# Normal function

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{{Short description|Function of ordinals in mathematics}}
{{one source |date=March 2024}}
In [axiomatic set theory](/source/axiomatic_set_theory), a function {{math|''f'' : [Ord](/source/ordinal_number) → Ord}} is called '''normal''' (or a '''normal function''') if it is [continuous](/source/continuous_function) (with respect to the [order topology](/source/order_topology)) and [strictly monotonically increasing](/source/monotonic_function). This is equivalent to the following two conditions:

# For every [limit ordinal](/source/limit_ordinal) {{mvar|γ}} (i.e. {{mvar|γ}} is neither zero nor a [successor](/source/successor_ordinal)), it is the case that {{math|1=''f''{{hairsp}}(''γ'') = [sup](/source/supremum){{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](/source/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](/source/aleph_number)s <math>f(\alpha) = \aleph_\alpha</math>, which connect ordinal and [cardinal number](/source/cardinal_number)s, and by the [beth number](/source/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 [non-empty](/source/empty_set) 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](/source/supremum). For "{{math|≤}}", consider three cases:
* 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=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 {{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](/source/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](/source/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](/source/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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Adapted from the Wikipedia article [Normal function](https://en.wikipedia.org/wiki/Normal_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.