The Sacks density theorem and Σ₂-bounding

Abstract

Abstract The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P − + B Σ 2 . The proof has two components: a lemma that in any model of P − + B Σ 2 , if B is recursively enumerable and incomplete then I Σ 1 holds relative to B and an adaptation of Shore's [9] blocking technique in α -recursion theory to models of arithmetic.