Normal function
Template:Short description Template:One source In axiomatic set theory, a function Template:Math 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 Template:Mvar (i.e. Template:Mvar is neither zero nor a successor), it is the case that Template:Math.
- For all ordinals Template:Math, it is the case that Template:Math.
Examples
A simple normal function is given by Template:Math (see ordinal arithmetic). But Template:Math 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 Template:Mvar is a fixed ordinal, then the functions Template:Math, Template:Math (for Template:Math), and Template:Math (for Template:Math) 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 Template:Mvar is normal, then for any ordinal Template:Mvar,
- Template:Math.<ref>Template:Harvnb</ref>
Proof: If not, choose Template:Mvar minimal such that Template:Math. Since Template:Mvar is strictly monotonically increasing, Template:Math, contradicting minimality of Template:Mvar.
Furthermore, for any non-empty set Template:Mvar of ordinals, we have
Proof: "≥" follows from the monotonicity of Template:Mvar and the definition of the supremum. For "Template:Math", consider three cases:
- if Template:Math, then Template:Math and Template:Math;
- if Template:Math is a successor, then Template:Math is in Template:Mvar, so Template:Math is in Template:Math, i.e. Template:Math;
- if Template:Math is a nonzero limit, then for any Template:Math there exists an Template:Mvar in Template:Mvar such that Template:Math, i.e. Template:Math, yielding Template:Math.
Every normal function Template:Mvar has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function Template:Math, called the derivative of Template:Mvar, such that Template:Math is the Template:Mvar-th fixed point of Template:Mvar.<ref>Template:Harvnb</ref> For a hierarchy of normal functions, see Veblen functions.