A Π¹₁-uniformization principle for reals

We introduce a Π 1 1 \Pi ^1_1 -uniformization principle and establish its equivalence with the set-theoretic hypothesis ( ω 1 ) L = ω 1 (\omega _1)^L=\omega _1 . This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π 1 1 \Pi ^1_1 -maximal chains and thin maximal antichains in the Turing degrees. We also use the Π 1 1 \Pi ^1_1 -uniformization principle to study Martin’s conjectures on cones of Turing degrees, and show that under V = L V=L the conjectures fail for uniformly degree invariant Π 1 1 \Pi ^1_1 functions.