papalex on Nostr: after all, I have the impression there are two possible meanings of whether “a ...

after all, I have the impression there are two possible meanings of whether “a model of PA in a nonstandard model of ZFC is standard or not” (likely there is only one by some definition, if anyone knows that: please drop a reference 🙏). However, in either the solution to the problem becomes quite tautological.

First, the notion of standard model in a theory is anchored by the theory. In ZFC for example the axiom of infinity is that anchor, but there is no initial model of PA in ZFC. Standard is more like the informal „common“ rather than the formal „canonical“, in that it’s us identifying this axiom as the anchor, but there is not a canonical rule it seems to identify the anchor in an arbitrary theory.

Now, if we change from „model of PA in ZFC“ to „model of PA in __model of__ ZFC“ I wonder what’s the appropriate adaptation of the notion of standard. Here are the two options I see:

1/n