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The Feferman–Schütte ordinal, commonly denoted by $$\Gamma_0$$[1][2] (pronounced "gamma-nought"), 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, $$\psi_0(\Omega^\Omega) = \psi_0(\psi_1(\psi_1(\psi_1(0))))$$ using Buchholz's psi function, $$\theta(\Omega,0)$$ using Feferman's theta function, and $$\vartheta(\Omega^2)$$ using Weiermann's theta function.

## Appearances in googology

Using the Veblen hierarchy:

## Sources

1. D. Madore, A Zoo of Ordinals (p.2). Retrieved 2021-06-10.
2. M. Rathjen, Proof theory: From arithmetic to set theory (p.13). Retrieved 2021-06-19.