User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-27513631-20170127192537

No, we don't have to prove *, just its relative consistency. And you seem to have missed that we don't have to manage theorems of the form 'not * implies...'. Consider the theories ZFC+'if there is an inaccessible cardinal, there is a Mahlo cardinal' and ZFC+'there is no I accessible cardinal'. The second theory is stronger, but can obviously be interpreted by the first...