> Also, “not finitely axiomatizable” doesn’t mean “having infinitely many truths in the (?) model”. Not sure what you meant by that.
If you take all truths in the standard model of arithmetic, and make them your axioms you'll get a complete and consistent system with infinitely many axioms, and an extension of Q.
If one uses Roser's trick, Godel says a system cannot be all of these at the same time:
* a conservative extension of Q ("minimal amount of arithmetic")
* finitely axiomatizable
* complete
* consistent
It is ok to have any <=3 of these 4.
Note that PA is a conservative extension of Q. So you can replace Q with PA above.
Godel's original proof doesn't generalize this much but using Roser's trick Godel immediately implies this.
Also, “not finitely axiomatizable” doesn’t mean “having infinitely many truths in the (?) model”. Not sure what you meant by that.