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In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and : Here, ''αM'' is the class of all sequences of length α whose elements are in M. Huge cardinals were introduced by . == Variants == In what follows, j''n'' refers to the ''n''-th iterate of the elementary embedding j, that is, j composed with itself ''n'' times, for a finite ordinal ''n''. Also, ''<αM'' is the class of all sequences of length less than α whose elements are in M. Notice that for the "super" versions, γ should be less than j(κ), not . κ is almost n-huge if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and : κ is n-huge if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and : Notice that 0-huge is the same as R> : κ is super n-huge if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ<j(κ), and : Notice that 0-huge is the same as measurable cardinal; and 1-huge is the same as huge. A cardinal satisfying one of the rank into rank axioms is ''n''-huge for all finite ''n''. The existence of an almost huge cardinal implies that Vopěnka's principle is consistent; more precisely any almost huge cardinal is also a Vopěnka 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「Huge cardinal」の詳細全文を読む スポンサード リンク
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