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 if \( \varepsilon_0 = \varphi_1(0) \), 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.