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{{Short description|Mathematical result on ordinals}}
The '''fixed-point lemma for normal functions''' is a basic result in [[axiomatic set theory]]; it states that any [[normal function]] has arbitrarily large [[fixed point (mathematics)|fixed point]]s. A formal version and proof (using the [[Zermelo-Fraenkel axioms]]) follow.▼
▲The '''fixed-point lemma for normal functions''' is a basic result in [[axiomatic set theory]]
== 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(\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<\lambda\}</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>.
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 <math>\beta</math> such that <math>f(\beta)=\beta</math>.
== Proof ==▼
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 <math>\alpha</math>, there exists an ordinal <math>\beta</math> such that <math>\beta\geq\alpha</math> and <math>f(\beta)=\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.
▲== Proof ==
The first step of the proof is to verify that <math>f(\gamma)\ge\gamma</math> for all ordinals <math>\gamma</math> and that <math>f</math> commutes with suprema. Given these results, inductively define an increasing sequence <math>\langle\alpha_n\rangle_{n<\omega}</math> by setting <math>\alpha_0 = \alpha</math>, and <math>\alpha_{n+1} = f(\alpha_n)</math> for <math>n\in\omega</math>. Let <math>\beta = \sup_{n<\omega} \alpha_n</math>, so <math>\beta\ge\alpha</math>. Moreover, because <math>f</math> commutes with suprema,
:<math>f(\beta) = f(\sup_{n<\omega} \alpha_n)</math>
:<math>\qquad = \sup_{n<\omega} f(\alpha_n)</math>
:<math>\qquad = \sup_{n<\omega} \alpha_{n+1}</math>
:<math>\qquad = \beta</math>
The last equality follows from the fact that the sequence <math>\langle\alpha_n\rangle_n</math> increases. <math> \square </math>
As an aside, it can be demonstrated that the <math>\beta</math> found in this way is the smallest fixed point greater than or equal to <math>\alpha</math>.
== Example application ==
==
{{refbegin}}
* {{cite book
| author = Levy, A.
| title = Basic Set Theory
| year = 1979
| publisher = Springer
| id = Republished, Dover, 2002.
| isbn = 978-0-387-08417-6
| url-access = registration
| url = https://archive.org/details/basicsettheory00levy_0
}}
*{{cite journal
| author= Veblen, O.
| authorlink = Oswald Veblen
| title = Continuous increasing functions of finite and transfinite ordinals
| journal = Trans. Amer. Math. Soc.
| volume = 9
| year = 1908
| pages = 280–292
| doi= 10.2307/1988605
| issue = 3
| jstor= 1988605
| issn= 0002-9947| doi-access = free
}}
{{refend}}
[[Category:Ordinal numbers]]
[[Category:Fixed-point theorems|Normal Functions]]
[[Category:Lemmas in set theory]]
[[Category:Articles containing proofs]]
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