Takeuti-Feferman-Buchholz ordinal

Using Buchholz's psi notation, the ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\), usually called the "Takeuti-Feferman-Buchholz ordinal", is a large countable ordinal that is the proof-theoretic ordinal of \(\Pi_1^1-\textrm{CA}+\textrm{BI}\), a subsystem of second-order arithmetic. It is also the proof-theoretic ordinal of \(\Pi_1^1\)-comprehension+transfinite induction In googology, the ordinal is abbreviated to TFBO. Readers should be careful that Takeuti-Feferman-Buchholz ordinal is different from Buchholz's ordinal, which is abbreviated to BO.

Property
It is the limit of Feferman's theta function, as well as the limit of Buchholz's psi function. It is the order type of \(D_1 0\) in Buchholz's ordinal notation \((OT,<)\).

It is also the ordinal measuring the strength of Buchholz hydras with \(\omega\) labels, as well as the upper bound of the SCG function.

It was named by David Madore under the nickname "Gro-Tsen" on wikipedia.