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

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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