The Sacks Density Theorem and S_2-Bounding

Abstract

The Sacks Density Theorem (Sacks 1964) states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P \Gamma +B \Sigma 2 . The proof has two components: a lemma that in any model of P \Gamma +B \Sigma 2 , if B is recursively enumerable and incomplete then I \Sigma 1 holds relative to B and an adaptation of Shore's (1976) blocking technique in ff-recursion theory to models of arithmetic. 1 Introduction Proofs using the priority method are the trademark of recursion theory. In this paper, we continue the line of inquiry in which we use subsystems of first order arithmetic to calibrate priority methods and the theorems in whose proofs they appear. In the hierarchy of Groszek and Slaman (unpublished), we classify priority constructions according to the syntactic complexity of the outcomes in its most complicated families of strategies. In a \Pi 1 -priority construction the strategies have outcomes that are descri..