Veblen hierarchy
The Veblen hierarchy is a sequence of ordinal-indexed functions \( \varphi_\alpha: \mathrm{Ord} \to \mathrm{Ord} \) which extend Cantor normal form by beginning with the base function \( \alpha \mapsto \omega^\alpha \) and iteratively taking fixed points at each next step. Formally, letting \( \mathrm{AP} := \{\omega^\alpha: \alpha \in \mathrm{Ord}\} \), we can define \( \varphi_\alpha(\beta) := \min\{\zeta \in \mathrm{AP}: \forall \gamma ((\gamma < \alpha \rightarrow \varphi_\gamma(\zeta) = \zeta) \land (\gamma < \beta \rightarrow \varphi_\alpha(\gamma) < \zeta))\} \). This is a definition by transfinite recursion, and is well-defined by the transfinite recursion theorem. Analogously to Cantor normal form, every ordinal can be written in its Veblen normal form, as a sum \( \varphi_{\alpha_1}(\beta_1) + \varphi_{\alpha_2}(\beta_2) + \cdots + \varphi_{\alpha_n}(\beta_n) \). Like how the limit of hereditary CNF is \( \varepsilon_0 = \varphi_1(0) \), the limit of hereditary of hereditary VNF is known as \( \Gamma_0 \). Ordinals beyond \( \Gamma_0 \) can either be written using a variadic extension of the Veblen hierarchy, or using ordinal collapsing functions.
Ordinals unreachable from below via Veblen normal form, the first of which is \( \Gamma_0 \), are typically known as gamma numbers or strongly critical ordinals. They are important in ordinal analysis due to the involvement of the Veblen function in cut-elimination. The enumeration function of the strongly critical ordinals is typically denoted \( \alpha \mapsto \Gamma_\alpha \), analogously to \( \alpha \mapsto \varepsilon_\alpha \) or \( \alpha \mapsto \zeta_\alpha \).
The first stage of the Veblen hierarchy is just the function \( \alpha \mapsto \omega^\alpha \). Then the next stage, \( \varphi_1(\beta) \) (alternatively written \( \varphi(1,\beta) \)) enumerates the fixed points of the first stage - these are just the epsilon numbers. This continues to the zeta numbers, followed by the eta numbers, although this convention is rare. At the \( \omega \)th stage, since \( \omega - 1 \) doesn't exist, instead \( \varphi(\omega,\beta) \) enumerates the ordinals that are simultaneous fixed points of \( \alpha \mapsto \varphi(n,\beta) \) for all \( n < \omega \).