True vs. Provable Gödel's first incompleteness theorem states that "...For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system".
What does it mean that a statement is true if it's not provable?
What is the difference between true and provable?

Suppose I made some kind of statement like "For all natural numbers $n$, $P(n)$ holds."
For my claim to be true, it must be that there is no counterexample. That is, there is no integer $n_0$ such that $P(n_0)$ fails. Even if this is the case, I may not be able to prove that it is so. So, my statement could be true, but unprovable.
