Fixed-point lemma for normal functions: Difference between revisions

Revision as of 03:02, 13 August 2011

Revision as of 03:23, 31 August 2011

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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 point (mathematics)|fixed point]]s (Levy 1979: p. 117). It was first proved by [[Oswald Veblen]] in 1908.

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A [[normal function]] is a [[proper class|class]] function ''f'' from the class Ord of [[ordinal numbers]] to itself such that:

* ''f'' is '''strictly increasing''': ''f''(&alpha;) < f(&beta;) whenever &alpha; < &beta;.

* ''f'' is '''continuous''': for every nonzero limit ordinal &lambda; (i.e. &lambda; is neither zero nor a successor), ''f''(&lambda;) = sup { f(&alpha;) : &alpha; < &lambda; }.

It can be shown that if ''f'' is normal then ''f'' commutes with [[supremum|suprema]]; for any nonempty set ''A'' of ordinals,

:''f''(sup ''A'') = sup {''f''(&alpha;) : &alpha; ∈ ''A'' }.