In mathematics, the Feferman–Schütte ordinal Γ₀ is a large countable ordinal

Large countable ordinal

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal...

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It is the proof theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion.
It is named after Solomon Feferman

Solomon Feferman

Solomon Feferman is an American philosopher and mathematician with major works in mathematical logic.He was born in New York City, New York, and received his Ph.D. in 1957 from the University of California, Berkeley under Alfred Tarski...

 and Kurt Schütte

Kurt Schütte

Kurt Schütte was a German mathematician who worked on proof theory and ordinal analysis. The Feferman-Schütte ordinal, which he showed to be the precise ordinal bound for predicativity, is named after him.-References:...

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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.

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 function

Veblen function

In mathematics, the Veblen functions are a hierarchy of normal functions , introduced by Oswald Veblen in...

s φ_α(β). That is, it is the smallest α such that φ_α(0) = α.