User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35392788-20180813124110/@comment-35470197-20180813131536

Thank you for the answer.

> By the way, these ordinals are not smaller than the BHO, which is (0,0)(1,1)(2,2). I believe you meant ψ(Ωω).

Oh, sorry. I fix it. I am not so familiar with names of ordinals.

> Using Bucholz's psi (or UNOCF's, for that matter), it's very easy to analyse pair sequence system.

So your answer states \(1 = 3\). Then I have additional questions.

On the other hand, we have \begin{eqnarray*} \psi_0(\psi_1(\psi_2(0))) \leq \psi_0(\psi_1(\psi_2(\psi_3(0)))) \leq \psi_0(\psi_1(\psi_3(0))) \leq \psi_0(\psi_2(0)) = \psi_0(\psi_1(\psi_2(0))) \end{eqnarray*} by Lemma 1.2 (c) and (d) in [Buc], and hence \begin{eqnarray*} \psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_2(0)). \end{eqnarray*} Moreover, we have \begin{eqnarray*} \psi_0(\psi_2(0)) < \psi_0(\psi_2(0)) \times \omega = \psi_0(\psi_2(0) + 1) \leq \psi_0(\psi_3(0)) \end{eqnarray*} by Lemma 1.6 in [Buc], and hence \begin{eqnarray*} \psi_0(\psi_1(\psi_2(\psi_3(0)))) < \psi_0(\psi_3(0)). \end{eqnarray*} It contradicts your answer \(1 = 3\). Maybe I am wrong, as you said that it is easy. Could you point out my mistake?

[Buh] W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32, 1986, pp. 195--207.