Definability in the Turing Degrees
نویسندگان
چکیده
Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with ≤T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The conditions in the forcing partial specify finite initial segments of the generic reals and impose a infinite constraint on further extensions. In section 3, this result is applied to show that any elementary function from D to D is an automorphism. Other applications are given toward the rigidity question for D. By observing that a single jump is all that is needed to meet the relevant dense sets, it is also shown that the recursively enumerable degrees can be defined from finitely many parameters in the structure consisting of the degrees below 0′ with ≤T .
منابع مشابه
Discontinuous Phenomena and Turing Definability
We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined.
متن کاملTuring Definability in the Ershov Hierarchy
We obtain the first nontrivial d.c.e. Turing approximation to the class of computably enumerable (c.e.) degrees. This depends on the following extension of the splitting theorem for the d.c.e. degrees: For any d.c.e. degree a, any c.e. degree b, if b < a, then there are d.c.e. degrees x0,x1 such that b < x0,x1 < a and a = x0 ∨ x1. The construction is unusual in that it is incompatible with uppe...
متن کاملGlobal Properties of the Turing Degrees and the Turing Jump
We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D.
متن کاملDegree Structures
The Turing degrees 3d were introduced by Kleene and Post ([9], 1954) to isolate and study those properties of the subsets of the natural numbers N which are expressed purely in terms of relative computability. Intuitively, we form 3d by identifying any pair of subsets of N which are mutually computable and ordering the resulting equivalence classes by relative computability. The natural hierarc...
متن کاملDefinability, automorphisms, and dynamic properties of computably enumerable sets
We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of FelixKlein’sErlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and theMartin Invariance Conjecture on their Tur...
متن کاملNatural 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 issues of getting definitions of interesting, apparently external, relations on degrees that are o...
متن کامل