Fixed-point lemma for normal functions: Difference between revisions

Revision as of 19:57, 7 September 2019

Revision as of 12:58, 23 February 2020

Line 2:

== Background and formal statement ==

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''(α\alpha) < f(β\beta)</math> whenever α < βmath>\alpha<\beta</math>.

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

Indeed, if <math>\sup ''A''</math> is a successor ordinal then <math>\sup ''A''</math> is an element of ''<math>A''</math> and the equality follows from the increasing property of ''<math>f''</math>. If <math>\sup ''A''</math> is a limit ordinal then the equality follows from the continuous property of ''<math>f''</math>.

A '''fixed point''' of a normal function is an ordinal β<math>\beta</math> such that ''<math>f''(β\beta) = β\beta</math>.

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<math>\alpha</math>, there exists an ordinal β<math>\beta</math> such that β ≥ α<math>\beta\geq\alpha</math> and ''<math>f''(β\beta) = β\beta</math>.

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