azul on Nostr: I think the problem is confusing. There are three levels here: Peano Arithmetic (the ...

I think the problem is confusing. There are three levels here: Peano Arithmetic (the theory), models of PA which live in ZFC set theory (the metatheory), and models of ZFC set theory which live in some ambient set/class theory (the meta-meta-theory). The "standard model of PA" is both a meta-theoretic idea that is internal to ZFC set theory (usually the initial model), and a meta-meta-theoretic idea that relates different models of set theory (N in V and its images).