A π1 1-uniformization principle for reals

Abstract

We introduce a Π1 1-uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω 1)L= ω1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π1 1-maximal chains and thin maximal antichains in the Turing degrees. We also use the Π1 1-uniformization principle to study Martin's conjectures on cones of Turing degrees, and show that under V = L the conjectures fail for uniformly degree invariant Π1 1 functions. © 2009 American Mathematical Society.