User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-5529393-20180624020054

@Ecl1psed276:

There are notations in the literature that go far beyond psi(e(K+1)), that could be considered "standard" I suppose. I believe these notations will turn out to be stronger than BMS and TON, but it doesn't look like anyone is close to figuring out exactly how they compare to each other.

How sure are you that UNOCF goes higher than DAN? It seems that perhaps psi(e(T+1)) is the level of PNAN in the R function, which are notations using one *. I think a reasonable guess would be that C(1;;;0) has the same strength as **, C(1;;;;0) the same strength as ***, and so on. In Nested Array Notaion, n *'s is equivalent to ,,n - this is equivalent to ,,,n in DAN. So perhaps the Small Dropping Ordinal can be expressed as some notation with ,,,1,2 in DAN. Just speculation on my part, but it does seem to me that the nesting that takes place in DAN is stronger than what is going on in UNOCF.

@P進大好きbot:

BMS was found to have certain expressions where the computation does not terminate; after this was found, BashicuHyudora created BMS2, which I believe he claims to have no such expressions. (Various other versions were created as well by several people.) But, it hasn't been proven that the computation always terminates. I think KurohaKafka claimed a proof for the original BMS, but of course that proof must have been faulty.

Hyp Cos's strong array notation hasn't been proven to always terminate either. As Wojowu said, Taranovsky has claimed a proof that a portion of TON is well-founded, but probably no one has looked at the proof critically. So, I would say the well-foundedness for all three of these notations are up in the air.

Concerning the notations created using large cardinals - as you have noted, the large cardinals can be replaced by their recursive analogues. So for psi(e(K+1)), we could replace weakly inaccessibles by recursively inaccessibles, weakly Mahlos by recursively Mahlos, and weakly compact cardinals by Pi-3 reflecting ordinals, and so on. The apparent issue with doing this is that proofs can become much harder. So, I don't know if, for example, a full ordinal analysis of KP + Pi-3 Reflection has been done using an ordinal notation with recursive analogues. But, that does not seem necessary to having an ordinal notation be "completely defined". If we use the recursive analogues for psi(e(K+1)), we will have a recursive notation that refers to a countable set of ordinals, with nothing independent of ZFC being used. So it should satisfy "completely defined", I would think.

It doesn't look like UNOCF has any ruleset yet. This should be a big priority for UNOCF I would think.