Gödel proved that there are true sentences that cannot be proved.
Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)
Is that logically possible? (And, no, I haven't proved it!)
(In reply to My thought is....
Let's be clear about what Godel's theorem asserts. It says that in any formal system of axioms and numbers rich enough to include all the integers, multiplication, and addition, and a few other familiar properties, that there exist statements in that system that are true in that system, but cannot be formally derived from the axioms of that system.
In fact, Godel demonstrated how to construct such a statement--lets call it G. Godel showed that G is true by creating a correspondence between statements about numbers and the numbers themselves. This, essentially, "steps outside" the formal system to demonstrate the truth of G. Statement G depends on what set of axioms you choose as the basis of your system. G is a very long statement, but its existence only demonstrates that statements exist which are true but have no formal proof within their own system.
Is the Goldbach conjecture such a statement? Until someone proves it or finds a counter-example, we'll never know.