Normal function

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 mononotically increasing. This is equivalent to the following two conditions:

1. For every infinite limit ordinal γ, f(γ) = sup {f(ν) : ν < γ}.
2. For all ordinals α < β, f(α) < f(β).

Facts

If f is normal, then for any γ ∈ Ord,

f(γ) ≥ γ.

(Proof: if this was not the case, we could choose a minimal γ with f(γ) < γ; then, since f is strictly monotonically increasing, f(f(γ)) < f(γ), which is a contradiction to γ being minimal.)

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 "≤", we set δ = sup S and distinguish three cases:

• if δ = 0, then S={0} and sup f(S) = f(0);
• if δ = ν + 1, then there exists s in S with ν < s, implying δ ≤ s and so f(δ) ≤ f(s), which implies f(δ) ≤ sup f(S);
• if δ is an infinite limit ordinal, we pick any ν < δ, then find s in S with ν < s (since δ = sup S) and hence f(ν) < f(s), implying f(ν) < sup f(S) and therefore f(δ) = sup { f(ν) : ν < δ } ≤ sup f(S) as desired.)

Every normal function f has arbitrarily large fixed points; see Fixed-point lemma for normal functions for a proof. One can thus create a new function g : Ord → Ord, colloquially described as "g(r) is the r-th fixed point of f". The function g is again normal.