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

David Schrittesser (Department of Mathematical Sciences, University of Copenhagen, Denmark; and Kurt Gödel Research Center, University of Vienna, Austria)

Asger Törnquist (Department of Mathematical Sciences, University of Copenhagen, Denmark)

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

Published 1 February 2019