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 Γ₀ itself has order type less than Γ₀. ^[4]

References

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