In the mathematical fields of

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

and

proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...

, the Takeuti–Feferman–Buchholz ordinal (TFBO) 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 releva ...

, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function. It was named by David Madore, after

Gaisi Takeuti was a Japanese mathematician, known for his work in proof theory. After graduating from Tokyo University, he went to Princeton to study under Kurt Gödel. He later became a professor at the University of Illinois at Urbana–Champaign. Takeu ...

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 Wilfried Buchholz. It is written as

\psi_0(\varepsilon_)

using Buchholz's psi function, an

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 ...

invented by Wilfried Buchholz, and

\theta_(0)

in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories: *

\Pi_1^1 -CA + BI

, a subsystem of second-order arithmetic *

\Pi_1^1

-comprehension + transfinite induction * ID_ω, the system of ω-times iterated inductive definitions Despite being one of the largest

large countable ordinals 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 relev ...

and recursive ordinals, it is still vastly smaller than 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 ZFC.

Definition

* Let

\Omega_\alpha

represent the smallest

uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...

ordinal with

cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

\aleph_\alpha

. * Let

\varepsilon_\beta

represent the

\beta

th epsilon number, equal to the

1+\beta

th fixed point of

\alpha \mapsto \omega^\alpha

* Let

\psi

represent Buchholz's psi function

References

{{DEFAULTSORT:Takeuti-Feferman-Buchholz ordinal Proof_theory Ordinal_numbers Set_theory