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In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. ==Formal definition== If λ is any ordinal, κ is λ-supercompact means that there exists an elementary embedding ''j'' from the universe ''V'' into a transitive inner model ''M'' with critical point κ, ''j''(κ)>λ and : That is, ''M'' contains all of its λ-sequences. Then κ is supercompact means that it is λ-supercompact for all ordinals λ. Alternatively, an uncountable cardinal κ is supercompact if for every ''A'' such that |''A''| ≥ κ there exists a normal measure over ()< κ, in the following sense. ()< κ is defined as follows: : An ultrafilter ''U'' over ()< κ is ''fine'' if it is κ-complete and , for every . A normal measure over ()< κ is a fine ultrafilter ''U'' over ()< κ with the additional property that every function such that is constant on a set in . Here "constant on a set in ''U''" means that there is such that . 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「Supercompact cardinal」の詳細全文を読む スポンサード リンク
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