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.