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, the former of whom suggested the name Γ₀.

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: $\psi(\Omega^\Omega)$, $\theta(\Omega)$, $\varphi_\Omega(0)$, or $\varphi(1,0,0)$.

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, 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 Γ₀ itself has order type less than Γ₀.Nachum Dershowitz
Termination of Rewriting
(pp.98--99), Journal of Symbolic Computation (1987). Accessed 3 October 2022.

References

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Proof theory
Ordinal numbers

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