User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213035846/@comment-1605058-20180214064545

Taranovsky himself has proven well-foundedness in ZFC + measurable cardinal.

It's worth pointing out that it is not possible to verify well-foundedness using a computer program - the definition of well-foundedness involves all the possible subsets or over all sequences. There are uncountably many of them, and a program clearly can't do that.