User blog comment:Bubby3/Guesses about the strength of BMS and the Catching Hierarchy./@comment-32697988-20180419222547/@comment-30754445-20180419233211

As far as I know, the limit of DAN hasn't been found yet.

According to Hyp Cos' own analysis, pDAN probably reaches stable ordinals. If we believe this analysis (and I'm not sure that I do) then DAN would go beyond that.

Since we don't have any ordinal notation that goes this high, further analysis is pretty much impossible. And the same, of-course, goes for Taranovsky C.

As for BMS, I'll be very very surprised if it reaches stable ordinals at any point, let alone (0,0,0)(1,1,1)(2,2,0). It's not the kind of thing that can happen by accident. Especially given the fact that BMS strings are basically hydras, and it's difficult to see how hydras would get us this far.