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 \( \Gamma_0 \), aka the Feferman-Schütte ordinal. It is equal to a moderately powerful system of second-order arithmetic, containing a second-order analogue of Peano arithmetic known as arithmetical comprehension, with an additional axiom that allows one to do transfinite recursions. As such it has been described as the least truly impredicative ordinal - the least ordinal which truly can not described from below and any definition must involve some form of self-reference. However, this description has been challenged on multiple occassions.

Ordinals beyond \( \Gamma_0 \) can be written using an extension of the Veblen hierarchy. This extension takes the form of \(\varphi(...,\alpha_3,\alpha_2,\alpha_1,\alpha_0)\) for finite amounts of entries. This reaches the Small Veblen ordinal, and can be extended, through transfinite amounts of entries, to the Large Veblen ordinal, and through "rows" and "planes" of entries to reach the Bachmann-Howard ordinal.