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| In [[axiomatic set theory]], a function ''f'' : [[ordinal number|Ord]] → Ord is called '''normal''' (or a '''normal function''') [[iff]] it is [[continuous function#Continuous functions between partially ordered sets|continuous]] (with respect to the [[order topology]]) and [[monotonic function|strictly monotonically increasing]]. This is equivalent to the following two conditions:
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| # For every [[limit ordinal]] γ (i.e. γ is neither zero nor a successor), ''f''(γ) = [[supremum|sup]] {''f''(ν) : ν < γ}.
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| # For all ordinals α < β, ''f''(α) < ''f''(β).
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| == Examples ==
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| A simple normal function is given by ''f''(α) = 1 + α; note however that ''f''(α) = α + 1 is ''not'' normal. If β is a fixed ordinal, then the functions ''f''(α) = β + α, ''f''(α) = β × α and ''f''(α) = β<sup>α</sup> (for β > 1) are all normal.
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| More important examples of normal functions are given by the [[aleph number]]s <math>f(\alpha) = \aleph_\alpha</math> which connect ordinal and [[cardinal number]]s, and by the [[beth number]]s <math>f(\alpha) = \beth_\alpha</math>.
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| == Properties ==
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| If ''f'' is normal, then for any ordinal α,
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| :''f''(α) ≥ α.<ref>{{harvnb|Johnstone|1987|loc=Exercise 6.9, p. 77}}</ref> | |
| '''Proof''': If not, choose γ minimal such that ''f''(γ) < γ. Since ''f'' is strictly monotonically increasing, ''f''(''f''(γ)) < ''f''(γ), contradicting minimality of γ.
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| Furthermore, for any non-empty set ''S'' of ordinals, we have
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| :''f''(sup ''S'') = sup ''f''(''S'').
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| '''Proof''': "≥" follows from the monotonicity of ''f'' and the definition of the [[supremum]]. For "≤", set δ = sup ''S'' and consider three cases:
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| * if δ = 0, then ''S'' = {0} and sup ''f''(''S'') = ''f''(0);
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| * if δ = ν + 1 is a [[successor ordinal|successor]], then there exists ''s'' in ''S'' with ν < ''s'', so that δ ≤ ''s''. Therefore, ''f''(δ) ≤ ''f''(''s''), which implies ''f''(δ) ≤ sup ''f''(''S'');
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| * if δ is a nonzero limit, pick any ν < δ, and an ''s'' in ''S'' such that ν < ''s'' (possible since δ = sup ''S''). Therefore ''f''(ν) < ''f''(''s'') so that ''f''(ν) < sup ''f''(''S''), yielding ''f''(δ) = sup {''f''(ν) : ν < δ} ≤ sup ''f''(''S''), as desired.
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| Every normal function ''f'' has arbitrarily large fixed points; see the [[fixed-point lemma for normal functions]] for a proof.
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| One can create a normal function ''g'' : Ord → Ord, called the derivative of ''f'', where ''g''(α) is the α-th fixed point of ''f''.<ref>{{harvnb|Johnstone|1987|loc=Exercise 6.9, p. 77}}</ref>
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{citation
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| |first=Peter
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| |last=Johnstone
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| |authorlink=Peter Johnstone (mathematician)
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| |year=1987
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| |title=Notes on Logic and Set Theory
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| |publisher=[[Cambridge University Press]]
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| |isbn=978-0-521-33692-5}}.
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| [[Category:Set theory]]
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| [[Category:Ordinal numbers]]
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