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。