About: Feferman–Schütte ordinal   Sponge Permalink

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

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  • Feferman–Schütte ordinal
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  • 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 \( heta(\Omega,0)\) using the Feferman theta function.
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  • 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 \( heta(\Omega,0)\) using the Feferman theta function.
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