipated in 1877 as his celebrated but apparently unprovable ‘continuum hypothesis’) that the second or next highest power corresponds to the second number class, and so on. ‘Every set of the power of the first class is countable by numbers of the second number class and only by such numbers; every set can always be given a succession of its elements such that it can be counted in this succession by an arbitrarily chosen number of the second number class, which number gives the Anzahl of the elements of the set with respect to that succession’ (39).
Reference to the distinction of power and Anzahl also allows Cantor to specify ‘the essential difference between finite and infinite sets [...]: a finite set presents the same Anzahl of elements for every succession which one can give its elements; in contrast, a set consisting of infinitely many elements will in general give rise to different Anzahlen, depending on the succession that one gives the elements. The power of a set is, as we saw, a property independent of the ordering; but the Anzahl of the set reveals itself to be, in general (as soon as one has anything to do with infinite sets) a property dependent on a given succession of elements. Nevertheless, there is even for infinite sets a certain connection between the power of a set and the Anzahl of its elements determined by a given succession’ (38). (In § 3 of Cantor’s original article, which is omitted from Milner’s translation, he further insists on this point, stipulating that ‘it is always possible to bring any well-defined set into the form of a well-ordered set – a law which seems to me fundamental and momentous and quite astonishing by reason of its general validity’, before going on to show how the fundamental operations for both finite and infinite numbers can be derived directly and with ‘apodictic certainty’ from the ‘concept of well-ordered set’).
What most strikes Cantor about his discovery is ‘how the concept of integer or whole number, which in the finite has only the background of Anzahl, as it were splits into two concepts when we ascend to the infinite – one of power [Machtigkeit] which is independent of the ordering which a set is given and one of Anzahl which is necessarily bound to a lawlike ordering of the set by virtue of which it becomes well-ordered [wohlgeordneten]. And when I descend again from the infinite to the finite I see clearly how the two concepts become one again and flow together to from the concept of finite integer’ (47).
Cantor reiterates the essential point of his article in § 4, when he begins to engage directly with the traditional (Aristotelian) assumption that numerical determination is limited to the domain of the finite alone.
I believe, however, that I have proved [...] that determinate countings can be carried out just as well for infinite sets as for finite ones, provided that one gives the set a determinate law that turns them into well-ordered sets. That without such a lawlike succession of the elements of a set it cannot be counted – this lies in the nature of the concept of counting. Finite sets too can be counted only if we have a determinate ordering [Aufeinanderfolge] of the counted elements; but here we encounter a particular property of finite sets, namely, that the result of the counting – the Anzahl – is independent of the particular ordering; while for infinite sets, as we have seen, such an independence does not in general hold. On the contrary, the Anzahl of an infinite set is an infinite integer which is codetermined by the law of the counting; it is precisely here, and here alone, that the essential difference, founded in nature itself and therefore never to be abolished, between the finite and the infinite is located. Never again will the existence of the infinite be denied because of this difference, but on the contrary the existence of the finite [itself] can now be upheld (42).
The second half of Cantor’s article deals in large part with earlier philosophical efforts to grapple with the infinite. Ever since Aristotle, ‘number’ has only ever referred to something that was intrinsically finite, and philosophers have generally sought to assert and preserve the apparent ‘limits to the flight of mathematical speculation, to show the domain within which the passion for mathematical thought will run no danger of falling into the abyss of the “transcendent” " – where, it is said with fear and wholesome alarm, “anything is possible.”’ In anticipation of the critique that Althusser and his students will later make of the epistemological obstacle that philosophy and ideology erect to obstruct the revolutionary development of a science, Cantor condemns such principles as ‘erroneous’ and stifling: ‘if they were actually to be followed, then science would be held back or banished into the narrowest confines’ (41). Aristotle wrongly assumed, Cantor argues, that counting or ordering could apply only to finite numbers. Aristotle also denied the existence of infinite numbers because he feared the annihilation or swallowing up of the finite in the infinite. But Aristotle’s fears can be proved groundless. If we begin with the finite, and add an infinite number to it, then the finite number is indeed consumed, while leaving the infinite number unaltered. But if we add the finite to the infinite, conceived on its own terms as a determinate number, then it is the infinite itself that changes, and the finite number that endures unaltered (42).
