Googology Wiki
Googology Wiki

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.


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]


  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. (see the end of "Going beyond the Bachmann-Howard ordinal")

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)