Feferman–Schütte ordinal

In mathematics, the Feferman–Schütte ordinal Γ₀ is a large countable ordinal. It is the proof theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte.

It is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". Sometimes an ordinal is said to be predicative if it is less than Γ₀.

Unfortunately there is no standard notation for ordinals at and beyond the Feferman–Schütte ordinal, so there are several ways of representing it, some of which use ordinal collapsing functions: $\psi(\Omega^\Omega)$ , $\theta(\Omega)$ or $\phi_\Omega(0)$

Definition

The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition and the Veblen functions φ_α(β). That is, it is the smallest α such that φ_α(0) = α.

References

Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8, MR 1026933
Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244

Large countable ordinals

First infinite ordinal ω Epsilon numbers ε₀ Feferman–Schütte ordinal Γ₀ Ackermann ordinal θ(Ω²) small Veblen ordinal θ(Ω^ω) large Veblen ordinal θ(Ω^Ω) Bachmann–Howard ordinal ψ(ε_Ω+1) Ψ₀(Ω_ω) Church–Kleene ordinal ω₁^CK

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