Three Special Topics

Abstract

Abstract The topics of this chapter are of more specialized interest and are not necessary for the results of our final chapter. They will probably be of more interest to the specialist (particularly the results of Section III) than to the general reader. We know from Chapter 2 that if all recursive sets are representable in S, then S is undecidable. We also know from Chapter 4 that if all recursively enumerable sets are representable in S, then S is not only undecidable but generative. We might also ask the question, If all recursive sets are representable in S, is S necessarily generative? Shoenfield [1961] answered this question negatively. He constructed an axiomatizable system in which all recursive sets are representable, and so the system is undecidable, but he showed that the system is not creative. Let us say that all recursive sets are uniformly representable in S if there is a recursive function g(x) such that for any number i, if ωi is a recursive set, then g(i) is the Godel number of a formula which represents ωi in S. We will show that if all recursive sets are uniformly representable in S, then S is generative. We also showed in Chapter 2 that if all recursive sets are definable in S and S is consistent, then the pair (P,R) of its nuclei is recursively inseparable. Under the same hypothesis, is the pair (P,R) necessarily effectively inseparable? The answer is no. In Shoenfield’s system all recursive sets are not only representable, but definable. However, the set P is not creative. Hence the pair (P, R) of nuclei of the system, though recursively inseparable, is not effectively inseparable.