The Notion of ‘Set’ in Cantor’s Transfinite Number Theory

Lucy K. Compton

Lucy K. Compton is currently in her final year at the University of Southampton, England, reading philosophy. Her main interests lie in the philosophies of mind, action and mathematics, as well as in metaphysics. If awarded funding, Lucy will begin study for the MPhil at King’s College London, England in autumn 2005, specialising in the philosophy of mind.

I. Introduction

Cantor defines an ‘aggregate’ as: ‘any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the “elements” of M’. [1] An ‘aggregate’ is in Cantor’s terminology what we would today, more familiarly, call a ‘set’.[2] A set is a collection of objects grouped together to form a whole, just as the definite and separate ‘elements’ m go to constitute the whole, or aggregate, M. Given our greater familiarity with the notion of a set, I will use the term ‘set’ rather than ‘aggregate’ when referring to objects which when grouped together are referred to as a whole.

In this essay I wish to argue that Cantor is wrong to assert that the set of real numbers (R) has more elements than the set of natural numbers (N). The statement 'R has more elements than N' is true if we assent to: Cantor's definition of 'set'; his infinite sets; and his set of real numbers, R. I intend to argue that we while we might assent to Cantor's definition of ‘set’, though not uncritically, we should not assent to infinite sets on Cantor’s account. Thus, I will claim that R does not have more elements than N because there is no set of real numbers, R, though there are real numbers. To this end I will also consider Cantor’s Platonism regarding numbers and will critique his use of the notion of ‘infinity’.

The cardinality, c, of a set, X, c(X) for example, is its cardinal number: ‘the number that measures the size of a set’.[3] The cardinality of a set tells us how many members it has. If two sets, X and Y, have the same cardinality then the elements of those sets can be put in a one-to-one correspondence with one another―for each element of X there is an element of Y: c(X) = c(Y). A ‘denumerable’ set is one whose cardinality is equal to that of N. Hence, if R is understood to contain more elements than N, as Cantor wishes it to be, then it is not a denumerable set.

Taking for granted, for now, that N, though infinite, has some discernable use because it is enumerable, I suggest that we should still not assent to the existence of R. This is because, as Cantor claims, R has a greater cardinality than N: it is a non-denumerable set. I will argue that R, given that it is infinite and non-denumerable, is not enumerable: the elements of R cannot be counted. I assume therefore that a set of elements is a ‘set’ if and only if (iff) it is enumerable. Thus, because R is not enumerable it is not a set.

II. Cantor’s Argument

Cantor claims that R and N are infinite sets, and also that they cannot be put in a one-to-one correspondence with one another, 'c(R) ≠ c(N)'.[4] Hence, Cantor argues that the cardinality of R is greater than the cardinality of N, 'c(R) > c(N)’. Cantor asserts that 'if a and b are any two cardinal numbers, then either a = b a < b or a > b'.[5] Hence, Cantor claims that 'c(R) > c(N)', iff, 'c(R) ≥ c(N)' and 'c(R) ≠ c(N)'.[6] To prove this, Cantor must show two things. Firstly, that 'c(R) ≥ c(N)' is true, which is proven iff there is a subset of R, R’, such that ‘c(R’) = c(N)’. Secondly, that 'c(R) ≠ c(N)', which is proven if Cantor can show that R is non-denumerable via his diagonal argument.

R is an infinite set because it can be put in a one-to-one correspondence with a proper subset of itself. N is one such proper subset of R. Hence, 'c(R) ≥ c(N)' is true, because there is a subset of R, R’, such that c(R’) = c(N). As required, Cantor uses the diagonal argument to show that 'c(R) ≠ c(N)'. Any attempt to put the real numbers between 0 and 1 in a one-to-one correspondence with the natural numbers will fail; there will always be a real number left unpaired with a natural number.[7]

Basically, the diagonal argument shows that a real number between 0 and 1, which does not belong to the given list of such real numbers, can always be produced by altering the first digit of the first number at the first decimal place, the second at the second… and so on ad infinitum.[8] There is no one-to-one correspondence between R and N because there is always this extra real number. R is therefore non-denumerable according to Cantor; the cardinality of R greater than that of N. Thus, 'c(R) ≠ c(N)' is true. Moreover, given that 'c(R) ≥ c(N)' and 'c(R) ≠ c(N)' are both true, 'c(R) > c(N)' is true. Since sets R and N are both infinite (transfinite), Cantor believes he has shown that there is more than one infinite cardinal number. According to Cantor’s transfinite number theory, the c(N) is א_o (aleph-null)_, while c(R) is 2^אo (2 to the power aleph-null).[9]

III. Critique of Cantor’s Notion of ‘Set’ and Infinite Sets

I do not however accept that ‘R > N’--that the number of elements in R is greater than the number in N--because I do not believe that R is a set. I will first critique Cantor’s notion of a ‘set’ and infinite sets. I will then argue, taking Cantor’s definition for granted, that R is not in fact a set. Moreover, I will suggest that while N is not a set either because it is not a determinate totality it is a justifiable and usable notion.

