# Takeuti–Feferman–Buchholz ordinal

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 (largest number definable using) Buchholz’s psi function and Feferman’s theta function. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as

${displaystyle psi (varepsilon _{Omega _{omega }+1})}$ in Buchholz’s psi function, an OCF invented by Wilfried Buchholz, and

${displaystyle theta _{varepsilon _{Omega _{omega }+1}}(0)}$ in Feferman’s theta function, an OCF invented by Solomon Feferman. It is the proof-theoretic ordinal of

${displaystyle Pi _{1}^{1}-CA+BI}$ , a subsystem of second-order arithmetic,

${displaystyle Pi _{1}^{1}}$ -comprehension + transfinite induction,IDω, the system of ω-times iterated inductive definitions and KPI, Kripke-Platek set theory with a recursively inaccessible universe.

Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC.

## . . . Takeuti–Feferman–Buchholz ordinal . . .

• Let
${displaystyle Omega _{alpha }}$ represent an uncountable ordinal with cardinality

${displaystyle aleph _{alpha }}$ .

• Let
${displaystyle varepsilon _{beta }}$ represent the

${displaystyle beta }$ th epsilon number, equal to the

${displaystyle 1+beta }$ th fixed point of

${displaystyle alpha mapsto omega ^{alpha }}$ • Let
${displaystyle psi }$ represent Buchholz’s psi function

• The TFBO is equal to
${displaystyle psi (varepsilon _{Omega _{omega }+1})}$ .

In other words, the TFBO is the smallest ordinal which cannot be expressed from

${displaystyle 0}$ ,

${displaystyle 1}$ ,

${displaystyle omega }$ and

${displaystyle Omega }$ using sums, products, exponentials, and the

${displaystyle psi }$ function itself, the latter of which only to previously constructed ordinals less than

${displaystyle varepsilon _{Omega _{omega }+1}}$

.