Normal function

<h2 id="examples">Examples</h2>
A simple normal function is given by f (α) = 1 + α (see <a href="/facts/Ordinal_arithmetic/satc9FHy">ordinal arithmetic</a>). But f (α) = α + 1 is not normal because it is not continuous at any limit ordinal; that is, the <a href="/facts/Inverse_image/KANefMAl">inverse image</a> of the one-point open set {λ + 1} is the set {λ}, which is not open when λ is a limit ordinal. If β is a fixed ordinal, then the functions f (α) = β + α, f (α) = β × α (for β ≥ 1), and f (α) = βα (for β ≥ 2) are all normal.
More important examples of normal functions are given by the <a href="/facts/Aleph_number/kwe9I4v5">aleph numbers</a> 
 
 
 
 f
 (
 α
 )
 =
 
 ℵ
 
 α
 
 
 
 
 {\displaystyle f(\alpha )=\aleph _{\alpha }}
 
, which connect ordinal and <a href="/facts/Cardinal_number/ccoKwU4R">cardinal numbers</a>, and by the <a href="/facts/Beth_number/zNRDWG5Q">beth numbers</a> 
 
 
 
 f
 (
 α
 )
 =
 
 ℶ
 
 α
 
 
 
 
 {\displaystyle f(\alpha )=\beth _{\alpha }}
 
.

<h2 id="properties">Properties</h2>
If f is normal, then for any ordinal α,

f (α) ≥ α.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
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 <a href="/facts/Empty_set/ffEk3eA6">non-empty</a> set S of ordinals, we have

f (sup S) = sup f (S).
Proof: "≥" follows from the monotonicity of f and the definition of the <a href="/facts/Supremum/oXCKVopz">supremum</a>. For "≤", set δ = sup S and consider three cases:

<ul><li>if δ = 0, then S = {0} and sup f (S) = f (0);</li>
<li>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);</li>
<li>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) f (S), yielding f (δ) = sup {f (ν) : ν < δ} ≤ sup f (S), as desired.</li></ul>
Every normal function f has arbitrarily large fixed points; see the <a href="/facts/Fixed-point_lemma_for_normal_functions/dvzlwXz2">fixed-point lemma for normal functions</a> 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.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> For a hierarchy of normal functions, see <a href="/facts/Veblen_function/XzaRJyjr">Veblen functions</a>.

<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Peter_Johnstone_(mathematician)/j3k2A3Xl">Johnstone, Peter</a> (1987), <a href="https://archive.org/details/notesonlogicsett0000john">Notes on Logic and Set Theory</a>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-33692-5</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Johnstone 1987, Exercise 6.9, p. 77 - Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5 <a href="https://archive.org/details/notesonlogicsett0000john" target="_blank">https://archive.org/details/notesonlogicsett0000john</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Johnstone 1987, Exercise 6.9, p. 77 - Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5 <a href="https://archive.org/details/notesonlogicsett0000john" target="_blank">https://archive.org/details/notesonlogicsett0000john</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Normal function open-in-new

Normal function