Double Recursion Theorems

12 Chapter 0. PrerequisitesTheorem 5. 13 If S is an axiomatizable u>-consistent system inwhich all true % 0 -sentences are provable, then S is incomplete.§6. Rosser Systems. We call S a Rosser system for sets iffor any r.e. sets A and 5, the set A-B is strongly separable fromB-A in S. More generally, for any positive integer n, we say thatS is a Rosser system for n-ary relations if for any r.e. relationsRI(XI, ... ,x n ) and ^(^i) • • • ? x n)) the relation R\-R-z (that is tosay RI A ~ #2) is strongly separable from R^-Ri- We call S aRosser system if it is a Rosser system for sets and for relations ofany number of arguments.If S is a Rosser system for sets, then obviously for any disjointr.e. sets A and B, the set A is strongly separable from B in S (sincethen A - B = A and B - A = B). The converse also happens tohold (as we will later see).Suppose now that S is a simply consistent axiomatizable Rossersystem for sets. Then the sets P* and R* are both r.e., and byconsistency, they are disjoint. R* is then strongly separable from P*in «S, and by Theorem 3.1, S is then incomplete. And so we have:Theorem 6. If S is a simply consistent axiomatizable Rosser systemfor sets, then S is incomplete.§7. The Systems P.A., (Q) and (R). We call a systemSi a subsystem of $3, and we say that £3 is an extension of «Si ifall the provable formulas of