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Georg Cantor's first proof of uncountability demonstrates that the set of all real numbers is uncountably, rather than countably, infinite. This proof differs from the more familiar proof that uses his diagonal argument. Cantor's first uncountability proof was published in 1874, in an article that also contains a proof that the set of real algebraic numbers is countable, and a proof of the existence of transcendental numbers.〔Cantor 1874. English translation: Ewald 1996, pp. 840–843.〕 Two points about which not all authors writing about Cantor's article have agreed are these: * Is Cantor's proof of the existence of transcendental numbers constructive or non-constructive?〔Gray 1994.〕 * Why did Cantor emphasize the countability of the real algebraic numbers rather than the uncountability of the real numbers?〔Dauben 1979, pp. 66–70; Ferreirós 2007, pp. 183–186.〕 In 1891 Cantor published his diagonal argument,〔Cantor 1891. English translation: Ewald 1996, pp. 920–922.〕 which produces an uncountability proof that is generally considered simpler and more elegant than his first proof. Both uncountability proofs contain ideas that can be used elsewhere. The diagonal argument is a general technique that is useful in mathematical logic and theoretical computer science, while Cantor's first uncountability proof can be generalized to any ordered set with the same order properties as the real numbers.〔The diagonal argument cannot be applied to sets that only have an ordering. Applying the diagonal argument to the real numbers requires a numeral system—such as the decimal representation system—and a numeral system uses the addition and multiplication properties of the real numbers.〕 ==The article== Cantor's article〔Cantor 1874. English translation: Ewald 1996, pp. 840–843.〕 begins with a discussion of the real algebraic numbers, and a statement of his first theorem: The collection of real algebraic numbers can be put into one-to-one correspondence with the collection of positive integers. Cantor restates this theorem in terms more familiar to mathematicians of his time: The collection of real algebraic numbers can be written as an infinite sequence in which each number appears only once. Next Cantor states his second theorem: Given any sequence of real numbers ''x''1, ''x''2, ''x''3, … and any interval (),〔The notation () denotes the set of real numbers that are ≥ ''a'' and ≤ ''b''.〕 one can determine numbers in () that are not contained in the given sequence. Cantor observes that combining his two theorems yields a new proof of the theorem: Every interval () contains infinitely many transcendental numbers. This theorem was first proved by Joseph Liouville.〔Liouville proved this theorem by constructing what are now known as Liouville numbers, and then proving that these numbers are transcendental.〕 He then remarks that his second theorem is: :''the reason why collections of real numbers forming a so-called continuum (such as, all real numbers which are ≥ 0 and ≤ 1) cannot correspond one-to-one with the collection (ν) (collection of all positive integers ); thus I have found the clear difference between a so-called continuum and a collection like the totality of real algebraic numbers.''〔Cantor 1874, p. 259. English translation: Gray 1994, p. 820.〕 The first half of this remark is Cantor's uncountability theorem. Cantor does not explicitly prove this theorem, which follows easily from his second theorem. To prove it, use proof by contradiction. Assume that the interval () can be put into one-to-one correspondence with the set of positive integers, or equivalently: The real numbers in () can be written as a sequence in which each real number appears only once. Applying Cantor's second theorem to this sequence and () produces a real number in () that does not belong to the sequence. This contradicts our original assumption, and proves the uncountability theorem. Cantor's second theorem is constructive and thereby separates the constructive content of his work from the proof by contradiction needed to establish uncountability.〔Gray 1994, p. 823.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cantor's first uncountability proof」の詳細全文を読む スポンサード リンク
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