Veblen function

Remarks

Each $\varphi_\gamma$ is a normal function, and this makes clauses 2 and 3 meaningful, since the fixed points of normal functions form a closed unbounded class. Example: $\varphi_1(\alpha) = \varepsilon_\alpha$, the $\alpha$-th epsilon number, so $\varphi_1(0) = \varepsilon_0$ is the least $\alpha$ with $\omega^\alpha = \alpha$. Check: $\varphi_1$ lists the fixed points of $\varphi_0$, and these are by definition the epsilon numbers. Non-example: the function $\alpha \mapsto \omega^\alpha + 1$ cannot start the hierarchy, since it is not normal and has no fixed points. The least ordinal $\alpha$ with $\varphi_\alpha(0) = \alpha$ is the Feferman–Schütte ordinal $\Gamma_0$. More generally, one may start from any normal function $f$ in place of $\alpha \mapsto \omega^\alpha$.