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 set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on
Cantor's argument
In 1891, he published "a much simpler proof ... which does not depend on considering the irrational numbers."[2] His new proof uses his diagonal argument to prove that there exists an infinite set with a larger number of elements (or greater cardinality) than the set of natural numbers N = {1, 2, 3, ...}. This larger set consists of the elements (x1, x2, x3, ...), where each xn is either m or w.[3] Each of these elements corresponds to a subset of N—namely, the element (x1, x2, x3, ...) corresponds to {n ∈ N: xn = w}. So Cantor's argument implies that the set of all subsets of N has greater cardinality than N. The set of all subsets of N is denoted by P(N), the power set of N.
Cantor generalized his argument to an arbitrary set A and the set consisting of all functions from A to {0, 1}.[4] Each of these functions corresponds to a subset of A, so his generalized argument implies the theorem: The power set P(A) has greater cardinality than A. This is known as Cantor's theorem.
The argument below is a modern version of Cantor's argument that uses power sets (for his original argument, see
- There exists at least one infinite set. This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinity. This axiom implies that N, the set of all natural numbers, exists.
- P(N), the set of all subsets of N, exists. In formal set theory, this is implied by the power set axiom, which says that for every set there is a set of all of its subsets.
- The concept of "having the same number" or "having the same cardinality" can be captured by the idea of equinumerous, and the correlation is called a one-to-one correspondence.
- A set cannot be put into one-to-one correspondence with its power set. This implies that N and P(N) have different cardinalities. It depends on very few assumptions of set theory, and, as John P. Mayberry puts it, is a "simple and beautiful argument" that is "pregnant with consequences".[6] Here is the argument:
- Let be a set and be its power set. The following theorem will be proved: If is a function from to then it is not onto. This theorem implies that there is no one-to-one correspondence between and since such a correspondence must be onto. Proof of theorem: Define the diagonal subset Since proving that for all will imply that is not onto. Let Then which implies So if then and if then Since one of these sets contains and the other does not, Therefore, is not in the image of , so is not onto.
- Let be a set and be its power set. The following theorem will be proved: If is a function from to then it is not
Next Cantor shows that is equinumerous with a subset of . From this and the fact that and have different cardinalities, he concludes that has greater cardinality than . This conclusion uses his 1878 definition: If A and B have different cardinalities, then either B is equinumerous with a subset of A (in this case, B has less cardinality than A) or A is equinumerous with a subset of B (in this case, B has greater cardinality than A).
Around 1895, Cantor began to regard the well-ordering principle as a theorem and attempted to prove it.[12] In 1895, Cantor also gave a new definition of "greater than" that correctly defines this concept without the aid of his well-ordering principle.[13] By using Cantor's new definition, the modern argument that P(N) has greater cardinality than N can be completed using weaker assumptions than his original argument:
- The concept of "having greater cardinality" can be captured by Cantor's 1895 definition: B has greater cardinality than A if (1) A is equinumerous with a subset of B, and (2) B is not equinumerous with a subset of A.[13] Clause (1) says B is at least as large as A, which is consistent with our definition of "having the same cardinality". Clause (2) implies that the case where A and B are equinumerous with a subset of the other set is false. Since clause (2) says that A is not at least as large as B, the two clauses together say that B is larger (has greater cardinality) than A.
- The power set has greater cardinality than which implies that P(N) has greater cardinality than N. Here is the proof:
- Define the subset Define which maps onto Since implies is a one-to-one correspondence from to Therefore, is equinumerous with a subset of
- Using proof by contradiction, assume that a subset of is equinumerous with . Then there is a one-to-one correspondence from to Define from to if then if then Since maps onto maps onto contradicting the theorem above stating that a function from to is not onto. Therefore, is not equinumerous with a subset of
Besides the axioms of infinity and power set, the axioms of
Reception of the argument
Initially, Cantor's theory was controversial among mathematicians and (later) philosophers. As
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.[16] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[17] Carl Friedrich 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."[18] 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
Wittgenstein did not object to mathematical formalism wholesale, but had a finitist view on what Cantor's proof meant. The philosopher maintained that belief in infinities arises from confusing the intensional nature of mathematical laws with the extensional nature of sets, sequences, symbols etc. A series of symbols is finite in his view: In Wittgenstein's words: "...A curve is not composed of points, it is a law that points obey, or again, a law according to which points can be constructed."
