Getting around the Halting Problem

Let {\Theta}(p,n) be a program that either loops or returns only true or false. The parameter p is the index of a program P and n is its input. Suppose that {\Theta}(p,n) = true if and only if P halts on n. It follows that if {\Theta}(P,n) = false then P does not halt on n. Let furthermore {\Theta}^(n) = {\Theta}(n,n) and {\Theta}*() = {\Theta}^({\theta}*). The claim is that in the class {\Theta} there exists a program H such that H(h,h*) = false, that is, H proves that it does not prove that H* halts. This has implications for solving the liar paradox and for generalization of G\"odel incompleteness theorem to formal systems other than PA.