About Gödel's Incompleteness Theorem A short time ago, I've been thinking about what statements in Number Theory are true but not provable. I've seen the proof of the Incompleteness Theorem (in the Gödel's works) and he gave an example of one statements which is true but not provable, but that statement is too weird (maybe because the proof was constructive, and Gödel constructed this statement, I think).My Logic professor said that, in past, mathematicians believe that Fermat's conjeture was other of those statements, but later Andrew Wiles gave a demonstration of the conjeture. So, I have two big questions.


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*Could Goldbach's conjeture be true but not provable?

*What statements in Number Theory are true but not provable?


Thanks so much for all your ideas!

Firstly, by 'true' (and I wish people would stop using that word in connection with Godels theorem) we mean valid in the standard interpretation. Such statements are usually shown with the aid of set theory. 
Second, Fermats last has not yet been shown to be a consequence of Peano's axioms. Can Wiles proof be formalized in PA ?
To answer your questions, 


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*Yes, at the time of this writing it could be but we dont know.

*There are a number of natural and elegant theorems known not to be provable in PA, Paris-Harrington, Goodstein, and Hercules and the Hydra. 
