Definable maximal discrete sets in forcing extensions

TL;DR

This paper demonstrates the existence of definable maximal discrete sets in certain forcing extensions and explores their implications for maximal orthogonal families of measures, revealing differences based on the type of forcing and reals added.

Contribution

It establishes the existence of $oldsymbol{ riangle^1_2}$ maximal $oldsymbol{ e}$-discrete sets in Sacks and Miller extensions and shows the non-existence of $oldsymbol{oldsymbol{ e}}^2$ mofs if a Mathias real exists over $L$.

Findings

Existence of $oldsymbol{ riangle^1_2}$ maximal $oldsymbol{ e}$-discrete sets in Sacks and Miller extensions.

Existence of a $oldsymbol{oldsymbol{ e}}^1$ maximal orthogonal family of Borel probability measures in these extensions.

No $oldsymbol{ e}^2$ mofs if a Mathias real exists over $L$.

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 $\Delta^1_2$ 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 $\Sigma^1_2$ mofs.