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

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) = β^α (for β ≥ 2) are all normal.

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

...
Wikipedia