A Bounded Jump for the Bounded Turing Degrees

A Bounded Jump for the Bounded Turing Degrees

We define the bounded jump of A by A = {x ∈ ω | ∃i ≤ x[φi(x) ↓ ∧ Φ φi(x) x (x)↓]} and let A denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT ) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, fo...

Turing Degrees and Muchnik Degrees of Recursively Bounded DNR Functions

On the Turing Degrees of Divergence Bounded Computable Reals

The d-c.e. (difference of c.e.) and dbc (divergence bounded computable) reals are two important subclasses of ∆2-reals which have very interesting computability-theoretical as well as very nice analytical properties. Recently, Downey, Wu and Zheng [2] have shown by a double witness technique that not every ∆2-Turing degree contains a d-c.e. real. In this paper we show that the classes of Turing...

Global 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.

Jump Embeddings in the Turing Degrees

Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if (P, ≤P ) is a partial ordering of cardinality at most א1 which is locally countable—each point has at most coun...