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In mathematics, an uncountable set (or uncountably infinite set)〔(Uncountably Infinite — from Wolfram MathWorld )〕 is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. ==Characterizations== There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions holds: * There is no injective function from ''X'' to the set of natural numbers. * ''X'' is nonempty and every ω-sequence of elements of ''X'' fails to include at least one element of ''X''. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than . The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles. 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「uncountable set」の詳細全文を読む スポンサード リンク
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