Comment by ozankabak

So if we have a theory expressive enough to make statements about ordinary (Peano) arithmetic, we can always form a self-referential statement within the framework of this theory which we can not prove or disprove. So far, so good. Here is my question: What happens if we restrict/weaken the theory to preclude self-referential statements? Obviously, we will lose our ability to express certain arithmetic statements which correspond to self-referential statements in the original theory. But what else? Is that the only class of statements we lose? Also, are there any other kinds of statements that still make the theory incomplete?