Fixed-point lemma for normal functions: Difference between revisions

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The '''fixed-point lemma for normal functions''' is a basic result in [[axiomatic set theory]]; it statesstating that any [[normal function]] has arbitrarily large [[fixed point (mathematics)|fixed point]]s and(Levy can1979: oftenp. be used to construct [[ordinal number]]s with interesting properties117). AIt formalwas versionfirst andproved proof (using theby [[Zermelo-FraenkelOswald axiomsVeblen]]) followsin 1908.

== Background and formal statement ==

== Formal version ==

A [[normal function]] is a [[proper class|class]] function ''f'' from the class Ord of [[ordinal numbers]] to itself so that:

Let ''f'' : [[ordinal number|Ord]] → Ord be a [[normal function]]. Then for every α ∈ Ord, there exists a β ∈ Ord such that β ≥ α and ''f''(β) = β.

* ''f'' is '''increasing''': ''f''(α) ≤ f(β) whenever α ≤ β.

* ''f'' is '''continuous''': for every limit ordinal &lambda;, ''f''(&lambda;) = sup { f(&alpha;) : &alpha; < &lambda; }.

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

:''f''(sup ''A'') = { sup ''f''(&alpha;) : &alpha; ∈ ''A'' }.

A '''fixed point''' of a normal function is an ordinal &beta; such that ''f''(&beta;) = &beta;.

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, there exists an ordinal β such that β ≥ α and ''f''(β) = β.

The continuity of the normal function implies the set 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 ==

WeThe knowfirst step of the proof is to verify that ''f''(γ) ≥ γ for all ordinals γ and that ''f'' commutes with suprema. WeGiven nowthese declareresults, inductively define an increasing sequence &lt;α_''n''&gt; (''n'' &lt; ω) by setting α₀ = α, and α_''n''+1 = ''f''(α_''n'') for ''n'' &lt;∈ ω,. and defineLet β = sup &lt;{α_''n'' : ''n'' ∈ &gtomega;. Clearly}, so β ≥ α. SinceMoreover, because ''f'' commutes with [[supremum|suprema]], we have

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

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

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

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

(The last stepequality usesfollows from the fact that the sequence &lt;α_''n''&gt; increases).

== Example application ==

It is easily checked that theThe function ''f'' : Ord → Ord, ''f''(α) = א_α is normal (see [[aleph number]]);. thusThus, there exists an ordinal Θ such that Θ = א_Θ. In fact, the above lemma shows that there areis a closed, unbounded infinitelyclass manyof such Θ.

==References==

* {{cite book

{{unreferenced||date=June 2006}}

| author = Levy, A.

| title = Basic Set Theory

| year = 1979

| publisher = Springer

| id= Republished, Dover, 2002. ISBN 0-486-42079-5}}

*{{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

| id = Available via [http://links.jstor.org/sici?sici=0002-9947%28190807%299%3A3%3C280%3ACIFOFA%3E2.0.CO%3B2-1 JSTOR]. }}

[[Category:Set theory]]