In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function.[1][2] It was named by David Madore,[2] after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as using Buchholz's psi function,[3] an ordinal collapsing function invented by Wilfried Buchholz,[4][5][6] and in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman.[7][8] It is the proof-theoretic ordinal of several formal theories:
- ,[9] a subsystem of second-order arithmetic
- -comprehension + transfinite induction[3]
- IDω, the system of ω-times iterated inductive definitions[10]
Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC.[11]
Definition
- Let
represent the smallest uncountable ordinal with cardinality
.
- Let
represent the
th epsilon number, equal to the
th fixed point of
- Let
represent Buchholz's psi function