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Properties: Make proof clearer

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	Furthermore, for any [[empty set\|non-empty]] set {{mvar\|S}} of ordinals, we have		Furthermore, for any [[empty set\|non-empty]] set {{mvar\|S}} of ordinals, we have
	:{{math\|1=''f''{{hairsp}}(sup ''S'') = sup ''f''{{hairsp}}(''S'')}}.		:{{math\|1=''f''{{hairsp}}(sup ''S'') = sup ''f''{{hairsp}}(''S'')}}.
	'''Proof''': "≥" follows from the monotonicity of {{mvar\|f}} and the definition of the [[supremum]]. For "{{math\|≤}}", ~~set {{math\|1=''δ'' = sup ''S''}} and~~ consider three cases:		'''Proof''': "≥" follows from the monotonicity of {{mvar\|f}} and the definition of the [[supremum]]. For "{{math\|≤}}", consider three cases:
	* if {{math\|1=''δ'' = 0}}, then {{math\|1=''S'' = {{mset\|0}}}} and {{math\|1=sup ''f''{{hairsp}}(''S'') = ''f''{{hairsp}}(0)}};		* if {{math\|1=sup ''S'' = 0}}, then {{math\|1=''S'' = {{mset\|0}}}} and {{math\|1=sup ''f''{{hairsp}}(''S'') = ''f''{{hairsp}}(0) = ''f''{{hairsp}}(sup ''S'')}};
	* if {{math\|1=''δ'' = ''ν'' + 1}} is a successor, then ~~there exists {{mvar\|s}} in {{mvar\|S}} with~~ {{math\|''~~ν'' < ''s~~''}}~~, so that~~ {{~~math~~\|~~''δ'' ≤ ''s''~~}}~~. Therefore~~, {{math\|''f''{{hairsp}}(''δ'') ≤ ''f''{{hairsp}}(''s'')}}, ~~which implies~~ {{math\|''f''{{hairsp}}(δ) ≤ sup ''f''{{hairsp}}(''S'')}};		* if {{math\|1=sup ''S'' = ''ν'' + 1}} is a successor, then {{math\|1=sup ''S''}} is in {{mvar\|S}}, so {{math\|1=''f''{{hairsp}}(sup ''S'')}} is in {{math\|''f''{{hairsp}}(''S'')}}, i.e. {{math\|''f''{{hairsp}}(sup ''S'') ≤ sup ''f''{{hairsp}}(''S'')}};
	* if {{~~mvar~~\|δ}} is a nonzero limit, ~~pick~~ any {{math\|''ν'' < ''δ''}}~~, and~~ an {{mvar\|s}} in {{mvar\|S}} such that {{math\|''ν'' < ''s''}} ~~(possible since {{math\|1=''δ'' = sup ''S''}})~~. ~~Therefore,~~ {{math\|''f''{{hairsp}}(''ν'') < ''f''{{hairsp}}(''s'')~~}} so that {{math\|''f''{{hairsp}}(''ν'') <~~ sup ''f''{{hairsp}}(''S'')}}, yielding {{math\|1=''f''{{hairsp}}(''δ'') = sup {{mset\|''f''{{hairsp}}(ν) : ''ν'' < ''δ''}} ≤ sup ''f''{{hairsp}}(''S'')}}~~, as desired~~.		* if {{math\|1=sup ''S''}} is a nonzero limit, then for any {{math\|''ν'' < sup ''S''}} there exists an {{mvar\|s}} in {{mvar\|S}} such that {{math\|''ν'' < ''s''}}, i.e. {{math\|''f''{{hairsp}}(''ν'') < ''f''{{hairsp}}(''s'') ≤ sup ''f''{{hairsp}}(''S'')}}, yielding {{math\|1=''f''{{hairsp}}(sup ''S'') = sup {{mset\|''f''{{hairsp}}(ν) : ''ν'' < sup ''S''}} ≤ sup ''f''{{hairsp}}(''S'')}}.

	Every normal function {{mvar\|f}} has arbitrarily large fixed points; see the [[fixed-point lemma for normal functions]] for a proof. One can create a normal function {{math\|''f{{hairsp}}′'' : Ord → Ord}}, called the '''derivative''' of {{mvar\|f}}, such that {{math\|''f{{hairsp}}′''(''α'')}} is the {{mvar\|α}}-th fixed point of {{mvar\|f}}.<ref>{{harvnb\|Johnstone\|1987\|loc=Exercise 6.9, p. 77}}</ref> For a hierarchy of normal functions, see [[Veblen function]]s.		Every normal function {{mvar\|f}} has arbitrarily large fixed points; see the [[fixed-point lemma for normal functions]] for a proof. One can create a normal function {{math\|''f{{hairsp}}′'' : Ord → Ord}}, called the '''derivative''' of {{mvar\|f}}, such that {{math\|''f{{hairsp}}′''(''α'')}} is the {{mvar\|α}}-th fixed point of {{mvar\|f}}.<ref>{{harvnb\|Johnstone\|1987\|loc=Exercise 6.9, p. 77}}</ref> For a hierarchy of normal functions, see [[Veblen function]]s.

