@mdjoknaan You're welcome!

Hi; all has been said about the topos-theoretic account of your question, so I just wanted to expand a bit about the actual syntactic notion of interp...

@morgan-rogers Oh I never denied some changes or generalizations were possible, I was just unfolding what you had actually written, not thinking more ...

Wonderful remark! It made me try to rephrase it as I enjoy to do most as soon as I can, i.e. in terms of metamathelatical, syntactic interpretations...