>>12524308Yeah that was exactly the point of my post. By "up to equivalence" I meant that some axiomatizations look different but give the same models (technically that's an abuse of language. What is meant here is that they satisfy the same sentences). A simple example of this is PA with strong induction vs weak induction, which turn out to be equivalent, even though they look different.
There are of course also inconsistent axiomatizaions, and you can view them all as equivalent with the model being empty.
