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?

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*Assuming your formal system is consistent, Gödel shows there is a statement in that system whose interpretation is true but that is unprovable in the system.  The statement is actually provable, but not in that system: you need the additional assumption that the system in consistent, and that is not provable in the system (unless the system happens to be inconsistent!).

*There's no "arbitrarily judgement" here. If there were a proof of "a", you could use that to produce a proof of 0=1. Thus if the system is consistent, "a" is not provable in the system.