Cantor then refers to several modern philosophers, Locke, Descartes, Spinoza, and Leibniz (43). He devotes scant attention to the first two, but spends some time with Spinoza and Leibniz. Cantor endorses the Spinozist thesis omnis determinatio est negatio: ‘all determination is a negation’. Spinoza’s error, in his view, was to conceive of only two levels, an undetermined absolute and then the determined domain of the finite. Cantor accepts that there is a maximum or supra-numerical level – an undetermined absolute that we can correlate withGod – but insists that this does not affect the existence of the transfinite. Everything finite or infinite can be determined, except God (44). Cantor denies that the apparent finitude of the human intellect is an obstacle to his demonstration of a determinate infinite. ‘However limited in truth human nature may be, still very much of the infinite adheres to it, and I even assert that if it were not itself in many respects infinite, the solid confidence and certainty in the being of the Absolute, about which we know we all agree, would be inexplicable’ (44).
Cantor then points to a familiar difficulty in Spinoza’s system: the relation of finite to infinite modes. How can the finite maintain itself in the face of the infinite? Here Cantor returns to the argument he previously mobilized against Aristotle. This relation is only a problem if the finite is granted primacy. ‘If is the first number of the second class, then we have; by contrast, where is a number entirely distinct from. Everything depends, therefore, as we can clearly see here, on the position of the finite in its relation to the infinite; if the finite precedes, it passes into the infinite and disappears there; if it cedes the way however, and takes place after the infinite, it subsists and combines itself with an infinite that is new, because it is modified’ (45). Cantor accepts the primacy accorded to determination in Spinoza’s system, but argues for the anteriority of the infinite in all instances, even in the case of the human intellect. According to Martial Gueroult, whose lectures on Spinoza were important for several Cahiers authors, Cantor’s own arguments on this score are crucially anticipated by Spinoza himself, the differences between the thinkers having more to do with the nomenclature of number than the content of their positions (cf. Gueroult, pp. 583-4).
Leibniz’s shortcoming, for Cantor, is that he grants too much power to the ‘infinite improperly understood’, positing divisibility all the way down, and developing a mathematics which tolerates this regress. Cantor rejects the Leibnizian concept of infinitesimals (cf. Dauben, pp. 129-30), insisting that ‘all attempts aiming to transform by a coup de force the infinitely small into the infinitely small properly understood should at last be abandoned, and their futility acknowledged’ (40). Cantor detects a complicity between Leibniz and the atomism of certain materialist traditions, such as that of Democritus, which he also disavows, ‘notwithstanding all the useful discoveries permitted up to a certain point by this fiction’ (45).
Cantor goes on to laud the Czech mathematician and philosopher Bernard Bolzano as the nineteenth-century’s greatest defender of the infinite proper. In his work, The Paradoxes of the Infinite (1851), Bolzano identified the error that marred all attempts to make sense of, or circumvent, the ‘infinite improperly understood’. This kind of infinite – the one constituted by an incessant addition that never reaches an end – is what the scholastics called a ‘syncategorematic infinite’, that is, an infinite that names nothing but a relation. The ‘paradoxes of the infinite’ evaporate, however, once the infinite is recognized as a categorematic concept, that is, the concept for an entity regarded, not as relational but as an entity in itself. All that Bolzano lacked in order to make a positive breakthrough on this front was the distinction between numéral (ordinal) and power (cardinal). This distinction within the domain of number itself is what Cantor’s regards as the signal innovation of his own effort.
