Fixed-point lemma for normal functions: Difference between revisions

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

* <math>f</math> is '''strictly increasing''': <math>f(\alpha)<f(\beta)</math> whenever <math>\alpha<\beta</math>.

* <math>f</math> is '''continuous''': for every limit ordinal <math>\lambda</math> (i.e. <math>\lambda</math> is neither zero nor a successor), <math>f(\lambda)=\sup\{f(\alpha):\alpha<\kappalambda\}</math>.

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

:<math>f(\sup A)=\sup f(A) = \sup\{f(\alpha):\alpha \in A\}</math>.