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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

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. p進大好きbot, ペア数列の停止性, Japanese Googology Wiki user blog.
6. p進大好きbot, List of common mistakes in googology/FGH/Is the least catching ordinal well-defined?, Googology Wiki user blog.