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# Mathematical Research Letters

## Volume 25 (2018)

### Number 5

### Definable maximal discrete sets in forcing extensions

Pages: 1591 – 1612

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n5.a11

#### Authors

#### 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 [7] 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.

The authors gratefully acknowledge the generous support from Sapere Aude grant no. 10-082689/FNU from Denmark’s Natural Sciences Research Council, and the first author gratefully acknowledges generous support from the DNRF Niels Bohr Professorship of Lars Hesselholt.

Received 26 October 2015

Accepted 19 April 2018

Published 1 February 2019