User blog comment:Boboris02/Breakthrough! Traditional OCF definition for TON/@comment-30754445-20180414193201/@comment-30118230-20180415123557

But either way, I already talked about this with Wojowu and Julian and it turns out that a theory can be too weak to prove TON well-founded but still be able to prove that my C = Taranovsky's C, assuming it can't give results for well-foundedness on either. However, for this specific case it may not be that way since my C has a standard OCF definition which is (relatively) easy to prove well-founded in ZFC, and if ZFC can prove my C to be well-founded but not Taranovsky, then it can't prove them to be the same since one would imply the other to be well-founded.