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 \(\Pi_1^1-\textrm{CA}+\textrm{BI}\)Buchholz, Feferman, Pohlers, Sieg, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies (1981), a subsystem of second-order arithmetic. It is also the proof-theoretic ordinal of \(\Pi_1^1\)-comprehension+transfinite inductionD. Madore, A Zoo of Ordinals (#1.21) (2017, accessed 2020-11-25) In googology, the ordinal is abbreviated to TFBO.Fish, Abbreviation, Googology Wiki user page. 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.https://en.wikipedia.org/w/index.php?title=Ordinal_collapsing_function&oldid=206127084 (see the end of "Going beyond the Bachmann-Howard ordinal")
In the fast-growing hierarchy, \(f_{\psi_0(\varepsilon_{\Omega_\omega+1})}(n)\) is approximately {n,n [1 [1 [1 \2,2 3] 2] 2] 2} in Bird's array notation,Beyond Bird’s Nested Arrays V (Retrieved 25 April, 2022) and is approximately s(n,n {1 ,, 2,2} 2) in strong array notation according to Googology Wiki user hyp cos.Primary dropping array notation – Steps Towards Infinity! (Retrieved 25 April, 2022) However, strong array notation is known to have problems, and hence the comparison does not make sense. See strong array notation#Common misconceptions for more details.
== Sources ==
== See also ==
ja:竹内・フェファーマン・ブーフホルツ順序数
Category:Transfinite ordinals
Category:Buchholz's psi-functions