In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
Contents
- 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
Properties
If f is normal, then for any ordinal α,
f(α) ≥ 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 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:
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.