Normal function
Jump to navigation
Jump to search
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:
- \(\alpha<\beta \Leftrightarrow f(\alpha)<f(\beta)\)
- \(f(\alpha)=\sup \{ f(\beta) | \beta < \alpha \} \) if \(\alpha\) 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.