The Existential Theory of the Uppersemilattice of Turing Degrees with Least Element and Jump is (original) (raw)
Ndjfl, 2008
Let HF be the collection of the hereditarily finite well-founded sets and let the primitive language of set theory be the first-order language which contains binary symbols for equality and membership only. As announced in a previous paper by the authors, "Truth in V for ∃ * ∀∀-sentences is decidable," truth in HF for ∃ * ∀∀-sentences of the primitive language is decidable. The paper provides the proof of that claim.
On relative enumerability of Turing degrees
Archive for Mathematical Logic, 2000
Let d be a Turing degree, R[d] and Q[d] denote respectively classes of recursively enumerable (r.e.) and all degrees in which d is relatively enumerable. We proved in Ishmukhametov [1999] that there is a degree d containing differences of r.e.sets (briefly, d.r.e.degree) such that R[d] possess a least element m>0. Now we show the existence of a d.r.e. d such that R[ d] has no a least element. We prove also that for any REA-degree d below 0 the class Q[d] cannot have a least element and more generally is not bounded below by a non-zero degree, except in the trivial cases.
Extensions of embeddings below computably enumerable degrees
Transactions of the American Mathematical Society, 2012
Toward establishing the decidability of the two-quantifier theory of the Δ 2 0 \Delta ^0_2 Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular, the degree of the halting set 0 ′ \boldsymbol {0}’ . We obtain a good deal of sufficient and necessary conditions.
Chapter II: Embeddings and Extensions of Embeddings in the Degrees
1983
We define the degrees of unsolvability in this chapter, and show that these degrees from an uppersemilattice. Much of the rest of this book will be devoted to studying this upper semilattice. The study begins in this chapter, with sections on embedding theorems and on extensions of embeddings into the degrees. We also prove the decidability of a certain natural class of sentences about the degrees.
Low upper bounds in the Turing degrees revisited
Journal of Logic and Computation, 2010
We give an alternative proof of a result of Kučera and Slaman [KS09] on low bounds of ideals in the ∆ 0 2 Turing degrees. This is a characterization of the ideals in the ∆ 0 2 degrees which have a low upper bound. It follows that there is a low upper bound for the ideal of the K-trivial degrees. Our proof is direct, in the sense that it does not use universal classes of PA degrees.
The Journal of Symbolic Logic, 1996
We give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e. wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint. We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to the pm degrees require no new complexity-theoretic results. The fact that the pm-degrees of the recursive sets have a decidable two-quantifier theo...
Decidability and Definability in the 02Enumeration Degrees
Enumeration reducibility was introduced by Friedberg and Rogers in 1959 as a positive reducibility between sets. The enumeration degrees provide a wider context in which to view the Turing degrees by allowing us to use any set as an oracle instead of just total functions. However, in spite of the fact that there are several applications of enumeration reducibility in computable mathematics, until recently relatively little research had been done in this area. In Chapter 2 of my thesis, I show that the ∀∃∀-fragment of the first order theory of the Σ 0 2-enumeration degrees is undecidable. I then show how this result actually demonstrates that the ∀∃∀-theory of any substructure of the enumeration degrees which contains the ∆ 0 2-degrees is undecidable. In Chapter 3, I present current research that Andrea Sorbi and I are engaged in, in regards to classifying properties of non-splitting Σ 0 2-degrees. In particular I give proofs that there is a properly Σ 0 2-enumeration degree and that every ∆ 0 2-enumeration degree bounds a non-splitting ∆ 0 2-degree. Advisor: Prof. Steffen Lempp I am grateful to Steffen Lempp, my thesis advisor, for all the time, effort, and patience that he put in on my behalf. His insight and suggestions have been of great worth to me, both in and out of my research. I am especially grateful for his help in getting me back in school after my two-year leave of absence and for offering me a research assistantship so I could study for a year with him in Germany. I am also grateful to Andrea Sorbi for funding a visit to Siena, Italy that allowed me to do research with him, and for the friendship that has grown from our research contact. Hopefully we will be able to go running together in the mountains again. I would like to thank Todd Hammond for introducing me to mathematical logic, to Mirna Dzamonja for getting me excited about Computability Theory, and to Jerome Keisler, Ken Kunen, Arnie Miller, and Patrick Speissegger for teaching interesting logic classes. I would like to thank all of the wonderful teachers over the years who have encouraged my interest in mathematics, especially Patty Av3ery and Slade Skipper. Thanks also go to Eric Bach, Joel Robbin, and Mary Ellen Rudin for help they have given and for serving on my defense committee. I am very appreciative for my parents and sister, for the support and love they have given me over the past 31 years. The most appreciation, however, goes to my wonderful wife, Joy, for always being there for me. I could not have made it without her encouragement and unconditional love.