Feferman–Schütte ordinal

The Feferman–Schütte ordinal, commonly denoted by \(\Gamma_0\)D. Madore, A Zoo of Ordinals (p.2). Retrieved 2021-06-10.M. Rathjen, Proof theory: From arithmetic to set theory (p.13). Retrieved 2021-06-19. (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 and . 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:
*Fast-growing hierarchy: \(f_{\Gamma_0}(n)\) represents the pentational space in the climbing methods of the Extended Cascading-E Notation and BEAF, such as pentacthulhum and kungulus.
*Hardy hierarchy: \(H_{\Gamma_0}(n)\) catches up \(f_{\Gamma_0}(n)\) as \(H_{\omega^{\alpha}}(n) = f_{\alpha}(n)\) for \(\alpha < \varepsilon_0\).
*Slow-growing hierarchy: \(g_{\Gamma_0}(n) \approx f_{\omega+1}(n)\) in the FGH \(\approx \{n,n,1,2\}\) in BEAF, which has an expandal growth rate, such as Graham's number.
==Sources==
== See also ==
ja:フェファーマン・シュッテの順序数
zh:費佛曼-舒特序
Category:Transfinite ordinals