Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff it is continuous and strictly mononotically increasing. That is, a function is normal iff the following two conditions hold:

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

It is easy to show that normal functions have arbitrarily large fixed points; see Fixed-point lemma for normal functions for a proof.