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, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here C_n(_) 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.

翻译：1. 如果A(n,m)停止,那么Cn(m)则不同。 2. 对于所有 n, C_k(n) = A(n,n) 和 C_s(n) = C_k(s) 。 3. A(k) 停止,而对于所有 n, A(s) 则不同。这里,C_n(_) 是一个方案,在所有可能的方案中都列出了索引n。这对解决谎言悖论和将G\"odel不完全的理论推广到除PA之外的正式系统有影响。