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?

Consider this claim: John Smith will never be able to prove this statement is true.
If the statement is false, then John Smith will be able to prove it's true. But clearly that can't be, since it's impossible to prove that a false statement is true. (Assuming John Smith is sensible.)
If it's true, there's no contradiction. It just means John Smith won't be able to prove it's true. So it's true, and John Smith won't be able to prove it's true. This is a limit on what John Smith can do. (So if John Smith is sensible, there are truths he cannot prove.)
What Goedel showed is that for any sensible formal axiom system, there will be formal versions of "this axiom system cannot prove this claim is true". It will be a statement expressible in that formal system but, obviously, not provable within that axiom system.
