# Veblen function

The Veblen function is used to obtain larger ordinal numbers than those provided by exponentiation. It builds on a hierarchy of closed and unbounded classes:

• $Cr(0)$ is the additively indecomposable numbers, $\mathbb{H}$

• $Cr(Sn)=Cr(n)^{\prime}$ the set of fixed points of the enumerating function of $Cr(n)$

• $Cr(\lambda)=\bigcap_{\alpha<\lambda}Cr(\alpha)$

The Veblen function $\varphi_{\alpha}\beta$ is defined by setting $\varphi_{\alpha}$ equal to the enumerating function of $Cr(\alpha)$.

We call a number $\alpha$ strongly critical if $\alpha\in Cr(\alpha)$. The class of strongly critical ordinals is written $\mathbf{SC}$, and the enumerating function is written $f_{\mathbf{SC}}(\alpha)=\Gamma_{\alpha}$.

$\Gamma_{0}$, the first strongly critical ordinal, is also called the Feferman-Schutte ordinal.

Title Veblen function VeblenFunction 2013-03-22 13:29:10 2013-03-22 13:29:10 Henry (455) Henry (455) 4 Henry (455) Definition msc 03E10 msc 03F15 strongly critical Feferman-Schutte ordinal