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In set theory, a Rowbottom cardinal, introduced by , is a certain kind of large cardinal number. An uncountable cardinal number κ is said to be ''Rowbottom'' if for every function ''f'': ()<ω → λ (where λ < κ) there is a set ''H'' of order type κ that is quasi-homogeneous for ''f'', i.e., for every ''n'', the ''f''-image of the set of ''n''-element subsets of ''H'' has countably many elements. Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. In general, Rowbottom cardinals need not be large cardinals in the usual sense: Rowbottom cardinals could be singular. It is an open question whether ZFC + “ is Rowbottom” is consistent. If it is, it has much higher consistency strength than the existence of a Rowbottom cardinal. The axiom of determinacy does imply that is Rowbottom (but contradicts the axiom of choice). == References == * * 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「Rowbottom cardinal」の詳細全文を読む スポンサード リンク
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