No at all, you are absolutely correct and what you describe is known as omega-consistency! https://en.wikipedia.org/wiki/%CE%A9-consistent_theory https://en.wi

No at all, you are absolutely correct and what you describe is known as omega-consistency! https://en.wikipedia.org/wiki/%CE%A9-consistent_theory https://en.wikipedia.org/wiki/%CE%A9-consistent_theory However, your description is just another way of saying what I'm saying about nonstandard numbers. From outside of the theory you can prove P(1), P(2), P(n) for any N that is an actual natural number, but from inside of the theory you can't do that. Why not? Because from inside of the theory there is a model that is completely consistent with Peano arithmetic but contains some object E that is greater than any standard natural number. The theory can not eliminate the existence of this E and in fact it's possible to construct an E such that P(E) is false.