In mathematics, the Feferman–Schütte ordinal Γ
0 is a
large countable ordinal.
It is the
proof-theoretic ordinal
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.
If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has ...
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 Γ
0.
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 t ...
s:
,
,
, or
.
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 Γ
0.
Any recursive path ordering whose function symbols are well-founded with order type less than that of
itself has order type
.
[N. Dershowitz]
Termination of Rewriting
(pp.98--99), Journal of Symbolic Computation (1987). Accessed 3 October 2022.
References
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{{DEFAULTSORT:Feferman-Schutte ordinal
Proof theory
Ordinal numbers