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
 an OCF invented by Wilfried Buchholz, and
in Buchholz’s psi function,
 It is the proof-theoretic ordinal of
in Feferman’s theta function, an OCF invented by Solomon Feferman.
 a subsystem of second-order arithmetic,
IDω, the system of ω-times iterated inductive definitions and KPI, Kripke-Platek set theory with a recursively inaccessible universe.
-comprehension + transfinite induction,
Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC.
represent an uncountable ordinal with cardinality
th epsilon number, equal to the
th fixed point of
represent Buchholz’s psi function
- The TFBO is equal to
In other words, the TFBO is the smallest ordinal which cannot be expressed from
using sums, products, exponentials, and the
function itself, the latter of which only to previously constructed ordinals less than
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