User blog comment:Syst3ms/STON 2 : BMS and the Star/@comment-33713741-20190904131346/@comment-35392788-20190904143744

@Plain'N'Simple Yeah, we don't have any proofs of termination (or anything else for that matter) of BMS past PSS. So, when I'm comparing things to BMS, of course I need to assume its termination beforehand, otherwise it doesn't make sense. And no, I don't have any particularly solid basis to say that something like (0,0,0)(1,1,1)(2,2,2) terminates.

However please note this. If that matrix is as strong as it's claimed to be (i.e probably past most OCFs out there, although I don't know enough about Pi-w reflection to make a fair statement about OCFs based on it), then it would be extremely difficult to prove its termination. The most common way of proving the well-foundedness of a system is creating a mappings to ordinals in a suitably strong OCF, but if that matrix is indeed stronger than all standard OCFs, that way of proving it is just ruled out ; and of course, you're very much aware of how hard it is to make a powerful OCF, even with background in set theory.

On the other hand, successfully pulling off that strategy would show that BMS is a lot weaker than expected (or intended, for that matter).

Bottom line is, if I make comparisons to notations not proven to terminate, I implicitly assume they do for the sake of argument. But my claims about the strength aren't backed by anything right now, really, since analysis is stuck quite early in the beginning of my first definition, and I don't know stronger OCFs than Rathjen's well enough to continue "reasonable" analysis.