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In mathematics, the Feferman–Schütte ordinal Γ0 is a
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 rele ...
. It is the proof-theoretic ordinal of several mathematical theories, such as
arithmetical transfinite recursion Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in cont ...
. It is named after
Solomon Feferman Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to th ...
and
Kurt Schütte Kurt Schütte (14 October 1909, Salzwedel – 18 August 1998, Munich) 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 predicativi ...
, 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 function In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining ( notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger ...
s: \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 φα(β). That is, it is the smallest α such that φα(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 \Gamma_0 itself has order type <\Gamma_0. N. Dershowitz
Termination of Rewriting
(pp.98--99), Journal of Symbolic Computation (1987). Accessed 3 October 2022.


References

* * {{DEFAULTSORT:Feferman-Schutte ordinal Proof theory Ordinal numbers