Ψ 0(Ω ω)

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

In googology, the ordinal is widely called Buchholz's ordinal or BO, and is sometimes abbrviated to \(\psi(\Omega_{\omega})\). Readers should be careful that Buchholz's ordinal is different from Takeuti-Feferman-Buchholz ordinal. The abbreviation to \(\psi\) was originally invented by Rathjen and Weiermann to express the restriction \(\psi_0 |_{\varepsilon_{\Omega+1}}\), 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. 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 \(\textrm{Trans}(M)\). 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. 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.