User blog comment:Nedherman1/My Factorial vs HAN!!/@comment-35811215-20180703173610/@comment-30754445-20180707183535

I'll say that SAN has much more potential than BMS.

BM2 has just been proven to contain an infinite-loop, just a week or so ago (a fact which shouldn't have surprised anybody, given the complete lack of evidence that the system always terminates). So BMS as a whole isn't well-defined. The one thing it does have going for it, is that it is - by far - the simplest and most intuitive notation that reaches the BHO (Level 150).

SAN stands on much firmer grounds, at least up to a certain level. Hyp Cos made a thorough analysis of his system up to weakly compact cardinals (250). Unfortunately, we really have no way of knowing how strong the system is beyond that point, but that's strong enough to wipe clean most of the competitors.

At any rate, analyzing anything beyond 280 or so runs into a quite hairy problem: We can't analyze a notation without some yardstick to compare it too, and such a yardstick simply doesn't exist at this point.

To be more precise: The only "yardstick" we currently have for such large ordinals is that of abstract proof-theoretic concepts. Things like reflections and stable ordinals and nonprojectible ordinals and gods-knows-what-else. We know which of these properties is stronger (which is why I could order them on my semi-arbitrary scale) but there's no general way to test whether a given notation has a certain property.

So we're pretty much at the dark at these levels... At least until the professional proof-theorists find a way to actually build the ordinals beyond level 280 from the ground up (a thing, by the way, that they are trying very hard to acoomplish. Getting to 300, which corresponds to the PTO of 2nd Order Arithmetic, is one of the "holy grails" of proof theory)