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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 $$\Pi_1^1-\textrm{CA}+\textrm{BI}$$[1], a subsystem of second-order arithmetic. It is also the proof-theoretic ordinal of $$\Pi_1^1$$-comprehension+transfinite induction[2] In googology, the ordinal is abbreviated to TFBO.[3] 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.[4]

Sources

1. Buchholz, Feferman, Pohlers, Sieg, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies (1981)
2. D. Madore, A Zoo of Ordinals (#1.21) (2017, accessed 2020-11-25)
3. Fish, Abbreviation, Googology Wiki user page.
4. https://en.wikipedia.org/w/index.php?title=Ordinal_collapsing_function&oldid=206127084 (see the end of "Going beyond the Bachmann-Howard ordinal")