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Arithmetical Sacks Forcing
 Archive for Mathematical Logic
"... Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1generic 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 the ..."
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Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1generic 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 ngeneric 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 1generic 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 1generic 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 1generic a < 0 ′ ′ are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmunefree (minimal) degree could be below a 1generic one. The point here is that the construction of a hyperimmunefree degree by and large directly uses forcing with perfect trees, and is a much more “pure ” form of SpectorSacks 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 hyperimmunefree. Moreover, the results of Jockusch [5], in fact prove that for n ≥ 2, if 0 < a ≤ b and b is ngeneric, then a bounds a ngeneric 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.
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Ultrafilters with property (s)
, 2003
"... A set X ⊆ 2 ω has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 ω there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter i ..."
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A set X ⊆ 2 ω has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 ω there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on ω. It is not difficult to see that if U is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even Ppoints are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA) there exists an ultrafilter U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of ω. It is a well known classical result due to Sierpinski (see [1]) that a nonprincipal ultrafilter U on ω when considered as a subset of P(ω) = 2 ω cannot have the property of Baire or be Lebesgue measurable. Here we identify 2 ω and P(ω) by identifying a subset of ω with its characteristic function. Another very weak regularity property is property (s) of Marczewski (see Miller [7]). A set of reals X ⊆ 2 ω has property (s) iff for every perfect set P there exists a subperfect set Q ⊆ P such that either Q ⊆ X or Q ∩ X = ∅. Here by perfect we mean homeomorphic to 2 ω. It is natural to ask: Question. (Steprans) Can a nonprincipal ultrafilter U have property (s)? If U is an ultrafilter in a model of set theory V and W ⊇ V is another model of set theory then we say U generates an ultrafilter in W if for every z ∈ P(ω) ∩ W there exists x ∈ U with x ⊆ z or x ∩ z = ∅. This means that the filter generated by U (i.e. closing under supersets) is an ultrafilter in W. We begin with the following result: 1 Thanks to the Fields Institute, Toronto for their support during the time these results were proved and to Juris Steprans for helpful conversations and thanks to Boise State University for support during the time this paper was written.