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The Feferman–Schütte ordinal $$\Gamma_0$$ (pronounced "gamma-zero") is the first ordinal inaccessible 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.[citation needed] 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.