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:

1. For every limit ordinal Template:Mvar (i.e. Template:Mvar is neither zero nor a successor), it is the case that Template:Math.
2. 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; that is, the inverse image of the one-point open set Template:Math is the set Template:Math, which is not open when Template:Mvar is a limit ordinal. 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 $f(\alpha )={\mathrm{\aleph }}_{\alpha }$, which connect ordinal and cardinal numbers, and by the beth numbers $f(\alpha )={\beth }_{\alpha }$.

Properties

If Template:Mvar is normal, then for any ordinal Template:Mvar,

Template:Math.

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

Template:Math.

Proof: "≥" follows from the monotonicity of Template:Mvar and the definition of the supremum. For "Template:Math", set Template:Math and consider three cases:

• if Template:Math, then Template:Math and Template:Math;
• if Template:Math is a successor, then there exists Template:Mvar in Template:Mvar with Template:Math, so that Template:Math. Therefore, Template:Math, which implies Template:Math;
• if Template:Mvar is a nonzero limit, pick any Template:Math, and an Template:Mvar in Template:Mvar such that Template:Math (possible since Template:Math). Therefore, Template:Math so that Template:Math, yielding Template:Math, as desired.

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. For a hierarchy of normal functions, see Veblen functions.

References

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