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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.
== Background and formal statement ==
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Indeed, if sup ''A'' is a successor ordinal then sup ''A'' is an element of ''A'' and the equality follows from the increasing property of ''f''. If sup ''A'' is a limit ordinal then the equality follows from the continuous property of ''f''.
A '''fixed point''' of a normal function is an ordinal β such that ''f''(β) = β.
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 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 ''f''(γ) ≥ γ for all ordinals γ and that ''f'' commutes with suprema. Given these results, inductively define an increasing sequence <α<sub>''n''</sub>> (''n'' < ω) by setting α<sub>0</sub> = α, and α<sub>''n''+1</sub> = ''f''(α<sub>''n''</sub>) for ''n'' ∈ ω. Let β = sup {α<sub>''n''</sub> : ''n'' ∈ ω}, so β ≥ α. Moreover, because ''f'' commutes with suprema,
:''f''(β) = ''f''(sup {α<sub>''n''</sub> : ''n'' < ω})
: = sup {''f''(α<sub>''n''</sub>) : ''n'' < ω}
: = sup {α<sub>''n''+1</sub> : ''n'' < ω}
: = β.
The last equality follows from the fact that the sequence <α<sub>''n''</sub>> increases.
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| authorlink = Oswald Veblen
| title = Continuous increasing functions of finite and transfinite ordinals
| journal = Trans. Amer. Math Soc.
| volume = 9
| year = 1908
| pages = 280–292
| id = Available via [http://links.jstor.org/sici?sici=0002-9947%28190807%299%3A3%3C280%3ACIFOFA%3E2.0.CO%3B2-1 JSTOR].
| doi= 10.2307/1988605
| issue = 3
| |||