User blog:PsiCubed2/BMS, even if optimally defined, is much weaker than you thought

I've been investigating BMS for some time now, and I've reached the following conclusions:

(1) The source of the power of BMS is that it is equivalent to ordinal trees.

(2) Pair sequences, under a suitable definition, mimic buchholz hydras.

(3) Triad sequences up to (0,0,0)(1,1,1)(2,2,2), under a suitable (and quite unwieldy) definition, can mimic hydras of hydras, which get us to ψ(ψɪ(0)).

(Note: I highly doubt that any of the published versions of BMS actually reaches that far. They all contain too much patchwork which shows that they were created by trial and error. The odds of such an approach getting you the optimum result is virtually zero)

(4) It doesn't get much more powerful than that, simply because hydras alone cannot get you beyond inaccessibles.

If my analysis is correct (and I believe it is) that any version BMS must either taper off at these levels or reach an infinite loop.

Of-course, a set theory expert might be able to completely redefine the notation to make it keep track of Mahlos and beyond, but then it would not longer be BMS. As long as the good part/bad part thing is the basis of the notation, it will never go beyond inaccessibles.