User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-35470197-20180624023033

@Deedlit11

Thank you for the explanation!

> I think KurohaKafka claimed a proof for the original BMS

I think that he just assigned a direction of a possible proof of it, but did not claim a proof itself.

> 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.

Is the portion of TON sufficiently large? I would like to study the proof for the well-foundedness of TON. Could you have a reference?

> The apparent issue with doing this is that proofs can become much harder.

Exactly. Rathjen also mentioned the difficulty in his paper, as the original proof for the real weak Mahlo heavily depends on the method using cardinality.

> 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.

But if there is no proof of the well-foundedness, the transfinite definition of the map from the recursive ordinal notation to a countable set of ordinals do not work. Therefore even if the notation itself is well-defined, it does not present ordinal numbers.

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

I see.