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Normal function

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In <a href="Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>, a function f : <a href="Ordinal_number" title="Ordinal number">Ord</a> → Ord is called normal (or a normal function) if it is <a href="Continuous_function#Continuous_functions_between_partially_ordered_sets" title="Continuous function">continuous</a> (with respect to the <a href="Order_topology" title="Order topology">order topology</a>) and <a href="Monotonic_function" title="Monotonic function">strictly monotonically increasing</a>. This is equivalent to the following two conditions:

<ol><li>For every <a href="Limit_ordinal" title="Limit ordinal">limit ordinal</a> γ (i.e. γ is neither zero nor a <a href="Successor_ordinal" title="Successor ordinal">successor</a>), it is the case that f (γ) = <a href="Supremum" class="mw-redirect" title="Supremum">sup</a>{f (ν) : ν < γ}.</li>

<li>For all ordinals α < β, it is the case that f (α) < f (β).</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

A simple normal function is given by f (α) = 1 + α (see <a href="Ordinal_arithmetic" title="Ordinal arithmetic">ordinal arithmetic</a>). But f (α) = α + 1 is not normal because it is not continuous at any limit ordinal (for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\omega )=\omega +1\neq \omega =\sup\{f(n):n<\omega \}}">

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<annotation encoding="application/x-tex">{\displaystyle f(\omega )=\omega +1\neq \omega =\sup\{f(n):n<\omega \}}</annotation>

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</math><img src="./6f74a45be18f58237e096eae30fbda2d811199b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.909ex; height:2.843ex;" alt="{\displaystyle f(\omega )=\omega +1\neq \omega =\sup\{f(n):n<\omega \}}" loading="lazy">). 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="Aleph_number" title="Aleph number">aleph numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\alpha )=\aleph _{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle f(\alpha )=\aleph _{\alpha }}</annotation>

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</math><img src="./f5fcdeb3e28a550c319629977efeff65d5ecc93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.378ex; height:2.843ex;" alt="{\displaystyle f(\alpha )=\aleph _{\alpha }}" loading="lazy">, which connect ordinal and <a href="Cardinal_number" title="Cardinal number">cardinal numbers</a>, and by the <a href="Beth_number" title="Beth number">beth numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\alpha )=\beth _{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle f(\alpha )=\beth _{\alpha }}</annotation>

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</math><img src="./2f10c5c320d81db5fd2d616cb4b7eed7f50c850d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.508ex; height:2.843ex;" alt="{\displaystyle f(\alpha )=\beth _{\alpha }}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

If f is normal, then for any ordinal α,

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="Empty_set" title="Empty set">non-empty</a> set S of ordinals, we have

Proof: "≥" follows from the monotonicity of f and the definition of the <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a>. For "≤", set δ = sup S and consider three cases:

<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="Fixed-point_lemma_for_normal_functions" title="Fixed-point lemma for normal functions">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 href="#cite_note-2">[2]</a> For a hierarchy of normal functions, see <a href="Veblen_function" title="Veblen function">Veblen functions</a>.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFJohnstone1987">Johnstone 1987</a>, Exercise 6.9, p. 77

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFJohnstone1987">Johnstone 1987</a>, Exercise 6.9, p. 77

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFJohnstone1987" class="citation cs2"><a href="Peter_Johnstone_(mathematician)" title="Peter Johnstone (mathematician)">Johnstone, Peter</a> (1987), <a rel="nofollow" class="external text" href="https://archive.org/details/notesonlogicsett0000john">Notes on Logic and Set Theory</a>, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-33692-5</bdi></cite></li></ul>

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