What is adding Con(PA) to PA good for? Is there any non-trivial statement in PA + Con(PA), that was not already in PA? where PA = Peano Arithmetic
The standard model of PA (the basis of number theory) seems to be a model of PA + Con(PA) too. In this sense, is there a theorem in PA + Con(PA) that can't be proven in standard number theory?
 A: This is not quite an answer but it is related. Gentzen showed that you can prove the consistency of PA using primitive recursive arithmetic together with transfinite induction up to a countable ordinal called $\epsilon_0$. As I understand it, he argued that the existence of $\varepsilon_0$ was less controversial than the consistency of PA so this was a good reason to believe that PA was consistent. So instead of considering adding $\text{Con}(PA)$ let's consider adding the existence of $\varepsilon_0$.
Then there are some "natural" statements such as Goodstein's theorem, the strengthened finite Ramsey theorem, or the winnability of the (Kirby-Paris) hydra game which can be proven using transfinite induction up to $\varepsilon_0$ (and which I believe are essentially equivalent to the existence of $\varepsilon_0$ , although I can't find a reference explicitly stating this) but which are known to be independent of PA. (If they're equivalent to the existence of $\varepsilon_0$ then this follows since PA can't prove the existence of $\varepsilon_0$, by Gentzen.)
