Normal function

A normal function is an ordinal function that preserves limits and is strictly increasing. That is, \(f\) is normal if and only if it satisfies the following properties:

• \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\)
• \(f(\alpha)=\sup f(\beta)\) if and only if \(\beta<\alpha\) and \(\alpha\) is a limit ordinal.

Veblen's fixed point lemma, which is essential for constructing the Veblen hierarchy, guarantees that, not only does every normal function have a fixed point, but the class of fixed points is unbounded and their enumeration function is also normal.