Normal function


In axiomatic set theory, a function f : Ord → Ord is called normal if and only if it is continuous and strictly monotonically increasing. This is equivalent to the following two conditions:
  1. For every limit ordinal γ, f = sup.
  2. For all ordinals α < β, f < f.

    Examples

A simple normal function is given by f = 1 + α. But f = α + 1 is not normal. If β is a fixed ordinal, then the functions f = β + α, f = β × α, and f = βα are all normal.
More important examples of normal functions are given by the aleph numbers which connect ordinal and cardinal numbers, and by the beth numbers.

Properties

If f is normal, then for any ordinal α,
Proof: If not, choose γ minimal such that f < γ. Since f is strictly monotonically increasing, f < f, contradicting minimality of γ.
Furthermore, for any non-empty set S of ordinals, we have
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' is the α-th fixed point of f.