papalex on Nostr: thanks for the correction! I made an embarrassing confusion (with a good resolve): 1. ...

thanks for the correction! I made an embarrassing confusion (with a good resolve):

1. PA is not ℵ₀-categorical: there are countable models that are not isomorphic.

2. PA admits an initial model in ZFC: you are right, and I was wrong. I confused the two points.

This also means that my “interpretation 2” is at best morally fine. There is a canonical model and that is what it is. However, deconfusing all this led me to understand the following, which is anyways a way better formulation of the two interpretations fused to one, and I hope this time is (closer to) correct:

Not all minimal inductive sets in all models are equal: which is the crucial point for the discussion. Given some model of ZFC M and create a new model M’ by extension by some new element c. Add axioms c ∈ ω and c ≠ 0, c ≠ S(0), c ≠ S(S(0)),… (the latter one supposedly by completeness to deal with the infinitely many) So we bluntly force a new element into ω. Then, by definition every inductive set contains c for else c \not\in ω violating M’. Thus, by definition we have a model in which the standard PA via minimal inductive set is not the standard ω_0 (though this difference is only seen externally by observing that ω is not isomorphic to ω_0).

This is a countable model which is not isomorphic but, and that was my earlier confusion, the standard naturals do uniquely map into this set. So there is an initial model but the standard PA model in such nonstandard ZFC models can be a nonstandard PA model as seen from the outside.