Separating families and order dimension of Turing degrees

Abstract

We study families of functions and linear orders which separate countable subsets of the continuum from points. As an application, we show that the order dimension of the Turing degrees, denoted dimT, cannot be decided in ZFC. We also provide a combinatorial description of dimT and show that the Turing degrees have the largest order dimension among all locally countable partial orders of size continuum. Finally, we prove that it is consistent that the number of linear orders needed to separate countable subsets of the continuum from points is strictly smaller than the number of functions necessary for separating them.