Normal function explained

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:

1. For every limit ordinal (i.e. is neither zero nor a successor), it is the case that .
2. 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,

). 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

, 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.