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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.
Almost Weakly 2Generic Sets
 in Proceedings of the Sixth Annual IEEE Structure in Complexity Theory Conference
, 1996
"... There is a family of questions in relativized complexity theoryweak analogs of the Friedberg JumpInversion Theoremthat are resolved by 1generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2generic sets, i.e., sets which meet ever ..."
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There is a family of questions in relativized complexity theoryweak analogs of the Friedberg JumpInversion Theoremthat are resolved by 1generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2generic sets, i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2generic sets are very close to 1generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2generic set G such that NP G " coNP G 6` P G\PhiX . (On the other hand, if G is 1generic, then NP G " coNP G ` P G\PhiSAT , where SAT is the NP complete Satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2generic sets differ radically from those of 1generic sets: every degree above 0 0 contains an aw2generic set; no aw2generic set exists below any incomplete r.e. set; there is an aw2generic set which is the join of two Turing equivalent aw2generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0 0 is the jump of an aw2generic degree. 1
BEYOND THE ARITHMETIC
, 2005
"... Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees ..."
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Various results in different areas of Computability Theory are proved. First we work with the Turing degree structure, proving some embeddablity and decidability results. To cite a few: we show that every countable upper semilattice containing a jump operation can be embedded into the Turing degrees, of course, preserving jump and join; we show that every finite partial ordering labeled with the classes in the generalized high/low hierarchy can be embedded into the Turing degrees; we show that every generalized high degree has the complementation property; and we show that if a Turing degree a is either 1generic and ∆ 0 1, 2generic and arithmetic, nREA, or arithmetically generic, then the theory of the partial ordering of the Turing degrees below a is recursively isomorphic to true first order arithmetic. Second, we work with equimorphism types of linear orderings from the viewpoints of Computable Mathematics and Reverse Mathematics. (Two linear orderings are equimorphic if they can be embedded in each other.) Spector proved in 1955 that every hyperarithmetic ordinal is isomorphic to a recursive one. We extend his result and prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. From the viewpoint of Reverse Mathematics, we look at the strength of Fraïssé’s conjecture. From our results, we deduce that Fraïssé’s conjecture is sufficient and necessary to develop a reasonable theory of equimorphism types of linear orderings. Other topics we include in this thesis are the following: we look at structures for which Arithmetic Transfinite Recursion is the natural system to study them; we study theories of hyperarithmetic analysis and present a new natural example; we look at the complexity of the elementary equivalence problem for Boolean algebras; and we prove that there is a minimal pair of Kolmogorovdegrees. BIOGRAPHICAL SKETCH I was born and raised in Montevideo, Uruguay. I started to take mathematics
The Turing degrees below generics and randoms
, 2012
"... If x0 and x1 are both generic, the theories of the degrees below x0 and x1 are the same. The same is true if both are random. We show that the ngenericity or nrandomness of x do not su ¢ ce to guarantee that the degrees below x have these common theories. We also show that these two theories (for ..."
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If x0 and x1 are both generic, the theories of the degrees below x0 and x1 are the same. The same is true if both are random. We show that the ngenericity or nrandomness of x do not su ¢ ce to guarantee that the degrees below x have these common theories. We also show that these two theories (for generics and randoms) are di¤erent. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis. 1