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In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found near-universal^{[citation needed]} acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers.

Cantor's theorem that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}, has probably attracted more hostility^{[citation needed]} than any other mathematical argument, before or since. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument", asking, "what had it done to anyone to make them angry with 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 correspondence. This (purely definitional) assumption is sometimes known as Hume's principle. 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). Sets in such a correlation are often called equipollent, and the correlation itself is called a bijective function.

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. (For example, the subsets of the set {a, b} are { }, {a}, {b}, and {a, b}). This allows us to prove that there exists an infinite set which is not equipollent with 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 or "power", it follows that R has a different number than N. It does not prove, however, that the number of elements in R is in fact greater than the number of elements in N, for only the notion of two sets having different power has been specified; given two sets of different power, nothing so far has specified which of the two is greater.

Cantor presented a well-ordered sequence of cardinal numbers, the alephs, attempted to prove that the power of every well-defined set ("consistent multiplicity") is an aleph; therefore the ordering relation among alephs determines an order among the size of sets^[1]. However this proof was flawed, and as Zermelo wrote, "It is precisely at this point that the weakness of the proof sketched here lies… It is precisely doubts of this kind that impelled ... [my own] proof of the well-ordering theorem purely upon the axiom of choice…"^[1]

The assumption of the axiom of choice was later shown unnecessary by the Cantor-Bernstein-Schröder theorem, which makes use of the notion of injective functions from one set to another—a correlation which associates different elements of the former set with different elements of the latter set. The theorem shows that if there is an injective function from set A to set B, and another one from B to A, then there is a bijective function from A to B, and so the sets are equipollent, by the definition we have adopted. Thus it makes sense to say that the power of one set is at least as large as another if there is an injection from the latter to the former, and this will be consistent with our definition of having the same power. Since the set of natural numbers can be embedded in its power set, but the two sets are not of the same power, as shown, we can therefore say the set of natural numbers is of lesser power that its power set. However, despite its avoidance of the axiom of choice, the proof of the Cantor-Bernstein-Schröder theorem is still not constructive, in that it does not produce a concrete bijection in general.

Reception of the argument

At the start, Cantor's Theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

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's views on the subject can be paraphrased as: '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.

Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: "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). The development of schools of mathematics such as constructivism and the intuitionism were influenced by Cantor's infinitary ideas.

Objections to Hume's principle

Wittgenstein denies Hume's 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.

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)

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, Cantorians 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.)

Cantor's number is seen by the Zermelo constructivists as a representative model, without schema (code).

Objection to the axiom of infinity

A common objection 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".

Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl 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 [Cantor's] set theory …."
(Weyl, 1946)

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).

Other foundational controversies

Brouwer-Hilbert controversy

Notes

^ ^a ^b Cantor, letter to Dedekind, with comments by Zermelo, translated in van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.

References

Bishop, Errett; Bridges, Douglas (1985), Constructive Analysis, Grundlehren Der Mathematischen Wissenschaften, Springer, ISBN 978-0387150666
Cantor, Georg (1890), "Ueber eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 11, pp. 75–78
Frege, Gottlob; J.L. Austin (trans.) (1980), The Foundations of Arithmetic (2nd ed.), Northwestern University Press, ISBN 978-0810106055
Dauben, J. (1990), Cantor, Princeton, ISBN 0-691-02447-2{{citation}}: CS1 maint: location missing publisher (link)
Hilbert, David (1926), "Über das Unendliche", Mathematische Annalen, no. 95, pp. 161–90

Translated in Van Heijenoort, Jean, From Frege to Gödel: A source book in mathematical logic, 1879-1931, Harvard University Press {{citation}}: Cite has empty unknown parameter: |1= (help)

Hodges, Wilfrid (1998), "An editor recalls some hopeless papers", Bulletin of Symbolic Logic, vol. 4, no. 1, pp. 1–16
Kline, Morris (1982), Mathematics: The Loss of Certainty, Oxford, ISBN 0-19-503085-0{{citation}}: CS1 maint: location missing publisher (link)
Mayberry, J.P. (2000), The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications, vol. 82, Cambridge University Press
Poincaré, Henri (1908), The Future of Mathematics (PDF), Revue generale des Sciences pures et appliquees, vol. 23 (address to the Fourth International Congress of Mathematicians)
Sainsbury, R.M. (1979), Russell, London{{citation}}: CS1 maint: location missing publisher (link)
Weyl, Hermann (1946), "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell", American Mathematical Monthly, vol. 53, pp. 2–13
Wittgenstein, Ludwig; A. J. P. Kenny (trans.) (1974), Philosophical Grammar, Oxford{{citation}}: CS1 maint: location missing publisher (link)
Wittgenstein; R. Hargreaves (trans.); R. White (trans.) (1964), Philosophical Remarks, Oxford{{citation}}: CS1 maint: location missing publisher (link)
Wittgenstein (2001), Remarks on the Foundations of Mathematics (3rd ed.), Oxford{{citation}}: CS1 maint: location missing publisher (link)

External links

[fftg-1] Cantor, letter to Dedekind, with comments by Zermelo, translated in van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.

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@@ Line 27: / Line 27: @@
 Before Cantor, the notion of [[infinite set|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" ([[Henri Poincaré|Poincaré]] quoted from Kline 1982). [[Carl Gauss|Gauss]]'s views on the subject can be paraphrased as: '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.
-Cantor's ideas ultimately were largely accepted, strongly supported by [[David Hilbert]], amongst others. Although [[constructivism (mathematics)|constructivists]] and the [[intuitionism|intuitionists]] developed their schools of mathematics as a reaction to Cantor's infinitary ideas, most mathematicians do not have qualms about Cantor's Theory. It would appear that Hilbert's prediction has proved mostly accurate: "No one will drive us from the paradise which Cantor created for us" (Hilbert, 1926). To which  [[Ludwig Wittgenstein|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).
+Cantor's ideas ultimately were largely accepted, strongly supported by [[David Hilbert]], amongst others. Hilbert predicted: "No one will drive us from the paradise which Cantor created for us" (Hilbert, 1926). To which  [[Ludwig Wittgenstein|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). The development of schools of mathematics such as [[constructivism (mathematics)|constructivism]] and the [[intuitionism]] were influenced by Cantor's infinitary ideas.
 == Objections to Hume's principle ==