Feferman–Schütte ordinal

The Feferman–Schütte ordinal \(\Gamma_0\) (pronounced "gamma-zero") is the first ordinal inaccessible through the Veblen hierarchy. Formally, it is the first fixed point of \(\alpha \mapsto \varphi_{\alpha}(0)\), visualized as \(\varphi_{\varphi_{\varphi_{._{._..}.}(0)}(0)}(0)\). It's named after Solomon Feferman and Kurt Schütte.

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 Veblen function, \(\theta(\Omega,0)\) using the Feferman theta function, and \(\vartheta(\Omega^2)\) using Weiermann theta function.

Using the fundalemtal sequences for the extended Veblen function, we can define \(\Gamma_0[0]=0\) and \(\Gamma_0[n+1]=\varphi(\Gamma_0[n],0)\).