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In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. It was shown by Friedrich Hartogs in 1915, from ZF alone (that is, without using the axiom of choice), that there is a least well-ordered cardinal greater than a given well-ordered cardinal. To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: If ''X'' is any set, then the Hartogs number of ''X'' is the least ordinal α such that there is no injection from α into ''X''. If ''X'' cannot be well-ordered, then we can no longer say that this α is the least well-ordered cardinal ''greater'' than the cardinality of ''X'', but it remains the least well-ordered cardinal ''not less than or equal to'' the cardinality of ''X''. The map taking ''X'' to α is sometimes called Hartogs' function. ==Proof== Given some basic theorems of set theory, the proof is simple. Let . First, we verify that α is a set. #''X'' × ''X'' is a set, as can be seen in axiom of power set. # The power set of ''X'' × ''X'' is a set, by the axiom of power set. # The class ''W'' of all reflexive well-orderings of subsets of ''X'' is a definable subclass of the preceding set, so it is a set by the axiom schema of separation. # The class of all order types of well-orderings in ''W'' is a set by the axiom schema of replacement, as #::(Domain(''w''), ''w'') (β, ≤) #:can be described by a simple formula. But this last set is exactly α. Now because a transitive set of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into ''X'', then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into ''X''. Given β < α, β ∈ α so there is an injection from β into ''X''. 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「Hartogs number」の詳細全文を読む スポンサード リンク
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