Editing Fixed-point lemma for normal functions

{{Short description|Mathematical result on ordinals}}

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.&nbsp;117). It was first proved by [[Oswald Veblen]] in 1908.

== 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>.

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 ==
The function ''f'' : Ord → Ord, ''f''(''α'') = ω<sub>''α''</sub> is normal (see [[initial ordinal]]). Thus, there exists an ordinal ''θ'' such that ''θ'' = ω<sub>''θ''</sub>. In fact, the lemma shows that there is a closed, unbounded class of such ''θ''.

==References==
{{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&ndash;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]]