A second definition of ‘set’ given by Cantor is: ‘a “many, which can be thought of as one’.[10] I would argue that something that is a many or collection of definite elements cannot be combined into a whole or one.[11] As Moore says, Cantor’s definition of ‘set’ suggests that it has a determinate size, however when we consider infinite sets, this seems to lead to ‘paradoxes of the one and the many’: ‘it seems that for there to be infinitely many things of a given kind is precisely for them to resist being collected together in this way’.[12] Cantor’s notion of ‘set’ rests on the intuition that we seem to be able to ‘recognize unity in infinite diversity’ and from this assume that it is possible to unify a collection of elements into an infinite set. However, given the nature of the infinite this does not seem possible;[13] the infinite is not something that humans perceive in their thoughts, at least not independently of God.

Rucker points out that a line[14] can never be exhausted by any set of discrete points, however large the set.[15] Hence, there cannot be an infinite set containing all the points of a divided line, since it can always be divided further. Rather there is never-ending, infinite, series of points.[16] The line, or points, of any set are prior to it.[17] Indeed, Cantor does admit that ‘the elements of a set are “prior to” it’,[18] and yet he maintains that these elements can be determinate totalities, even though there will always be more elements to add to N and R. If infinite sets cannot be considered as ‘whole’ then they cannot be sets; I believe that Cantor is wrong to suggest that a collection is a whole.[19]

Cantor, I believe, maintains that N and R, as infinite sets, are determinate totalities because of his Platonism concerning number. Cantor believed that numbers have mind-independent existence and this belief of his is based on his belief in God. [20] It is because of his Platonism, and because he believes that infinite sets are determinate totalities, that Cantor believes his transfinite numbers to actually exist, perhaps in Plato’s World of the Forms.[21] Cantor believes the infinite to have actual existence. I think it important to consider the notion of ‘infinity’ at greater length here.

We must not mistake the notion of the mathematically infinite, with the metaphysically infinite; the infinite series of points on a continuous line illustrates mathematical rather than metaphysical infinity. I believe that Cantor, in trying to see infinite sets as determinate totalities, mistakes the metaphysical for the mathematical infinite. The metaphysically and mathematically infinite are both responses to different intuitions. The former, to the intuition that we can abstract unity from infinite diversity—it is conceived as wholeness—and the latter, to the intuition that we cannot do this given the nature of the infinite—it is conceived of as endlessness.[22]

Cantor conceives N and R as totalities because he wants to give them definite existence, but if we do not accept his Platonism it seems that Cantor is founding mathematical notions on a metaphysical basis which they cannot have because they do not exist metaphysically. N and R are not to be conceived as metaphysically but mathematically infinite, for they are endless series rather than determinate wholes.

Cantor conceives set of numbers as being totalities so that he can claim their actuality; Cantor requires actual infinity[23] because he wants to say that infinite sets are metaphysically real.[24] However, I do not accept Cantor’s Platonism because there are no infinite sets but only endless series of numbers,[25] such as the stick that can always be divided further. We cannot identify the actually infinite—actual completion—with the mathematically infinite. The mathematically infinite, infinite cardinal numbers, however many we say there are, can only be identified with the potentially infinite—with uncompletability.[26]

For Aristotle, the [potentially] infinite is untraversable. To traverse an infinitely dividable line is impossible; there is no end to the process of counting the natural numbers.[27] N and R could only be regarded as actually existing as totalities, in a Platonic realm, if the counting of them could be completed. There is no end to counting either N or R and so Cantor’s infinite cardinal numbers, א_o and 2^אo, can not actually exist as totalities, rather they exist potentially as never-ending series of numbers.

In both of his definitions of ‘set’, Cantor relies on ‘our thought’ to combine the collection of ‘definite and separate objects’ into a ‘whole’.[28] N is the collection into a whole of the ‘elements’ n, the natural numbers. That these elements are of our thoughts seems to suggest that they are subjective. However, Cantor believed that numbers have mind-independent existence given his Platonism concerning number. Hence, we have another reason to reject Cantor’s notion of ‘set’: a set, essentially, should be an objective notion, yet Cantor regards it as being of our subjective thoughts.

