Using Buchholz's function, the ordinal \(\psi_0(\Omega_{\omega})\) is a large countable ordinal that is the proof theoretic ordinal of \(\Pi_1^1\)-\(\text{CA}_0\)[1], a subsystem[2] of second-order arithmetic.

In googology, the ordinal is widely called Buchholz's ordinal or BO,[3] and is sometimes abbrviated to \(\psi(\Omega_{\omega})\)[citation needed]. Readers should be careful that Buchholz's ordinal is different from Takeuti-Feferman-Buchholz ordinal. The abbreviation to \(\psi\) was originally invented by Rathejen and Weiermann to express the restriction \(\psi_0 |_{\varepsilon_{\Omega+1}}\),[4] of \(\psi\), but somewhy several googologists tend to use the same abbreviation to express the full \(\psi_0\) and other sorts of ordinal collapsing functions with indices under confusion of them.


It is the order type of the segment bounded by \(D_0 D_{\omega} 0\) in Buchholz's ordinal notation \((OT,<)\).

The subcubic graphs, which are used in definition of SCG function, can be ordered so that we can make bijection between them and ordinals below \(\psi_0(\Omega_{\omega})\), as well as Buchholz hydras with \(\omega\) labels removed. Bird stated that SCG function and Buchholz hydras with \(\omega\) labels removed are both approximated to \(\psi_0(\Omega_{\omega})\) in the fast-growing hierarchy.[5] Since there is no actual proof, it might be based on a wrong belief that a notation which can "express" ordinals below a fixed ordinal \(\alpha\) gives a function approximated to \(\alpha\) in the fast-growing hierarchy with respect to some system of fundamental sequence (counterexamples to this belief exist, e.g. Irrational arrow notation).

Also, it is verified by a Japanese Googology Wiki user p進大好きbot that each standard pair sequence \(M\) corresponds to an ordinal \(\textrm{Trans}(M)\) below \(\psi_0(\Omega_{\omega})\) so that the expansion expansion of \(M\) gives a strictly increasing sequence of ordinals below \(\textrm{Trans}(M)\).[6] In particular, it implies that pair sequence system restricted to standard pair sequences gives a total computable function whose stractural well-ordering is of order type bounded by \(\psi_0(\Omega_{\omega})\). It is also strongly believed in this community that the structural well-ordering is of order type \(\psi_0(\Omega_{\omega})\).

Common misconception

It is believed to be the first ordinal \(\alpha\) for which \(g_{\alpha}(n)\) in the slow-growing hierarchy "catches up" with \(f_{\alpha}(n)\) the fast-growing hierarchy in this community, but the statement is just a common mistake as p進大好きbot pointed out.[7] The ambiguous statement itself is not meaningful unless we fix the definitions of a comparability and a system of fundamental sequences applicable to ordinals including \(\psi_0(\Omega_{\omega})\), although the choices are often omitted. For example, \(\omega\) is also the first ordinal \(\alpha\) for which \(g_{\alpha}(n)\) in the slow-growing hierarchy "catches up" with \(f_{\alpha}(n)\) the fast-growing hierarchy in some sense under a certain system of fundamental sequences.


  1. W. Buchholz, A new system of proof-theoretic ordinal functions (1984). Accessed 2021-06-19.
  2. S. Simpson, Subsustems of Second-Order Arithmetic, 2009
  3. Fish, Abbreviation, Googology Wiki user page.
  4. W. Buchholz, A survey on ordinal notations around the Bachmann-Howard ordinal (p.18)
  5. Bird, Chris. Fast-Growing Hierarchy in terms of Bird's Array Notation
  6. p進大好きbot, ペア数列の停止性, Japanese Googology Wiki user blog.
  7. p進大好きbot, List of common mistakes in googology/FGH/Is the least catching ordinal well-defined?, Googology Wiki user blog.

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'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)

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