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{{Short description|Function of ordinals in mathematics}}
{{one source |date=March 2024}}
In [[axiomatic set theory]], a function {{math|''f'' : [[ordinal number|Ord]] → Ord}} is called '''normal''' (or a '''normal function''') if 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:

# For every [[limit ordinal]] {{mvar|γ}} (i.e. {{mvar|γ}} is neither zero nor a [[successor ordinal|successor]]), it is the case that {{math|1=''f''{{hairsp}}(''γ'') = [[supremum|sup]]{{mset|''f''{{hairsp}}(''ν'') : ''ν'' < ''γ''}}}}.
# For all ordinals {{math|''α'' < ''β''}}, it is the case that {{math|''f''{{hairsp}}(''α'') < ''f''{{hairsp}}(''β'')}}.

== Examples ==
A simple normal function is given by {{math|1=''f''{{hairsp}}(''α'') = 1 + ''α''}} (see [[ordinal arithmetic]]). But {{math|1=''f''{{hairsp}}(''α'') = ''α'' + 1}} is ''not'' normal because it is not continuous at any limit ordinal (for example, <math>f(\omega) = \omega+1 \ne \omega = \sup \{f(n) : n < \omega\}</math>). If {{mvar|β}} is a fixed ordinal, then the functions {{math|1=''f''{{hairsp}}(''α'') = ''β'' + ''α''}}, {{math|1=''f''{{hairsp}}(''α'') = ''β'' × ''α''}} (for {{math|''β'' ≥ 1}}), and {{math|1=''f''{{hairsp}}(''α'') = ''β''<sup>''α''</sup>}} (for {{math|''β'' ≥ 2}}) are all normal.

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>.

== Properties ==
If {{mvar|f}} is normal, then for any ordinal {{mvar|α}},
:{{math|''f''{{hairsp}}(''α'') ≥ ''α''}}.<ref>{{harvnb|Johnstone|1987|loc=Exercise 6.9, p. 77}}</ref>
'''Proof''': If not, choose {{mvar|γ}} minimal such that {{math|''f''{{hairsp}}(''γ'') < ''γ''}}. Since {{mvar|f}} is strictly monotonically increasing, {{math|''f''{{hairsp}}(''f''{{hairsp}}(''γ'')) < ''f''{{hairsp}}(''γ'')}}, contradicting minimality of {{mvar|γ}}.

Furthermore, for any [[empty set|non-empty]] set {{mvar|S}} of ordinals, we have
:{{math|1=''f''{{hairsp}}(sup ''S'') = sup ''f''{{hairsp}}(''S'')}}.
'''Proof''': "≥" follows from the monotonicity of {{mvar|f}} and the definition of the [[supremum]]. For "{{math|≤}}", set {{math|1=''δ'' = sup ''S''}} and consider three cases:
* if {{math|1=''δ'' = 0}}, then {{math|1=''S'' = {{mset|0}}}} and {{math|1=sup ''f''{{hairsp}}(''S'') = ''f''{{hairsp}}(0)}};
* if {{math|1=''δ'' = ''ν'' + 1}} is a successor, then there exists {{mvar|s}} in {{mvar|S}} with {{math|''ν'' < ''s''}}, so that {{math|''δ'' ≤ ''s''}}. Therefore, {{math|''f''{{hairsp}}(''δ'') ≤ ''f''{{hairsp}}(''s'')}}, which implies {{math|''f''{{hairsp}}(δ) ≤ sup ''f''{{hairsp}}(''S'')}};
* if {{mvar|δ}} is a nonzero limit, pick any {{math|''ν'' < ''δ''}}, and an {{mvar|s}} in {{mvar|S}} such that {{math|''ν'' < ''s''}} (possible since {{math|1=''δ'' = sup ''S''}}). Therefore, {{math|''f''{{hairsp}}(''ν'') < ''f''{{hairsp}}(''s'')}} so that {{math|''f''{{hairsp}}(''ν'') < sup ''f''{{hairsp}}(''S'')}}, yielding {{math|1=''f''{{hairsp}}(''δ'') = sup {{mset|''f''{{hairsp}}(ν) : ''ν'' < ''δ''}} ≤ sup ''f''{{hairsp}}(''S'')}}, as desired.

Every normal function {{mvar|f}} has arbitrarily large fixed points; see the [[fixed-point lemma for normal functions]] for a proof. One can create a normal function {{math|''f{{hairsp}}′'' : Ord → Ord}}, called the '''derivative''' of {{mvar|f}}, such that {{math|''f{{hairsp}}′''(''α'')}} is the {{mvar|α}}-th fixed point of {{mvar|f}}.<ref>{{harvnb|Johnstone|1987|loc=Exercise 6.9, p. 77}}</ref> For a hierarchy of normal functions, see [[Veblen function]]s.

==Notes==
{{reflist}}

== References ==
{{refbegin}}
*{{citation
|first=Peter
|last=Johnstone
|authorlink=Peter Johnstone (mathematician)
|year=1987
|title=Notes on Logic and Set Theory
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-33692-5
|url-access=registration
|url=https://archive.org/details/notesonlogicsett0000john
}}
{{refend}}

[[Category:Set theory]]
[[Category:Ordinal numbers]]

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imported>Marcos: /* Properties */ Make proof clearer

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