He also described the diagonal argument as "hocus pocus" and not proving what it purports to do.
Objection to the axiom of infinity
A common objection to Cantor's theory of infinite number involves the axiom of infinity (which is, indeed, an axiom and not a logical truth). Mayberry 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 …"[21]
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 ..."[22]
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
- Preintuitionism
Notes
- ^ Dauben 1979, pp. 67–68, 165.
- ^ Cantor 1891, p. 75; English translation: Ewald p. 920.
- ^ Dauben 1979, p. 166.
- ^ Dauben 1979, pp.166–167.
- ^ Frege 1884, trans. 1953, §70.
- ^ Mayberry 2000, p. 136.
- ^ Cantor 1878, p. 242. Cantor 1891, p. 77; English translation: Ewald p. 922.
- ^ Hallett 1984, p. 59.
- ^ Cantor 1891, p. 77; English translation: Ewald p. 922.
- ^ Moore 1982, p. 42.
- ^ Moore 1982, p. 330.
- Absolute infinite, well-ordering theorem, and paradoxes. Part of Cantor's proof and Zermelo's criticism of it is in a reference note.
- ^ a b Cantor 1895, pp. 483–484; English translation: Cantor 1954, pp. 89–90.
- S2CID 14897182
- ^ Wolchover, Natalie. "Dispute over Infinity Divides Mathematicians". Scientific American. Retrieved 2 October 2014.
- ^ Zenkin, Alexander (2004), "Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum", The Review of Modern Logic, vol. 9, no. 30, pp. 27–80
- ^ (Poincaré quoted from Kline 1982)
- ISBN 9780140147391.
- ^ (Hilbert, 1926)
- ^ (RFM V. 7)
- ^ Mayberry 2000, p. 10.
- ^ Weyl, 1946
References
- ISBN 978-0-387-15066-6
- Cantor, Georg (1878), "Ein Beitrag zur Mannigfaltigkeitslehre", Journal für die Reine und Angewandte Mathematik, 84: 242–248
- Cantor, Georg (1891), "Ueber eine elementare Frage der Mannigfaltigkeitslehre" (PDF), Jahresbericht der Deutschen Mathematiker-Vereinigung, 1: 75–78
- Cantor, Georg (1895), "Beiträge zur Begründung der transfiniten Mengenlehre (1)", Mathematische Annalen, 46 (4): 481–512, S2CID 177801164, archived from the originalon April 23, 2014
- Cantor, Georg; ISBN 978-0-486-60045-1
- ISBN 0-674-34871-0
- ISBN 978-0140147391
- Ewald, William B., ed. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, Oxford University Press, ISBN 0-19-850536-1
- ISBN 978-0-8101-0605-5
- Hallett, Michael (1984), Cantorian Set Theory and Limitation of Size, Clarendon Press, ISBN 0-19-853179-6
- S2CID 121888793
- "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können."
- Translated in Van Heijenoort, Jean, On the infinite, Harvard University Press
- ISBN 0-19-503085-0)
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: 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
- Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development & Influence, Springer, ISBN 978-1-4613-9480-8
- Poincaré, Henri (1908), The Future of Mathematics (PDF), Revue generale des Sciences pures et appliquees, vol. 23, archived from the original (PDF) on 2003-06-29 (address to the Fourth International Congress of Mathematicians)
- Sainsbury, R.M. (1979), Russell, London
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- Wittgenstein; R. Hargreaves (trans.); R. White (trans.) (1964), Philosophical Remarks, Oxford
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: CS1 maint: location missing publisher (link) - Wittgenstein (2001), Remarks on the Foundations of Mathematics (3rd ed.), Oxford
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