User blog comment:Alemagno12/BM2 has a problem/@comment-30754445-20180724073311/@comment-35870936-20180725034527

@PsiCubed2

You seem to think that even a very good and detailed analysis can be completely useless, just because it's informal. You said "We have absolutely no idea what-so-ever how strong BM2 is", but in fact, we have EXTREMELY DETAILED analyses of BM2, like Aarex's, and you can't just say "well it's not formal, therefore it's completely useless". We DO have a good idea of how powerful BM2 is, and it's '''REALLY. FUCKING. POWERFUL.''' It's probably the second strongest notation anyone has ever made, second only to TON. (maybe it even goes past TON, but I doubt it). Of course, all that is under the assumption that BM2 actually terminates, but some people, including Nish and Koteitan, think that BM2 is likely to have an infinite loop somewhere past (0,0,0,0)(1,1,1,1). The newly defined BM2.3 is more likely to actually terminate, and keeps all the power of BM2.

You do realize that BM1 pair sequence expands quite differently than BM2 pair sequence? So when we found that (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1) in BM1 doesn't terminate, how does that affect whether the BHO in BM2 terminates? In fact, (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1) just happens to be an expression that BM2 expands differently than BM1. This is because BM1 had a problem, and BM2 fixed it. That is not a coincidence.

I thought the community pretty much agreed that (0,0)(1,1)(2,2) is the BHO, and the limit of pair sequence is psi(W_w). I would literally give you irl money if it turns out that the limit of pair sequence is something other than psi(W_w).

By the way, I have an idea for proving termination for pair sequence BMS. There are a few notations that are apparently extremely similar to pair sequence BMS, most notably UNOCF. The expression (a_1,b_1)(a_2,b_2)...(a_k,b_k) in BMS corresponds to a UNOCF expression, where each term (a,b) in BMS is equivalent to a psi_b function, at a depth of a. For example, (0,0)(1,1)(2,2)(3,1)(2,0) would be psi(psi_1(psi_2(psi_1(0))+psi_0(0))). If we could prove this correspondence, and if we formalize UNOCF up to psi(W_w) (which shouldn't be that hard tbh), the we have proved that the limit of pair sequence is psi(W_w).

And again, stop imposing an "analysis must be formal" rule on us. If we actually followed this rule, googology would have not NEARLY advanced as far as it has recently. We would be forced to slow down, and prove all our steps, and we would never get anywhere. We in the discord know exactly how powerful pDAN is (it's psi(e(T+1)) in UNOCF), and even full SAN (it's psi(C(1{w}0)).  We know this because we don't restrict ourselves to formal analysis.  Maybe we will turn out to be wrong, but if that's the case, then so be it.  It's just that as of now, there's no evidence at all that DAN is something other than psi(C(1{w}0)).