TITLE: How to prove a non-provable statement? That is weird...
QUESTION [20 upvotes]: a) there are mathematical statements, eg. formulated in Peano, which are known to be true but not provable. If not provable, how do we "prove" they are true? Is it not fishy? (I think Gödel does not address this exactly in this form. He says there is something which is true, but he does not say that we can know it is true.)
b) Suppose I add to Peano all those statements which can be known to be true theoretically. Will I have a statement then, which can be known to be true but not provable? I think Gödel does not address this exactly in this form. He says there will be something which is true, but he does not say that we can know it is true.

REPLY [5 votes]: There are two points here:

In the context of arithmetic, the term "true" means "true in the structure $\Bbb N$. So when Gödel's theorem states that there is a "true statement which is not provable", it means "true in the natural numbers" and not provable from $\sf PA$, or whatever relevant axiomatic system we're talking about.
But truth, even in the natural numbers, might depend on your meta-theory. This is a fickle question, since the choice of meta-theory depends a lot on the mathematician. But I guess most people would agree that since $\sf PA$ is consistent, the statement "$\sf PA$ is consistent" (which can be formulated as the non-existence of a solution for a Diophantine equation, for concreteness) is true, but it is not provable.

The incompleteness theorem does not state "every sufficiently complicated system is incomplete". Rather it states that "every sufficiently complicated system which can be recognized by a computer is incomplete". If we add all the true statements, then we get a complete theory, but this theory is no longer recognizable by a computer. Namely, we cannot write an algorithm to tell us whether a certain statement is an axiom of our theory or not.
This is bad, because the fact we can use a computer to verify our axioms means that we can use a computer to verify our proofs. And so the incompleteness theorem tells us that if we want to forego the incompleteness, we can no longer use machinated proof verification. And since humans tends to make mistakes, well, that's not ideal.


In any case, you prove things with a stronger theory. The statement $x\cdot y=y\cdot x$ is not provable from just the axioms of the group. After all, not every group is Abelian. But you can prove this once you add it as an axiom, or add a stronger axiom like $x\cdot x=1$.