Definable maximal discrete sets in forcing extensions

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Let  be a Σ11 binary relation, and recall that a set A is -discrete if no two elements of A are related by . We show that in the Sacks and Miller forcing extensions of L there is a Δ12 maximal -discrete set. We use this to answer in the negative the main question posed in [5] by showing that in the Sacks and Miller extensions there is a Π11 maximal orthogonal family ("mof") of Borel probability measures on Cantor space. A similar result is also obtained for Π11 mad families. By contrast, we show that if there is a Mathias real over L then there are no Σ12 mofs.
TidsskriftMathematical Research Letters
Udgave nummer5
Sider (fra-til)1591-1612
StatusUdgivet - 2018


ID: 184033460