Feferman–Schütte ordinal

The Feferman–Schütte ordinal, commonly denoted by \(\Gamma_0\) (pronounced "gamma-zero"), is the first ordinal unreachable through the two-argument Veblen hierarchy. Formally, it is the first fixed point of \(\alpha \mapsto \varphi_{\alpha}(0)\), visualized as \(\varphi_{\varphi_{\varphi_{._{._..}.}(0)}(0)}(0)\) or \(\varphi(\varphi(\varphi(...),0),0),0)\), where \(\varphi\) denotes the Veblen function. It's named after Solomon Feferman and Kurt Schütte. A common fundamental sequence of \(\Gamma_0\) is defined as \(\Gamma_0[0]=0\) and \(\Gamma_0[n+1]=\varphi_{\Gamma_0[n]}(0)\), and this is from this system.

The Feferman–Schütte ordinal is significant as the proof-theoretic ordinal of ATR0 (arithmetical transfinite recursion, a subsystem of second-order arithmetic). It is \(\varphi(1,0,0)\) using the extended finitary Veblen function, \(\psi(\Omega^\Omega)\) using Buchholz's psi function, \(\theta(\Omega,0)\) using Feferman's theta function, and \(\vartheta(\Omega^2)\) using Weiermann's theta function.