User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-1605058-20170122222538

I think that you are the one who is incorrect about interpretability. Let me use this post as a reference about definitions. If you use different ones, please give reference for them.

In these definitions, the only part which actually references the theory (as opposed to just its language) is: [\(T\) proves] \(i(\phi)\) for each axiom \(\phi\) of \(S\). If you have theories \(S_1\subseteq S_2\) in the same language, then the fact that \(T\) proves (the interpretation of) every axiom of \(S_2\) without a doubt implies that \(T\) proves (the interpretation of) every axiom of \(S_1\).

Also, if ZFC- means ZFC without infinity, I'm pretty sure it doesn't interpret SOA. ZFC- is equiconsistent with PA, and if it interpreted SOA, it could prove PA, hence itself, is consistent.

As for the first part of your comment: I am unsure what you mean with "class operations", and if by the Wikipedia article you mean the NBG article, I don't see where it defines them. However, thanks to the explanation, I think I can better see now how to do the interpretation of NBG in ZFC.