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, the former of whom suggested the name Γ₀.^[1]

There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing functions: ${\displaystyle \psi (\Omega ^{\Omega })}$, ${\displaystyle \theta (\Omega )}$, ${\displaystyle \varphi _{\Omega }(0)}$, or ${\displaystyle \varphi (1,0,0)}$.

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.

Properties

This ordinal is sometimes said to be the first impredicative ordinal,^[2][3] 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.

Any recursive path ordering whose function symbols are well-founded with order type less than that of ${\displaystyle \Gamma _{0}}$ itself has order type ${\displaystyle <\Gamma _{0}}$.^[4]

References

• Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, doi:10.1007/978-3-540-46825-7, ISBN 3-540-51842-8, MR 1026933
• Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244, Bibcode:2005math......9244W