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In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model ''M'' of cardinality κ of ZFC-minus-power set such that κ is in ''M'' and ''M'' contains all its sequences of length less than κ, there is a non-trivial elementary embedding ''j'' of ''M'' into a transitive model with the critical point of ''j'' being κ and ''j''(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model ''M'' of cardinality κ of ZFC-minus-power set such that κ is in ''M'' and ''M'' contains all its sequences of length less than κ, there is a non-trivial elementary embedding ''j'' of ''M'' into a transitive model "N" with the critical point of ''j'' being κ, ''j''(κ) ≥ λ, and V(λ) is a subset of ''N''. Without loss of generality, we can demand also that ''N'' contains all its sequences of length λ. Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ. These properties are essentially weaker versions of strong and supercompact cardinals, consistent with V = L. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom. A Ramsey cardinal is unfoldable, and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however. In L, any unfoldable cardinal is strongly unfoldable; thus unfoldables and strongly unfoldables have the same consistency strength. A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact. A κ+ω-unfoldable cardinal is totally indescribable and preceded by a stationary set of totally indescribable cardinals. ==References== * ''Unfoldable Cardinals and the GCH'', Joel David Hamkins. The Journal of Symbolic Logic, Vol. 66, No. 3 (Sep., 2001), pp. 1186–1198 * ''Strongly unfoldable cardinals made indestructible'', Thomas A. Johnstone. J. Symbolic Logic, Volume 73, Issue 4 (2008), 1215-1248. * ''Diamond (on the regulars) can fail at any strongly unfoldable cardinal'', Joel David Hamkins (The City University of New York), Mirna Džamonja (University of East Anglia). (Submitted to arxiv (http://arxiv.org/abs/math/0409304) on 17 Sep 2004) 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「Unfoldable cardinal」の詳細全文を読む スポンサード リンク
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