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In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph () (though in older mathematics books the letter aleph is often printed upside down by accident,〔For example, in the letter aleph appears both the right way up and upside down〕 partly because a Monotype matrix for aleph was mistakenly constructed the wrong way up ). The cardinality of the natural numbers is (read ''aleph-naught'' or ''aleph-zero''; the German term ''aleph-null'' is also sometimes used), the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as described below. The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line. ==Aleph-naught== (aleph-naught, also aleph zero or the German term Aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called ω or ω0, has cardinality . A set has cardinality if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are * the set of all square numbers, the set of all cubic numbers, the set of all fourth powers, ... * the set of all perfect powers, the set of all prime powers, * the set of all even numbers, the set of all odd numbers, * the set of all prime numbers, the set of all composite numbers, * the set of all integers, * the set of all rational numbers, * the set of all constructible numbers (in the geometric sense), * the set of all algebraic numbers, * the set of all computable numbers, * the set of all definable numbers, * the set of all binary strings of finite length, and * the set of all finite subsets of any given countably infinite set. These infinite ordinals: ω, ω+1, ω·2, ω2, ωω and ε0 are among the countably infinite sets. For example, the sequence (with ordinality ω·2) of all positive odd integers followed by all positive even integers : is an ordering of the set (with cardinality ) of positive integers. If the axiom of countable choice (a weaker version of the axiom of choice) holds, then is smaller than any other infinite cardinal. 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「Aleph number」の詳細全文を読む スポンサード リンク
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