papalex on Nostr: Indeed, here are several levels at play causing confusion. I'll have a try to ...

Indeed, here are several levels at play causing confusion. I'll have a try to deconfuse:

1. In any model of ZFC we have a standard model of PA defined via the axiom of infinity. Call this inductively defined set ℕ.
2. Any nonstandard model of ZFC has a meta theory strictly richer than ZFC. This meta theory is the axiomatic (or theoretical) shadow of the nonstandard structure of this model. Reasoning in this meta theory is in a sense reasoning about the actual structure of the model.
3. In the standard model of ZFC ℕ only contains finite sets (or bounded numbers).
4. IST is conservative over ZFC, thus every model of IST is by reduct also a model of ZFC. Reasoning in the IST metatheory is just the god's eye view onto the models structure as visible from outside ZFC.
5. In IST by idealization every infinite set contains non-standard elements. Thus, in IST ℕ contains infinite sets (or unbounded numbers) and conversely this means that ℕˢᵗ is finite and thus not the same as ℕ in any meaningful way (neither in ZFC nor in IST).
6. Thus, the standard model of PA in a non-standard model of ZFC (via IST) is not isomorphic to the standard model of PA in the canonical ZFC model—because idealization forces ℕ itself to contain unbounded naturals.