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

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]ID_{ω}, 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
- Let
- Let
- The TFBO is equal to

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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