In the next section, Cantor affirms the ‘realist’ quality of his conception of number, but admits that it involves the distinction of two kinds of reality, an ‘intrasubjective or immanent reality’ and a ‘transsubjective’ or ‘transcendent’ reality (47). Though Cantor acknowledges the ‘conjoined’ nature of these realities, ‘transcendent reality’, he says, is a question for metaphysics, not mathematics. The power of mathematical thought results from the fact that, ‘in order to constitute its notional material, it must take into consideration uniquely and solely the immanent reality of its concepts, and as a consequence it is in no way obliged to test itself from the point of view of their transcendent reality’ (48). Cantor’s position here anticipates the orientation of the Cahiers project as a whole. Rather than worry about the correspondence between its concepts and transcendent reality, ‘mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real’ (48).
Since for Cantor ‘the essence of mathematics resides precisely in its freedom’ (49), he prefers the expression ‘free mathematics’ to ‘pure mathematics’. Making a claim that will be particularly important later on for Jean Cavaillès, Cantor insists that mathematics’ truth value is precisely a result of its being wholly conditioned by its own notional historical development, to the exclusion of anything extrinsic; in this way, its ‘immanent’ truth is guaranteed in a kind of ongoing self-validation. The domain of ‘applied mathematics’, by contrast, is best understood as a category of ‘metaphysics’ that ‘degenerates into a “description of nature” which necessarily compromises at once the living breath of free mathematical thought and the capacity to explain and establish natural phenomena’ (49).
The final section of Cantor’s text correlates his first two classes of numbers to two principles of numerical generation. The first class, the class of the finite integers, is generated by ‘adding a unity to an already formed and existing number’, i.e. through numerical succession. Cantor defines the ‘second principle of generation of integers [...] as follows: if there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i.e. is defined as the next greater number than all of them’ (50). Furthermore, ‘by following both principles of generation one obtains every again new numbers and sequences if numbers which have a fully determinate succession.’ (Cantor also suggests, in another evocation of his continuum hypothesis, that observation of a further ‘restricting or limiting principle’ should ensure that the ‘second number-class (II), defined with its assistance, not only has a higher power than (I), but precisely the next higher, that is the second power’ ). The new numbers generated along these lines, Cantor concludes, ‘are then always of the same determinacy and objective reality as the earlier ones. I therefore do not see anything that should hold us back from this activity of forming such new numbers, as soon as it seems that the introduction of a new number-class from these [already-established] innumerable number-classes is desirable or even indispensable for the progress of the sciences’ (52).
References to this text in other articles in the Cahiers pour l’Analyse:
- Cantor, Georg. ‘Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Investigation into the Theory of the Infinite’, trans. William Ewald, in Ewald, ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II. Oxford University Press, 1996, pp. 878-920.
- Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip Jourdain. London: Dover, 1915 (translation of ‘Beiträge zur Begründung der transfiniten Mengenlehre’, parts 1 and 2 [1895, 1897]). Online at http://www.archive.org/details/contributionstot003626mbp.
Selected secondary sources:
- Badiou, Alain. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
- Badiou, Alain. Number and Numbers, trans. Robin Mackay. London: Polity Press, 2008. Chapter 6.
- Cavaillès, Jean. Remarques sur la formation de la théorie abstraite des ensembles (1938) in Idem., Oeuvres complètes de philosophie des sciences, ed. Bruno Huisman. Paris: Hermann 1994.
- Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton: Princeton University Press, 1990.
- Dauben, Joseph Warren, Conceptual Revolutions and the History of Mathematics . In Revolutions in Mathematics, ed. Donald Gillies. Oxford: Oxford University Press, 1992. 49-71.
- Grattan-Guinness, Ivor. The Search for Mathematical Roots 1870–1940. Princeton: Princeton University Press, 2000.
- Gueroult, Martial. Spinoza I: Dieu. Paris: Aubier Montaigne, 1968, Appendices 9 & 17.
- Hallett, Michael, Cantorian Set Theory and Limitation of Size. Oxford: Oxford University Press, 1984.
- Tiles, Mary. The Philosophy of Set Theory: An Introduction to Cantor’s Paradise. Oxford: Basil Blackwell, 1989.