Sets of real numbers closed under Turing equivalence: Applications to fields, orders and automorphisms

TL;DR

This paper investigates the structure of sets of real numbers closed under Turing equivalence, exploring their algebraic, measure-theoretic, and order properties, and examines implications for automorphisms of Turing degrees and their interaction with 1-generic degrees.

Contribution

It provides new insights into the algebraic and order-theoretic structure of Turing-closed sets of reals and links these structures to automorphisms of Turing degrees.

Findings

Characterization of sets of reals closed under Turing equivalence in terms of algebra, measure, and order

Restrictions on how automorphisms of Turing degrees can interact with 1-generic degrees

New connections between Turing degree automorphisms and classical constructions in degree theory

Abstract

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and order. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with $1$-generic degrees.