Getting around the Halting Problem
Abstract
There are numbers k and s and a URM program A(n,m) satisfying the following conditions. 1. If A(n,m) halts, then Cn(m) diverges. 2. For all n, Ck(n) = A(n,n) and Cs(n) = Ck(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here Cn() is a program with index n in some exhaustive enumeration of all possible programs. This has implications for solving the liar paradox and for generalization of G\"odel incompleteness theorem to formal systems other than PA.
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