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 the range of 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 ${\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}$ using Buchholz's psi function,^[3] an ordinal collapsing function invented by Wilfried Buchholz,^[4][5][6] and ${\displaystyle \theta _{\varepsilon _{\Omega _{\omega }+1}}(0)}$ in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman.^[7][8] It is the proof-theoretic ordinal of several formal theories:

• ${\displaystyle \Pi _{1}^{1}-CA+BI}$,^[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]

Definition

• Let ${\displaystyle \Omega _{\alpha }}$ represent the smallest 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