Controversy over Cantor's theory

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Introduction

The pure mathematicians and applied mathematicians who object to Cantor's theory of sets claim that Georg Cantor introduced into mathematics an element of fantasy which should be expunged. The basic argument was stated most elegantly and concisely by Hermann Weyl when he wrote:

"...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory ..." (Weyl, 1946)

In other words, Weyl amongst others claimed that Cantor built a theory starting with a few assumptions about sets which can be proven true when our domain of discourse is finite sets, and then claimed (without actual proof) that we can retain logical consistency if we extend our domain of discourse to include infinite sets.

Since the original inception of Cantor's set theory it has been rigourously formalised in a number of different ways and these various rigourisations are collectively referred to as Axiomatic set theory. The most common formulation of Cantor's theory is known as ZF. The fact that this axiomatic theory cannot be proved consistent is today understood in terms of Godel's incompleteness theorems and it is not generally regarded as reason to doubt the theory.

Cantor's argument that there are sets that have a cardinality (or "power" or "number") that is greater than the (already infinite) cardinality of the whole numbers 1,2,3,... has probably attracted more hostility than any other theoretical argument, before or since. Logician Wilfrid Hodges has commented on the energy devoted to refuting this "harmless little argument". What had it done to anyone to make them angry with it?

This article summarises the argument and examines some of the objections that have been raised against it.

Cantor's argument

Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numbers), which has a larger number of elements, or as he puts it, has a greater 'power' (Mächtigkeit), than the infinite set of finite whole numbers 1, 2, 3, ...

There are a number of steps implicit in his argument, as follows

That the elements of no set can be put into one-to-one correspondence with all of its subsets. This is known as Cantor's theorem. It depends on very few of the assumptions of set theory, and, as John P. Mayberry puts it, is a "simple and beautiful argument" that is "pregnant with consequences". Few have seriously questioned this step of the argument.

That the concept of "having the same number" can be captured by the idea of one-to-one correlation. This (purely definitional) assumption is sometimes known as Hume's principle. Cantor argues (1883 §1) that every well-defined set has a determinate power, and that two sets have the same power if they can be correlated with one another, element for element. As Frege says, "If a waiter wishes to be certain of laying exactly as many knives on a table as plates, he has no need to count either of them; all he has to do is to lay immediately to the right of every plate a knife, taking care that every knife on the table lies immediately to the right of a plate. Plates and knives are thus correlated one to one" (1884, tr. 1953, §70).

That there exists at least one infinite set of things, usually identified with the set of all finite whole numbers or "natural numbers". This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinity. This assumption allows us to prove, together with Cantor's theorem, that there exists at least one set that cannot be correlated one-to-one with all its subsets. It does not prove, however, that there in fact exists any set corresponding to "all the subsets".

That there does indeed exist a set of all subsets of the natural numbers is captured in formal set theory by the power set axiom, which says that for every set there is a set of all of its subsets. This allows us to prove Cantor's assertion that there exists a set with a greater number of elements than the set of natural numbers. The set N of natural numbers exists (by the axiom of infinity), and so does the set R of all its subsets (by the power set axiom). By Cantor's theorem, R cannot be one-to-one correlated with N, and by Cantor's definition of number of "power", it can be shown that R has a greater number than N. QED.

Reception of the argument

From the start, Cantor's Theory was controversial among mathematicians and (later) philosophers.

I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there (Kronecker)

Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.

Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already (Poincaré quoted from Kline 1982)

Gauss views on the subject (paraphrased): "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics". In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.

Many mathematicians, along with Leopold Kronecker argued that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Cantor's ideas ultimately were accepted, strongly supported by David Hilbert, amongst others. Even constructivists and the intuitionists, who developed their schools of mathematics as a reaction to Cantor's infinitary ideas, generally no longer argue that mathematicians should abandon Cantor's Theory. It would appear that Hilbert's prediction has proved accurate:

"No one will drive us from the paradise which Cantor created for us" (Hilbert, 1926)

(To which Wittgenstein replied "if one person can see it as a paradise of mathematicians, why should not another see it as a joke? (RFM V. 7)).

Naïve objections

Objections to Cantor's proof (together with objections to Gödel's theorem) are a standard feature of mathematical discussions. These are generally flawed in some way.

Many of these objections depend on objections to step two of the argument. These typically use applications of the pigeonhole principle, or other assumptions that require "counting" all the natural numbers. Thus they rely on the assumption that we can "count" all such numbers by a process that at some point comes to an end. This is what Cantorians deny. They say this begs the question. Of course we can count finite numbers, indeed this constitutes one definition of a finite number. But who is to say that all numbers are finite, given that Hume's principle shows that infinite sets can also be compared in size? Cantor argued in his philosophical writing and correspondence that all objections based on the finitude of our "normal" concept of number, thus involve a petitio principii.

All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on general grounds as well. It is their "initial falsehood" that from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices. (Letter to Gustac Enestrom, quoted in Dauben p. 125)

Other objections depend on the idea that it is possible to define a function mapping every whole number on to some subset of whole numbers (though not one which captures every such subset). Cantor's theorem shows that at least one diagonal set is left over. We can then define a new function that does capture this diagonal set, and then define a further function that captures this set and so on ad infinitum. The problem with this objection is that it assumes there is a set yielded by this infinite process. Since Cantor's argument proves there can be no such set (indeed it is a necessary consequence of set theory – see Absolute infinite) – the objection once more begs the question.

