Kurt Gödel Research Center

2016 seminar talk: Maximal discrete sets with large continuum

Talk held by David Schrittesser (University of Copenhagen, Denmark) at the KGRC seminar on 2016-01-07.

Abstract

In a previous talk at the KGRC, I showed how to construct definable maximal discrete sets in forcing extensions of $L$, in particular in the Sacks and Miller extension. In particular, the existence of such sets is consistent with $V \neq L$.

In this talk I shall show the stronger result that the existence of definable discrete sets is consistent with large continuum. In the process, I show an interesting generalization of Galvin's theorem. In particular, this applies to the example of maximal orthogonal families of measures (mofs).

One might hope for a simpler way of constructing a mof in a model with large continuum: to find an indestructible such family in $L$. While such an approach is possible e.g. for maximal cofinitary groups, this is impossible for mofs.