Large Turing independent sets

Abstract

For a set of reals X X and 1 ≤ n > ω 1 \leq n > \omega , define X X to be n n -Turing independent iff the Turing join of any n n reals in X X does not compute another real in X X . X X is Turing independent iff it is n n -Turing independent for every n n . We show the following: (1) There is a non-meager Turing independent set. (2) The statement “Every set of reals of size continuum has a Turing independent subset of size continuum” is independent of ZFC plus the negation of CH. (3) The statement “Every non-meager set of reals has a non-meager n n -Turing independent subset” holds in ZFC for n = 1 n = 1 and is independent of ZFC for n ≥ 2 n \geq 2 (assuming the consistency of a measurable cardinal). We also show the measure analogue of (3).