Cantor would be unconcerned by this criticism because his Platonism objectifies his notion of a ‘set’. The subjectivity suggested by the fact that we can only think of an infinite set as a whole would not concern Cantor because he believed that the transfinite numbers are actually infinite, and so believed that they exist independently of our minds in a Platonic realm. Hence, they are objective. Furthermore, Cantor’s Platonism makes his notion of a ‘set’ objective because it is based on belief in God, who objectifies his notion of ‘set’.

According to Cantor, God could conceive the elements of a set as a whole because He is omnipotent and omniscient. Hence, for us to perceive in our thoughts a set of numbers as a whole is independent of our minds, but not independent of God. While we are only capable of thinking of potential infinity, God can conceive actual infinity thus objectifying N and R as determinate totalities existing in some Platonic realm. If we take away God though, we are left with no way to objectify the notion of a ‘set’ that Cantor provides.

Cantor’s definition of ‘set’ relies on faith, which he had,[29] but this does not justify it to everyone. Cantor’s notion of a ‘set’ requires proof of the existence of God if it is to be accepted by everyone. Without God to objectify our ability to abstract a whole from ‘definite and separate objects’ Cantor’s definition of ‘set’ is merely a subjective one, and cannot ground objects of mathematics.[30] Transfinite numbers would not be part of an objective mathematics but merely represent one possible structure for it.

If Cantor held that N and R existed mind-independently in a Platonic World of the Forms, where this belief was not based on God, it would be more tenable to accept that N and R exist actually as totalities rather than merely potentially as number series. Nevertheless, this position does not show how it is that humans can conceive an infinite series as a whole, given that it is essentially never-ending. Moreover, Platonism concerning numbers, including infinite ones, is―as is widely known―thwart with problems such as the infinite regression of Forms: it seems ridiculous for there to be a Form of the Form of the Form… and so on ad infinitum of 2 or א_o for example.

It could be argued, against my claim that N and R exist only potentially and not actually as determinate totalities, that we can think of infinite sets as determinate totalities just as we can large finite natural numbers. Bernardete argues, in objection to Wittgenstein, [31] that to say ‘there are infinitely many stars is much the same… as saying that there are a trillion’.[32] There is a confusion here because ‘infinitely many stars’ refers to metaphysical infinity, while a ‘trillion’ of them refers to mathematical infinity. To make use of Intuitionist thinking: is there a sense in which the construction of N is different from constructing all of the natural numbers with fewer than a trillion digits? I would argue, as Moore suggests, that, barring our mental and physical limitations, it is possible to construct the latter but we can never, necessarily, complete the task of constructing N.[33] Hence, we can think of very large finite natural numbers as determinate totalities but not infinite sets.[34] They are not sets because they are never complete. Moreover, according to Intuitionism generally, given that R and N can never be constructed we cannot say that ‘R > N’.[35]

IV. Argument that R is Not a Genuine Set

I do not think R, as a non-denumerable set, is tenable, assuming that we admit denumerable sets. Intuitionists do not recognize the infinite beyond what is denumerable; they do not accept infinite sets whose elements cannot be placed in a one-to-one correspondence with the first infinite ordinal, ω.[36] Ordinal numbers indicate the position of an element in a set: hence, the ordinal number of 1 is 1^st, the ordinal of 2 is 2^nd, and so on.[37] The first infinite ordinal according to Cantor, the first number after the entire set of natural numbers, is ω.

Poincaré argued that Cantor’s claim that ‘R > N’ could just as well be taken to mean that the diagonal method merely failed to pair off N with R, nothing more, or that R is not a genuine set because it is not a determinate totality. Peirce, concluded from ‘R > N’, unlike Cantor, that R did not exist as a completed whole and was at best potentially infinite.[38] I am in agreement with Poincaré and Peirce: R is not a completed whole, and hence is not a genuine set. That differing conclusions can be drawn from the diagonal method so blithely, only serves to emphasise the subjectivity of Cantor’s claim that ‘R > N’.

Now I have argued that N, as well as R, is not a determinate totality. It is here that my position departs from standard Intuitionist thought. I believe that Brouwer, though being an Intuitionist, would have considered N to be a genuine set because it is enumerable. Brouwer recognises only denumerable sets, those set which are enumerable: finite sets or infinite sets with cardinality equal to that of N.[39] Thus obviously, according to Brouwer N is a set. I however do not believe N is a set because, as I have argued, it is not a determinate totality. N is not a determinate totality because it does not exist actually, only potentially as a never-ending series of numbers.

