User blog comment:Boboris02/Breakthrough! Traditional OCF definition for TON/@comment-30754445-20180414193201/@comment-1605058-20180414194929

Correct me if I'm wrong, but I believe what Taranovsky jas proven is well-foundedness of an old version of his system. The new one, which I believe Boris is basing on, is much stronger and has not been proven total in any system.

I haven't gone through all the details, but it seems to me that ZFC, and probably even many weaker theories, can prove that this OCF gives us a well-founded system. However, the crucial part is that we have to establish this OCF is really equivalent to TON. As is mentioned in the second paragraph, this cannot be done in weak theories. However, as of now, we don't know whether the two are equivalent even under the assumption of strong systems.