User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-27513631-20161225231441

If we stop using first-order logic, then it seems to me that we have to use Henkin semantics, as otherwise it seems impossible to compare the numbers properly if they're not proper extensions or defined in the same background logic. And, afaik, A model of ZFC interpreted with second-order Henkin semantics is just a model of NBG with all classes definible from parameters.

Note that NBG doesn't give us a truth predicate for first order logic. One could use MK, but I see no reason to suppose we can quantify over \(A\subseteq V\), and am hesitant to change viewpoint.

I personally think that the new way forward is becoming inventive with truth predicates for truth predicates etc. (However there may be a way of doing this using \(\text{Def]^\alpha(V)\), which is equivalent to going beyond Higher Order Logic, but still using Henkin semantics.)