Nevertheless, I think that N can be seen to have use in a Wittgensteinian light as it denotes the ‘endless possibilities’ of the finite, whereas R defies even this use because unlike N it is non-denumerable. Wittgenstein said that the [potentially] ‘infinite’, N, has use: it can be used ‘to characterize the form of finite things and to talk about the endless possibilities that finite things afford’.[40] For Wittgenstein, the uses of ‘infinity’ are straightforward and should be taken at face value: ‘0, 1, 2, …’ is not an abbreviation of the infinite sequence of natural numbers but part of mathematic symbolism. ‘0, 1, 2, …’ does not point beyond itself to a totality, N, but is itself mathematical reality; the endlessness of infinity is reality.[41] Milan Kundera provides a distinction between the external and the internal infinity which illustrates Wittgenstein’s meaning of the ‘infinite’. Infinity, which is external to us, such the number of stars in the sky, escapes our grasp, but internal infinity, the infinite pleasures and experiences afforded us by finite things such as other people, is something we can conceive and appreciate.[42]

Moreover, Kaufmann, who denied R, argued that N provided a convenient way of discussing finite sets, but given its non-denumerability R cannot do even this.[43] N is justifiable as a set because there is a rule for the determination of its totality, though it will never achieve completion, this being the principle of progression or ‘add 1’. This is at odds with traditional Intuitionist thinking as Intuitionists generally would require that N be completed, constructed, before it would accept that N existed.[44] R is not a genuine set, not even as N is, because there is nothing which could count as progression within its series; whatever infinitesimally small decimal expansion we tried to add to 0 we could always add a smaller one.

V. Concluding Remarks

In conclusion, the aggregate of real numbers does not have more elements than the aggregate of natural numbers.[45] While I accept Cantor’s definition of a ‘set’, where those sets are finite, I do not agree that this conception of ‘set’ justifies Cantor’s infinite sets: the transfinite numbers. Infinite sets are not sets because they cannot be regarded as determinate totalities, they are instead never-ending series. Hence, we should accept the notion of infinity only as it is regarded as being potential, and not as actual, as Aristotle argued. Furthermore, I do not accept that the set of real numbers is a genuine set because it is non-denumerable and unlike the set of natural numbers it cannot be regarded as an endless series of finite possibilities, because there is no rule whereby its series can be said to progress.

Works Cited:

Benacerraf, Paul. "What numbers could not be." The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983a. 272-94.

---. "Mathematical Truth." The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983b. 403-20.

Bernays, Paul. "On platonism in mathematics." The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983. 258-71.

Blackburn, Simon. The Oxford Dictionary of Philosophy. Oxford: Oxford University Press, 1996.

Boolos, George. "The iterative conception of set." The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983. 486-502.

Brouwer, L. E. J. "Intuitionism and formalism.", Trans. A. Dresden. The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam, Cambridge: Cambridge University Press, 1983a. 77-89.

---. "Consciousness, philosophy, and mathematics." The Philosophy of Mathematics: Selected Readings . 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983b. 90-6.

Brown, James Robert. Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures, London & New York: Routledge, 1999.

Cantor, Georg. Contributions to the Founding of the Theory of Transfinite Numbers, Trans. P. E. B. Jourdain, New York: Dover Publications Inc., 1955.

Frege, Gottlob. “The concept of number.” The Philosophy of Mathematics: Selected Readings. 2^nd ed. Trans. M. S. Mahoney, Eds. Paul Benacerraf & Hilary Putnam. Cambridge: Cambridge University Press, 1983. 130-59.

Gödel, Kurt. "What is Cantor's continuum problem?" The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam, Cambridge: Cambridge University Press, 1983. 470-85.

Hallett, M. Cantorian Set Theory and Limitation of Size. Oxford: Clarendon Press, 1984.

Huntington, E. V. The Continuum: And Other Types of Serial Order, With an Introduction to Cantor's Transfinite Numbers. 2^nd ed., New York: Dover Publications Inc., 1955.

Moore, A. W. The Infinite. London: Routledge, 1990. Chapters: Introduction, 2, 8, 9, 10, 14.

Plato. The Republic. Revised 2^nd ed. Trans. Desmond Lee, London: Penguin Books, 1987.

