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.

## Property

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.

## Sources

- ↑ W. Buchholz,
*A new system of proof-theoretic ordinal functions*(1984). Accessed 2021-06-19. - ↑ S. Simpson,
*Subsustems of Second-Order Arithmetic*, 2009 - ↑ Fish, Abbreviation, Googology Wiki user page.
- ↑ W. Buchholz, A survey on ordinal notations around the Bachmann-Howard ordinal (p.18)
- ↑ Bird, Chris. Fast-Growing Hierarchy in terms of Bird's Array Notation
- ↑ p進大好きbot, ペア数列の停止性, Japanese Googology Wiki user blog.
- ↑ 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)