# 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.[1][2] It was named by David Madore,[2] after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as

${displaystyle psi (varepsilon _{Omega _{omega }+1})}$

in Buchholz’s psi function,[3] an OCF invented by Wilfried Buchholz,[4][5][6] and

${displaystyle theta _{varepsilon _{Omega _{omega }+1}}(0)}$

in Feferman’s theta function, an OCF invented by Solomon Feferman.[7][8] It is the proof-theoretic ordinal of

${displaystyle Pi _{1}^{1}-CA+BI}$

,[9] a subsystem of second-order arithmetic,

${displaystyle Pi _{1}^{1}}$

-comprehension + transfinite induction,[3]IDω, the system of ω-times iterated inductive definitions[10] and KPI, Kripke-Platek set theory with a recursively inaccessible universe.[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]

## . . . 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

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

.

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

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