User blog comment:Ecl1psed276/A list of all BMS versions and their differences/@comment-30754445-20180725115828

You do realize that whether a given version terminates or not, is not the interesting question?

What matters is the strength of the subsystem that does terminate.

For example, BM1 is still a great notation for ordinals below Γ₀. It's simple, elegant, and it works up to (0,0)(1,1)(2,1)(3,1) and a little further. The fact that BM1 doesn't terminate after that point does not deminish its worth in the domain it does work.

Or another example:

If you could really show that some version of BMS reaches weakly compacts at (say) (0,0,0)(1,1,1)(2,2,0), that would be very very impressive, even if the system breaks down at quad arrays. Given that the people here seem to be sure (for whatever reason) that triad arrays gets you beyond PTO(Z2), who cares whether quad arrays terminate or not?

By the way: It is impossible to answer the "does it terminate?" question without giving an estimate of how strong the system is. Either you analyze the system completely (and arrive at the precise strength of the system) or you prove that the system terminates using some axiomatic system (in which case the PTO of that axiomatic system is an upper bound of your notation's strength). ANY evidence one might give for termination, would also tell us alot about the strength of the notation.