Hodges (1998 – see references) has written an entertaining paper outlining other attempts. These include

the claim that Cantor had chosen the wrong enumeration of the positive integers
the argument that Cantor had used the wrong positive integers
a denial that proof by contradiction is valid

For a list of anti-Cantor sites generally regarded as "cranky", see the external link below.

Objections to Hume's principle

As argued above, many naïve objections depend on implicitly denying Hume's principle, and are therefore question-begging. Wittgenstein explicitly denies the principle, arguing that our concept of number depends essentially on counting. "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one"

The expressions "divisible into two parts" and "divisible without limit" have completely different forms. This is, of course, the same case as the one in which someone operates with the word "infinite" as if it were a number word; because, in everyday speech, both are given as answers to the question 'How many?'(PR §173)

Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes. In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar. (PR §141).

He argues that the sign for a list of things is itself a list, and that a list is therefore inherently finite ("The symbol for a class is a list ... A cardinal number is an internal property of a list." (PR § 119)

Anti-Cantorians who propose that a "reality criterion" should be added to mathematics are also (in effect) denying that the concept of "number" truly applies to infinite sets. They argue that we must take steps to guarantee that formal conclusions reached in the world of abstractions can be translated back into assertions about the concrete world. Now that we have a microscope for mathematics (i.e. the computer), it makes sense to think of the world of computation as real and concrete; infinite sets and power sets of infinite sets (and hence, real numbers etc.) exist only as useful fictions (abstractions) which help us reason about the concrete reality underlying mathematics; axioms and the rules of inference for abstractions should guarantee that any statement about the infinite should have implications for approximations to the infinite. Statements which have no implications observable in the world of computation, are fictions.

They argue that it is not clear that anyone has produced a collection of axioms and rules of inference that satisfy these criteria, and are powerful enough to do all potentially useful mathematics. The constructivists have made progress towards that goal .

Others have argued that the mathematical logic that underpins set theory is essentially mathematical, and therefore lacks genuine logical underpinnings.

...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory ..." (Weyl, 1946)

We cannot use the modern axiomatic method to establish the theory of sets. We cannot, in particular, simply employ the machinery of modern logic, modern mathematical logic, in establishing the theory of sets (Mayberry 2000, 7)

If God has mathematics of his own that needs to be done, let him do it himself." (Errett Bishop (19XX))

Philosopher Hartley Slater, in a number of papers, has repeatedly argued against the concept of "number" that underlies set theory (see external link below).

In reply, Cantoreans quote Cantor's saying (now inscribed on his tombstone) that "the essence of mathematics lies entirely in its freedom" (Grundlagen §8).

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. (ibid.)

Objection to the axiom of infinity

One of the most common (and also the most respectable) objections to Cantor's theory of infinite number involves the axiom of infinity. It is generally recognised view by all logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury (1979, p.305) has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Bertrand Russell for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry (2000, p.10) has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor's axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all".

This approach is known as finitism.

Richard Arthur, philosopher and expert on Leibniz, has argued that Cantor's appeal to the idea of an actual infinite (formally captured by the axiom of infinity) is philosophically unjustified. Arthur argues that Leibniz' idea of a "syncategorematic" but actual infinity is philosophically more appealing. (See external link below for one of his papers).

The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis).

Objections to the power set axiom

References

Bishop, E. Introduction to Foundations of Constructive Analysis
Cantor, G. "Ueber eine elementare Frage der Mannigfaltigkeitslehre" Jahresbericht der Deutschen Mathematiker-Vereinigung 11 (1890-1), 75-78.
Frege, G. (1884) Die Grundlagen der Arithmetik, transl. as The Foundations of Arithmetic, J.L. Austin, 2nd edition 1953.
Dauben, G., Cantor.
Hilbert, D., 1926. "Über das Unendliche". Mathematische Annalen, 95: 161—90. Translated as "On the infinite" in van Heijenoort, From Frege to Gödel: A source book in mathematical logic, 1879-1931, Harvard University Press.
Hodges, W. "An Editor Recalls Some Hopeless Papers", The Bulletin of Symbolic Logic Volume 4, Number 1, March 1998.
Kline, M., 1982. Mathematics: The Loss of Certainty. Oxford, ISBN 0195030850.
Mayberry, J.P., The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, Vol. 82, Cambridge University Press, Cambridge, 2000
Poincare, H., 1908. "The Future of Mathematics". Address to the Fourth International Congress of Mathematicians . Published in Revue generale des Sciences pures et appliquees 23.
Sainsbury, R.M., Russell, London 1979
Weyl, H., 1946. "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell". American Mathematical Monthly 53, pages 2—13.
Wittgenstein, L. Philosophical Grammar, translated by A. J. P. Kenny, Oxford, 1974
Wittgenstein, L. Philosophical Remarks, translated by Hargreaves & White, Oxford, 1964
Wittgenstein, L. Remarks on the Foundations of Mathematics, 3rd ed. Oxford 2001

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