Controversy over Cantor's theory

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In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.

Cantor's theorem is that there are sets having cardinality greater than the (already infinite) cardinality of the set of whole numbers {1,2,3,...}.

Cantor's argument[edit]

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 put it, has a greater 'mightiness' (Mächtigkeit), than the infinite set of finite whole numbers {1, 2, 3, ...}.

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

  • That the elements of a set can not 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.[clarification needed] 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, and attempted to prove that the power of every well-defined set ("consistent multiplicity") is an aleph; and therefore that the ordering relation among alephs determines an order among the sizes 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 than 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[edit]

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"[citation needed]. 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. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?"[2] Contrary to Hodges assertion, others have also taken issue with Cantor's proof regarding the cardinality of the power set.[3][4] Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.” [5]

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.[6] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[7] 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'.[citation needed] 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".[8] To which Wittgenstein replied "if one person can see it as a paradise of mathematicians, why should not another see it as a joke?".[9] The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism.

Objection to the axiom of infinity[edit]

Further information: Finitism

A common objection to Cantor's theory of infinite number involves the axiom of infinity. It is a generally recognized view by logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury[10] 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". Mayberry has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees.[11] 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 …."[12]

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).

See also[edit]

Notes[edit]

  1. ^ a b Cantor, letter to Richard Dedekind, with comments by Ernst 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.
  2. ^ Hodges, Wilfrid (1998), An Editor Recalls Some Hopeless Papers, The Bulletin of Symbolic Logic (Association for Symbolic Logic) 4 (1): 1–16, doi:10.2307/421003, JSTOR 421003 
  3. ^ Perez, Juan A. (2010). "Addressing mathematical inconsistency: Cantor and Gödel refuted". arXiv:1002.4433.
  4. ^ Zenkin, Alexander. "Cantor's Diagonal Argument: A New Aspect". Dorodnitsyn Computing Center. Retrieved 2 October 2014. 
  5. ^ Wolchover, Natalie. "Dispute over Infinity Divides Mathematicians". Scientific American. Retrieved 2 October 2014. 
  6. ^ Zenkin, Alexander (2004), Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum, The Review of Modern Logic 9 (30): 27–80 
  7. ^ (Poincaré quoted from Kline 1982)
  8. ^ (Hilbert, 1926)
  9. ^ (RFM V. 7)
  10. ^ Sainsbury 1979, p. 305
  11. ^ Mayberry 2000, p. 10
  12. ^ Weyl, 1946

References[edit]

"Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können."
Translated in Van Heijenoort, Jean, On the infinite, Harvard University Press 
  • Kline, Morris (1982), Mathematics: The Loss of Certainty, Oxford, ISBN 0-19-503085-0 
  • Mayberry, J.P. (2000), The Foundations of Mathematics in the Theory of Sets, Encyclopedia of Mathematics and its Applications 82, Cambridge University Press 
  • Poincaré, Henri (1908), The Future of Mathematics, Revue generale des Sciences pures et appliquees 23  (address to the Fourth International Congress of Mathematicians)
  • Sainsbury, R.M. (1979), Russell, London 
  • 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 53: 2–13 
  • Wittgenstein, Ludwig; A. J. P. Kenny (trans.) (1974), Philosophical Grammar, Oxford 
  • Wittgenstein; R. Hargreaves (trans.); R. White (trans.) (1964), Philosophical Remarks, Oxford 
  • Wittgenstein (2001), Remarks on the Foundations of Mathematics (3rd ed.), Oxford 

External links[edit]