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In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case is strongly inaccessible). Weakly inaccessible cardinals were introduced by , and strongly inaccessible ones by and . The term "inaccessible cardinal" is ambiguous. Until about 1950 it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible. (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected. == Models and consistency == ZFC implies that the ''V''κ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe ''L''κ is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal. If ''V'' is a standard model of ZFC and κ is an inaccessible in ''V'', then: ''V''κ is one of the intended models of Zermelo–Fraenkel set theory; and Def (''V''κ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and ''V''κ+1 is one of the intended models of Morse–Kelley set theory. Here Def (''X'') is the Δ0 definable subsets of ''X'' (see constructible universe). However, κ does not need to be inaccessible, or even a cardinal number, in order for ''V''κ to be a standard model of ZF (see below). Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, ''V''κ is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then ''L''κ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals. The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC + "there is an inaccessible cardinal" implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent. There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by , is that the class of all ordinals of a particular model ''M'' of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending ''M''. 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「Inaccessible cardinal」の詳細全文を読む スポンサード リンク
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