In axiomatic set theory, a function is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
- For every limit ordinal (i.e. is neither zero nor a successor), it is the case that .
- For all ordinals , it is the case that .
Examples
A simple normal function is given by (see ordinal arithmetic). But is not normal because it is not continuous at any limit ordinal (for example, <math>f(\omega) = \omega+1 \ne \omega = \sup \{f(n) : n < \omega\}</math>). If is a fixed ordinal, then the functions , (for ), and (for ) are all normal.
More important examples of normal functions are given by the aleph numbers <math>f(\alpha) = \aleph_\alpha</math>, which connect ordinal and cardinal numbers, and by the beth numbers <math>f(\alpha) = \beth_\alpha</math>.
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.