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

Ivan Ongay-Valverde

Published 2021-06-23Version 1

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.