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

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

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

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

-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
  ${displaystyle Omega _{alpha }}$

  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

,

,

and

using sums, products, exponentials, and the

function itself, the latter of which only to previously constructed ordinals less than

.

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

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