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The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908. == Background and formal statement == A normal function is a class function ''f'' from the class Ord of ordinal numbers to itself such that: * ''f'' is strictly increasing: ''f''(α) < f(β) whenever α < β. * ''f'' is continuous: for every limit ordinal λ (i.e. λ is neither zero nor a successor), ''f''(λ) = sup . It can be shown that if ''f'' is normal then ''f'' commutes with suprema; for any nonempty set ''A'' of ordinals, :''f''(sup ''A'') = sup . Indeed, if sup ''A'' is a successor ordinal then sup ''A'' is an element of ''A'' and the equality follows from the increasing property of ''f''. If sup ''A'' is a limit ordinal then the equality follows from the continuous property of ''f''. A fixed point of a normal function is an ordinal β such that ''f''(β) = β. The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal α, there exists an ordinal β such that β ≥ α and ''f''(β) = β. The continuity of the normal function implies the class of fixed points is closed (the supremum of any subset of the class of fixed points is again a fixed point). Thus the fixed point lemma is equivalent to the statement that the fixed points of a normal function form a closed and unbounded class. 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「Fixed-point lemma for normal functions」の詳細全文を読む スポンサード リンク
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