Abstract
A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.
Highlights
The goal of this paper is to look at the hierarchy of Turing degrees introduced by Downey and Greenberg [16, 22]
This paper can be read by two audiences: ‚ A lay logician could read it to see some of the issues involved in a sub-area of mathematical logic. ‚ A worker in computability theory can read this in more detail to glean ideas about the development of this new classification hierarchy and its recent generalizations
The area of this work is modern computability theory. This is an area of mathematics devoted to understanding what part of mathematics can be performed by a machine
Summary
Vaughan had a massive influence on World and New Zealand mathematics, both through his work and his personal commitment. Vaughan had the vision of improving mathematics in New Zealand, and hoped his Fields Medal could be leveraged to this effect. Vaughan would attend the meetings, guide them, help with the washing up, talk to the students, help with all academic things. The rest, as they say, is history, and the New Zealand mathematics community is full of excellent world class mathematicians. Vaughan had great vision for mathematics and saw great value, for example, in mathematical logic He was a great mentor and friend to Downey. One of the Theorems was part of the third author’s MSc Thesis
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