Normal function

In axiomatic set theory, a function is called normal if it is continuous and strictly monotonically increasing. This is equivalent to the following two conditions:

For every limit ordinal, it is the case that.
For all ordinals, it is the case that.

Examples

A simple normal function is given by . But is not normal because it is not continuous at any limit ordinal. If is a fixed ordinal, then the functions,, and 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 is normal, then for any ordinal,
Proof: If not, choose minimal such that. Since is strictly monotonically increasing,, contradicting minimality of.
Furthermore, for any non-empty set of ordinals, we have
Proof: "≥" follows from the monotonicity of and the definition of the supremum. For "", consider three cases:

if, then and ;
if is a successor, then is in, so is in, i.e. ;
if is a nonzero limit, then for any there exists an in such that, i.e., yielding.

Every normal function has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function, called the derivative of, such that is the -th fixed point of. For a hierarchy of normal functions, see Veblen functions.