Fixed-point lemma for normal functions: Difference between revisions

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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]] stating that any [[normal function]] has arbitrarily large [[fixed point (mathematics)|fixed point]]s (Levy 1979: p. 117). It was first proved by [[Oswald Veblen]] in 1908.

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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>.

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>.

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== Proof ==

The first step of the proof is to verify that ''<math>f''(γ\gamma) ≥ γ\ge\gamma</math> for all ordinals γ<math>\gamma</math> and that ''<math>f''</math> commutes with suprema. Given these results, inductively define an increasing sequence &lt;α<submath>''\langle\alpha_n\rangle_{n''<\omega}</submath>&gt; (''n'' &lt; ω) by setting α_{0</submath>\alpha_0 = α\alpha</math>, and α<submath>''\alpha_{n''+1}} = ''f''(α<sub>''n''\alpha_n)</submath>) for ''<math>n'' ∈ ω\in\omega</math>. Let β<math>\beta = sup \sup_{α<sub>''n''<\omega} \alpha_n</submath> : ''n'' ∈ ω}, so β ≥ α<math>\beta\ge\alpha</math>. Moreover, because ''<math>f''</math> commutes with suprema,

:<math>f(\beta) = f(\sup_{n<\omega} \alpha_n)</math>

:''f''(β) = ''f''(sup {α_''n'' : ''n'' &lt; ω})

:<math>\qquad = \sup_{n<\omega} f(\alpha_n)</math>

:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = sup {''f''(α_''n'') : ''n'' &lt; ω}

:<math>\qquad = \sup_{n<\omega} \alpha_{n+1}</math>

:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = sup {α_''n''+1 : ''n'' &lt; ω}

:<math>\qquad = \beta</math>

:&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; = β.

The last equality follows from the fact that the sequence &lt;α<submath>''n''\langle\alpha_n\rangle_n</submath>&gt; increases. <math> \square </math>

As an aside, it can be demonstrated that the β<math>\beta</math> found in this way is the smallest fixed point greater than or equal to α<math>\alpha</math>.

== Example application ==

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

==References==

{{refbegin}}

* {{cite book

| author = Levy, A.

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| year = 1908

| pages = 280&ndash;292

| id = Available via [https://www.jstor.org/stable/1988605 JSTOR].

| 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]]