Definable maximal discrete sets in forcing extensions

Abstract

Let $\mathcal{R}$ be a ${\Sigma }_1^1$ binary relation, and recall that a set $A$ is $\mathcal{R}$-discrete if no two elements of $A$ are related by $\mathcal{R}$. We show that in the Sacks and Miller forcing extensions of $L$ there is a ${\Delta }_2^1$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in \cite{Fischer2010} by showing that in the Sacks and Miller extensions there is a ${\Pi }_1^1$ 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 ${\Sigma }_2^1$ mofs.

17 pages; small corrections

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