FANDOM

9,740 Pages

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 $$\varphi_{\varphi_{\varphi_{._{._..}.}(0)}(0)}(0)$$.

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, and $$\theta(\Omega,0)$$ using the Feferman theta function.