Fixed-point lemma for normal functions: Difference between revisions

Revision as of 20:00, 5 October 2007

Revision as of 20:32, 5 October 2007

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* ''f'' is '''continuous''': for every limit ordinal &lambda;, ''f''(&lambda;) = sup { f(&alpha;) : &alpha; < &lambda; }.

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

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

A '''fixed point''' of a normal function is an ordinal &beta; such that ''f''(&beta;) = &beta;.

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 set 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 [[club set|closed and unbounded]] class.

== Proof ==