User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-30754445-20180813155654/@comment-35470197-20180813221451

Actually, I formally proved the termination of pair sequence system for specific three versions of BMS. (It is done in a personal note, and hence might contain mistakes.)

After that, I tried to analyse the strength using a correspondence to ordinals below psi_0(Omega_omega), which I used to prove the termination. Then I noticed that the correspondence might have serious skips. Although they ensure the termination, but do not ensure the strength.

So I checked other analyses. Then I guessed that several analysts seem to be assuming wrong equalities on Buchholz's psi function. That is why I asked these questions. If I found contradictions within their analyses, then I do not have to think about what actual versions they are considering.

For example, at least five googologists said that 1=3 with respect to the questions above. As I wrote below, I think that it is incorrect. I copy the computation here. \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*} You can skip psi_0(psi_1(psi_3(0))) because of psi_1(psi_2(psi_3(0))) < psi_2(0). I wrote it in order to argue in a step-by-step way.

Of course, I might be wrong. Do you have any correction to my computation?