Rucker, Rudy. "Excursion 1 The Transfinite Cardinals." Infinity and the Mind: The Science and Philosophy of the Infinite. Brighton: Harvester Press Limited, 1982. 221-65.

Russell, Bertrand. “Selections from Introduction to Mathematical Philosophy.” The Philosophy of Mathematics: Selected Readings. 2^nd ed. Eds. Paul Benacerraf & Hilary Putnam, Cambridge: Cambridge University Press, 1983. 160-82.

Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Trans. G. E. M. Anscombe, Eds. G. H. von Wright, R. Rhees and G. E. M. Anscombe, Oxford: Basil Blackwell, 1956.

---. Philosophical Investigations. Trans. G. E. M. Anscombe, Oxford: Blackwell Publishing, 2001.

---

[1] Cantor, 1955, p.85.

[2] Russell, 1983, p.167.

[3] Blackburn, 1996, p.54.

[4] Cantor avoids the paradoxes of the infinitely big when discussing infinite sets because he uses only ‘correlation’ criteria for comparing the size of sets, and not ‘subset’ criteria as well. Moore, 1990, p.118.

[5] Cantor, 1955, p.90.

[6] Rucker, 1982, p.225.

[7] Moore, 1990, p.119.

[8] Moore, ibid. pp.119-20.

[9] Moore, ibid. pp.152-4.

[10] Boolos, 1983, p.486.

[11] Boolos, ibid.

[12] Moore, 1990, p.10.

[13] Moore, ibid.

[14] An absolute continuous line, given by Haussdorf, seems to incorporate the idea advocated by Moore: ‘that there are more points on a line than can be gathered into any set’. Moore, ibid. pp.156-7.

[15] Rucker, 1982, p.244.

[16] Wittgenstein said that the paradox of the divided stick is set up wrong, that the stick is infinitely divisible but cannot be infinitely divided. Moore, 1990, p.138.

[17] Moore, ibid. p.158.

[18] Boolos, 1983, p.486.

[19] I may still refer to the infinite sets, R and N, on Wittgenstein’s precedent that the word ‘infinity’ be used to ‘characterize… the endless possibilities that finite things afford’, but I do not take them to be determinate totalities. Moore, 1990, p.137.

[20] Hallett, 1984, p.21.

[21] See Plato, 1987, particularly Part Seven: The Philosopher Ruler, pp.200-64, traditional notation: [471c-521b].

[22] Moore, 1990. p.10.

[23] Aristotle distinguished between ‘potential’ and ‘actual’ infinity, and moreover that nothing was actually, only potentially, infinite. Moore, ibid. pp.39-40.

[24] As Hallett argues, for Cantor there are no essential mathematical differences between objects of the transfinite and finite realms. Hallett, 1984, p.39.

[25] Bernays, 1983, pp.261-2.

[26] Moore, 1990, p.2. For Aristotle, the mathematically and potentially infinite were the same, see Moore, ibid. p.44.

[27] Moore, ibid. pp.35-41.

[28] Cantor, 1955, p.85.

[29] Moore, 1990, p.128.

[30] Hallett, 1984, p.21.

[31] Wittgenstein said that the endless series of natural numbers is mathematical reality (there are no determinate totalities lying beyond it). Moore, 1990, p.140.

[32] Moore, ibid. p.143.

[33] Moore, ibid. pp.212-4.

[34] It has been suggested to me by Ray Monk that we can think of the number of points in a continuous line, and so wonder how many of them there are. However, I think that it is unintelligible to ask ‘how many points there are on an infinite line?’ because there is no number of points that could count as an answer.

[35] For intuitionists, ‘R > N’ is not proven by the law of the excluded middle, only by the construction of the sets R and N; it is not proven by a logical principle.

[36] Brouwer, 1983a, p.81.

[37] Blackburn, 1996, p.266.

[38] Moore, 1990, p.122.

[39] Brouwer, 1983a, p.81. Also see Rucker, 1982, p.225.

[40] Moore, 1990, p.137.

[41] Moore, ibid. p.140. In accord with Wittgenstein, Russell said [in 1919] that ‘[w]e cannot enumerate all the natural numbers: they are 0, 1, 2, 3, and so on. At some point we must content ourselves with “and so on”’. See Russell, 1983, p.168.

[42] Moore, ibid. p.145.

[43] Moore, ibid. p.136.

[44] See Brouwer 1983a and 1983b.

[45] Discussion of this question could be greatly extended; I would have liked to have mentioned the paradox of the Set of all Sets, the Continuum Hypothesis, and different degrees of Platonism that could be adopted.