Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:

1. For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the case that fTemplate:Hairsp(γ) = sup{fTemplate:Hairsp(ν) : ν < γ}.
2. For all ordinals α < β, it is the case that fTemplate:Hairsp(α) < fTemplate:Hairsp(β).

Examples

A simple normal function is given by fTemplate:Hairsp(α) = 1 + α (see ordinal arithmetic). But fTemplate:Hairsp(α) = α + 1 is not normal because it is not continuous at any limit ordinal (for example, $f(\omega )=\omega +1\ne \omega =sup\{f(n):n<\omega \}$). If β is a fixed ordinal, then the functions fTemplate:Hairsp(α) = β + α, fTemplate:Hairsp(α) = β × α (for β ≥ 1), and fTemplate:Hairsp(α) = β^α (for β ≥ 2) are all normal.

More important examples of normal functions are given by the aleph numbers $f(\alpha )={\mathrm{\aleph }}_{\alpha }$, which connect ordinal and cardinal numbers, and by the beth numbers $f(\alpha )={\beth }_{\alpha }$.

Properties

If f is normal, then for any ordinal α,

fTemplate:Hairsp(α) ≥ α.

Proof: If not, choose γ minimal such that fTemplate:Hairsp(γ) < γ. Since f is strictly monotonically increasing, fTemplate:Hairsp(fTemplate:Hairsp(γ)) < fTemplate:Hairsp(γ), contradicting minimality of γ.

Furthermore, for any non-empty set S of ordinals, we have

fTemplate:Hairsp(sup S) = sup fTemplate:Hairsp(S).

Proof: "≥" follows from the monotonicity of f and the definition of the supremum. For "≤", consider three cases:

• if sup S = 0, then S = {0} and sup fTemplate:Hairsp(S) = fTemplate:Hairsp(0) = fTemplate:Hairsp(sup S);
• if sup S = ν + 1 is a successor, then sup S is in S, so fTemplate:Hairsp(sup S) is in fTemplate:Hairsp(S), i.e. fTemplate:Hairsp(sup S) ≤ sup fTemplate:Hairsp(S);
• if sup S is a nonzero limit, then for any ν < sup S there exists an s in S such that ν < s, i.e. fTemplate:Hairsp(ν) < fTemplate:Hairsp(s) ≤ sup fTemplate:Hairsp(S), yielding fTemplate:Hairsp(sup S) = sup {fTemplate:Hairsp(ν) : ν < sup S} ≤ sup fTemplate:Hairsp(S).

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 fTemplate:Hairsp′ : Ord → Ord, called the derivative of f, such that fTemplate:Hairsp′(α) is the α-th fixed point of f. For a hierarchy of normal functions, see Veblen functions.

References

• Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5