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, $ℍ$

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

The Veblen function ${\phi }_{\alpha }\beta$ is defined by setting ${\phi }_{\alpha }$ 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 $\mathrm{𝐒𝐂}$, and the enumerating function is written $f_{\mathrm{𝐒𝐂}}(\alpha )={\mathrm{\Gamma }}_{\alpha }$.

${\mathrm{\Gamma }}_0$, the first strongly critical ordinal, is also called the Feferman-Schutte ordinal.