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In `F33f`_`[axiomatic set theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Axiomatic_set_theory]`_`f, a function `*f`* : `F33f`_`[Ord`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ordinal_number]`_`f → Ord is called `!normal`! (or a `!normal function`!) if it is `F33f`_`[continuous`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Continuous_function]`_`f (with respect to the `F33f`_`[order topology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Order_topology]`_`f) and `F33f`_`[strictly monotonically increasing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monotonic_function]`_`f. This is equivalent to the following two conditions:

1. For every `F33f`_`[limit ordinal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_ordinal]`_`f γ (i.e. γ is neither zero nor a `F33f`_`[successor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Successor_ordinal]`_`f), it is the case that `*f`* (`*γ`*) = `F33f`_`[sup`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Supremum]`_`f{`*f`* (`*ν`*) : `*ν`* < `*γ`*}.

2. For all ordinals `*α`* < `*β`*, it is the case that `*f`* (`*α`*) < `*f`* (`*β`*).

>>Contents

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Examples

A simple normal function is given by `*f`* (`*α`*) = 1 + `*α`* (see `F33f`_`[ordinal arithmetic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ordinal_arithmetic]`_`f). But `*f`* (`*α`*) = `*α`* + 1 is `*not`* normal because it is not continuous at any limit ordinal (for example, f ( ω ω ) = ω ω + 1 ≠ ≠ ω ω = sup { f ( n ) : n < ω ω } {\\displaystyle f(\\omega )=\\omega +1\\neq \\omega =\\sup\\{f(n):n<\\omega \\}} ). If β is a fixed ordinal, then the functions `*f`* (`*α`*) = `*β`* + `*α`*, `*f`* (`*α`*) = `*β`* × `*α`* (for `*β`* ≥ 1), and `*f`* (`*α`*) = `*β`*`*α`* (for `*β`* ≥ 2) are all normal.

More important examples of normal functions are given by the `F33f`_`[aleph numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Aleph_number]`_`f f ( α α ) = ℵ ℵ α α {\\displaystyle f(\\alpha )=\\aleph _{\\alpha }} , which connect ordinal and `F33f`_`[cardinal numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cardinal_number]`_`f, and by the `F33f`_`[beth numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Beth_number]`_`f f ( α α ) = ℶ ℶ α α {\\displaystyle f(\\alpha )=\\beth _{\\alpha }} .

>>Properties

If f is normal, then for any ordinal α,

`*f`* (`*α`*) ≥ `*α`*.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

`!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.

Furthermore, for any `F33f`_`[non-empty`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Empty_set]`_`f set S of ordinals, we have

`*f`* (sup `*S`*) = sup `*f`* (`*S`*).

`!Proof`!: "≥" follows from the monotonicity of f and the definition of the `F33f`_`[supremum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Supremum]`_`f. For "≤", set `*δ`* = sup `*S`* and consider three cases:

• if `*δ`* = 0, then `*S`* = {0} and sup `*f`* (`*S`*) = `*f`* (0);

• if `*δ`* = `*ν`* + 1 is a 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`*);

• 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.

Every normal function f has arbitrarily large fixed points; see the `F33f`_`[fixed-point lemma for normal functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fixed-point_lemma_for_normal_functions]`_`f for a proof. One can create a normal function `*f ′`* : Ord → Ord, called the `!derivative`! of f, such that `*f ′`*(`*α`*) is the α-th fixed point of f.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f] For a hierarchy of normal functions, see `F33f`_`[Veblen functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Veblen_function]`_`f.

>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `F33f`_`[Johnstone 1987`#citerefjohnstone1987]`_`f, Exercise 6.9, p. 77

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Johnstone 1987`#citerefjohnstone1987]`_`f, Exercise 6.9, p. 77

>>References

• `:citerefjohnstone1987`a`F33f`_`[Johnstone, Peter`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Peter_Johnstone_(mathematician)]`_`f (1987), `*Notes on Logic and Set Theory`*, `F33f`_`[Cambridge University Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cambridge_University_Press]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-521-33692-5

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