Definable maximal discrete sets in forcing extensions

TY - JOUR

T1 - Definable maximal discrete sets in forcing extensions

AU - Schrittesser, David

AU - Törnquist, Asger Dag

PY - 2018

Y1 - 2018

N2 - Let R be a Σ 1 1 binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ 1 2 maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π 1 1 maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ 1 2 mofs.

AB - Let R be a Σ 1 1 binary relation, and recall that a set A is R-discrete if no two elements of A are related by R. We show that in the Sacks and Miller forcing extensions of L there is a ∆ 1 2 maximal R-discrete set. We use this to answer in the negative the main question posed in [7] by showing that in the Sacks and Miller extensions there is a Π 1 1 maximal orthogonal family (“mof”) of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over L then there are no Σ 1 2 mofs.

U2 - 10.4310/MRL.2018.v25.n5.a11

DO - 10.4310/MRL.2018.v25.n5.a11

M3 - Journal article

SN - 1073-2780

VL - 25

SP - 1591

EP - 1612

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 5

ER -