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Parameter Definability in the Recursively Enumerable Degrees
"... The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly defin ..."
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The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...
Definability and Global Degree Theory
 Logic Colloquium '90, Association of Symbolic Logic Summer Meeting in Helsinki, Berlin 1993 [Lecture Notes in Logic 2
"... Gödel's work [Gö34] on undecidable theories and the subsequent formalisations of the notion of a recursive function ([Tu36], [K136] etc.) have led to an ever deepening understanding of the nature of the noncomputable universe (which as Gödel himself showed, includes sets and functions of every ..."
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Gödel's work [Gö34] on undecidable theories and the subsequent formalisations of the notion of a recursive function ([Tu36], [K136] etc.) have led to an ever deepening understanding of the nature of the noncomputable universe (which as Gödel himself showed, includes sets and functions of everyday significance). The nontrivial aspect of Church's Thesis (any function not contained within one of the equivalent definitions of recursive/Turing computable, cannot be considered to be effectively computable) still provides a basis not only for classical and generalised recursion theory, but also for contemporary theoretical computer science. Recent years, in parallel with the massive increase in interest in the computable universe and the development of much subtler concepts of 'practically computable', have seen remarkable progress with some of the most basic and challenging questions concerning the noncomputable universe, results both of philosophical significance and of potentially wider technical importance. Relativising Church's Thesis, Kleene and Post [KP54] proposed the now
POSSIBLEWORLDS SEMANTICS FOR MODAL NOTIONS CONCEIVED AS PREDICATES
"... Abstract. If � is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where � is conceived as a predicate, i.e., an expression that yields a formula if it is applied to ..."
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Abstract. If � is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where � is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possibleworlds semantics for predicates of sentences has not been developed very far. By introducing possibleworlds semantics for the language of arithmetic plus the unary predicate �, we tackle both problems. Given a frame 〈W, R 〉 consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret � at every world in such a way that ��A � holds at a world w ∈ W if and only if A holds at every world v ∈ W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several ‘paradoxes ’ (like Montague’s Theorem, Gödel’s Second Incompleteness Theorem, McGee’s Theorem on the ωinconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not
Contemporary Mathematics Natural Definability in Degree Structures
"... A major focus of research in computability theory in recent years has involved definability issues in degree structures. There has been much success in getting general results by coding methods that translate first or second order arithmetic into the structures. In this paper we concentrate on the i ..."
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A major focus of research in computability theory in recent years has involved definability issues in degree structures. There has been much success in getting general results by coding methods that translate first or second order arithmetic into the structures. In this paper we concentrate on the issues of getting definitions of interesting, apparently external, relations on degrees that are ordertheoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal definition of natural but we offer some guidelines, examples and suggestions for further research.
Annals of Mathematics FirstOrder Theory of the Degrees of Recursive Unsolvability
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of