Arithmetical sacks forcing

10.1007/s00153-006-0004-4Archive for Mathematical Logic456715-72

Arithmetical Sacks Forcing

Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions resulting in various forms of Cohen genericity, and forcing with perfect trees, resulting in various forms of minimality. Whilst these constructions are clearly incompatible, this paper was motivated by the general question of “How can minimality and (Cohen) genericity interact?”. Jockusch [5] showed that for n ≥ 2, no n-generic degree can bound a minimal degree, and Haught [4] extended earlier work of Chong and Jockusch to show that that every nonzero Turing degree below a 1-generic degree below 0 ′ was itself 1generic. Thus, it seemed that these forcing notions were so incompatible that perhaps no minimal degree could even be comparable with a 1-generic one. However, this conjecture was shown to fail independently by Chong and Downey [1] and by Kumabe [7]. In each of those papers, a minimal degree below m < 0 ′ and a 1-generic a < 0 ′ ′ are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmune-free (minimal) degree could be below a 1-generic one. The point here is that the construction of a hyperimmune-free degree by and large directly uses forcing with perfect trees, and is a much more “pure ” form of Spector-Sacks forcing [10] and [9]. This means that it is not usually possible to use tricks such as full approximation or forcing with partial computable trees, which are available to us when we only wish to construct (for instance) minimal degrees. For instance, minimal degrees can be below computably enumerable ones, whereas no degree below 0 ′ can be hyperimmune-free. Moreover, the results of Jockusch [5], in fact prove that for n ≥ 2, if 0 < a ≤ b and b is n-generic, then a bounds a n-generic degrees and, in particular, certainly is not hyperimmune free. This contrasts quite strongly with the main result below. In this paper we will answer Jockusch’s question, proving the following result