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In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states: :There is no set whose cardinality is strictly between that of the integers and the real numbers. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in the year 1900. Τhe answer to this problem is independent of ZFC set theory (that is, Zermelo–Fraenkel set theory with the axiom of choice included), so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term ''the continuum'' for the real numbers. It is abbreviated CH. ==Cardinality of infinite sets== (詳細はcardinality'' or ''cardinal number'' if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets ''S'' and ''T'' to have the same cardinality means that it is possible to "pair off" elements of ''S'' with elements of ''T'' in such a fashion that every element of ''S'' is paired off with exactly one element of ''T'' and vice versa. Hence, the set has the same cardinality as . With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not take account of the fact that all three sets are infinite. It turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (''cardinality'') as the set of integers: they are both countable sets. Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (see Cantor's first uncountability proof and Cantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. Equivalently, as the cardinality of the integers is ("aleph-naught") and the cardinality of the real numbers is (i.e. it equals the cardinality of the power set of the integers), the continuum hypothesis says that there is no set for which : Assuming the axiom of choice, there is a smallest cardinal number greater than , and the continuum hypothesis is in turn equivalent to the equality : A consequence of the continuum hypothesis is that every infinite subset of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH) which says that for all ordinals : That is, GCH asserts that the cardinality of the power set of any infinite set is the smallest cardinality greater than that of the set. 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「Continuum hypothesis」の詳細全文を読む スポンサード リンク
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