In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
For every limit ordinal γ (i.e. γ is neither zero nor a successor), f(γ) = sup {f(ν) : ν < γ}.
For all ordinals α < β, f(α) < f(β).
Examples
A simple normal function is given by f(α) = 1 + α (see ordinal arithmetic). But f(α) = α + 1 is not normal. 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 aleph numbers f ( α ) = ℵ α {\displaystyle f(\alpha )=\aleph _{\alpha }} f(\alpha )=\aleph _{\alpha } which connect ordinal and cardinal numbers, and by the beth numbers f ( α ) = ℶ α {\displaystyle f(\alpha )=\beth _{\alpha }} f(\alpha )=\beth _{\alpha }.
Properties
If f is normal, then for any ordinal α,
f(α) ≥ α.[1]
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 non-empty set S of ordinals, we have
f(sup S) = sup f(S).
Proof: "≥" follows from the monotonicity of f and the definition of the supremum. 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 fixed-point lemma for normal functions 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.[2]
Notes
Johnstone 1987, Exercise 6.9, p. 77
Johnstone 1987, Exercise 6.9, p. 77
References
Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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