User blog comment:P進大好きbot/Relation between an OCF and an Ordinal Notation/@comment-30754445-20180810102658/@comment-30754445-20180811181217

@Alemagno12

"TON has only been proven to terminate up to C(C(Ω2ω,0),0) though (but Boboris02 is working on another proof of termination of TON)"

Depends on waht you mean by "proven" (and I should have been more clear).

TON has been proven to terminate using some pretty heavly large cardinal axioms. So I guess that the question of whether you'll trust this proof depends on how much "faith" you have in such axioms.

And no offence to Boboris02, but I highly doubt any wiki member here is going to prove the termination of TON in (say) ZFC. As usual, many people here are grossly underestimating the complexity of the topics we're dealing with.

The only faint hope of this ever happening, is if TON is much weaker than even the doubters believe. In that case, Hyp Cos or Deedlit might be able to prove termination by "simply" analyzing the notation to completion (and even then it would be a very remote possibility, because TON is known to be very difficult to analyze properly).