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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 and can often be used to construct [[ordinal number]]s with interesting properties. A formal version and proof (using the [[Zermelo-Fraenkel axioms]]) follow.
== Formal version ==
== Proof ==
FirstWe of all, it is clearknow that for any α ∈ Ord, ''f''(&alphagamma;) ≥ &alphagamma;. Iffor thisall wasordinals notγ. theWe case, wenow coulddeclare choosean aincreasing minimalsequence <α with <sub>''fn''(</sub>&alphagt;) (''n'' < αomega;) then,by sincesetting ''f''α<sub>0</sub> is= normalα, and thus monotone, α<sub>''fn''(+1</sub> = ''f''(α)<sub>''n''</sub>) <for ''fn''( &alphalt; ω), whichand isdefine aβ contradiction= tosup <α<sub>''n''</sub>>. beingClearly, minimalβ ≥ α. Since ''f'' commutes with [[supremum|suprema]], we have
# :''f''(β ) = 0. Then we have''f''(sup {α<sub>''n''</sub> = 0 for all: ''n'' , and< thus ''f''(& betaomega; }) = 0.▼: = sup {''f''(α<sub>''n''</sub>) : ''n'' < ω}
: = sup {α<sub>''n''+1</sub> : ''n'' < ω}
: = β
(The last step uses that the sequence <α<sub>''n''</sub>> increases).
== Example application ==
We now declare a 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, and define β = sup <α<sub>''n''</sub>>. There are three possible cases:
▲# β = 0. Then we have α<sub>''n''</sub> = 0 for all ''n'', and thus ''f''(β) = 0.
# β = δ + 1 for an ordinal number δ. Then there exists ''m'' < ω such that for all ''n'' ≥ ''m'', α<sub>''n''</sub> = δ + 1. It follows that ''f''(δ + 1) = ''f''(α<sub>''m''</sub>) = α<sub>''m'' + 1</sub> = δ + 1, and thus ''f''(β) = β.
# β is a [[limit ordinal]]. We first observe that sup <''f''(ν) : ν < β> = sup <''f''(α<sub>''n''</sub>) : ''n'' < ω>. "≥" is trivial; for "≤", we choose ν < β, then find an ''n'' with α<sub>''n''</sub> > ν, and since ''f'' is monotone, we have ''f''(α<sub>''n''</sub>) > ''f''(ν). Now we have ''f''(β) = sup <''f''(ν) : ν < β> (since ''f'' is continuous), and thus ''f''(β) = sup <''f''(α<sub>''n''</sub>) : ''n'' < ω> = sup < α<sub>''n''</sub> : ''n'' < ω > = β.
== Notes ==
It is easily checked that the function ''f'' : Ord → Ord, ''f''(α) = א<sub>α</sub> is normal; thus, there exists an ordinal Θ such that Θ = א<sub>Θ</sub>. In fact, the above lemma shows that there are infinitely many such Θ.
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