How could a statement be true without proof? Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions;
1) How can we say that a statement is true without a proof?
2) What has the self reference to do with this? Godel sentence "G" can say that SUB(a,a, no prove) but could be this just arbitrarily judgement about non-provability of "a" because it may simply has a proof which is not yet revealed or discovered?

As is convenient in discussions of the completeness theorems, let's focus on the theory of arithmetic over the natural numbers. We have a well-defined notion of truth in the natural numbers under the usual arithmetic operations: e.g., the sentence $\forall i\cdot \exists a, b, c\cdot \ a > i \land a^2 + b^2 = c^2$ asserts the existence of arbitrarily large Pythagorean triples and we "know" what that means (specifically it means that it is valid in what model theorists call the structure comprising the natural numbers and the usual arithmetic operations).
Now let's think about methods for deducing which sentences are true in this theory, e.g., we might take our method to be first-order logic and Peano's axioms. What  Gödel's incompleteness theorem says is that (under some very reasonable hypotheses on the method), there is no method that will let us deduce all true sentences and no false sentences. The theorem does not say "there is no proof", it says "a specific method of finding proofs will never find one".
The philosophical interest is that the theorem applies to an enormous range of "specific methods of finding proofs" and indicates that there is no chance of finding an adequate foundation system for mathematics that can prove its own consistency. 
