User blog comment:Googleaarex/Bashicu Matrix System Analysis (Part 1)/@comment-30754445-20170724222847/@comment-30754445-20170725044119

Oh, I agree that *if* BM is well-founded, then it is a very very strong system.

I guess what I'm asking is, how certain can we be that the above analysis is correct? Can we really be sure that (0,0,0)(1,1,1)(2,2,1)(3,2,0) is at the level of ψ(ψᵢ(0))? Can we even be sure that (0,0,0)(1,1,1)(2,2,1)(3,2,0) terminates?

It's not like Aarex provided us with proofs or even an intuitive explanation for why his estimates are correct. And we know from previous experience, how easy it get these things wrong. After all, there was a claimed proof for the well-foundedness of BM1 for years, until Hyp Cos provided us with counter-examples.

In short, I think that much more work will have to be done, before we can say things like "|(0,0,0)(1,1,1)(2,2,1)(3,2,0)|=ψ(ψᵢ(0))" with any kind of confidence.

And on a somewhat related/unrelated note:

What is SAN? I've never heard of it before, and I wonder how much work was done to show that (a) SAN itself is well-founded and (b) that is exactly as strong as it is claimed to be.

I really feel bad for being that guy who "rains on the parade", but this community has a very long history of notations which were claimed to reach crazy heights and later found to be ill-defined and/or not as strong as people thought they were. So I think we should be more cautious with our claims.

(I also think we need to have an agreed-upon universal notation for ordinals beyond ψ(ψᵢ(0)) before we even attempt to make claims at those levels)

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