Kurt Gödel Research Center

2015 seminar talk: Maximal discrete sets in arboreal forcing extensions

Talk held by David Schrittesser (University of Copenhagen, Denmark) at the KGRC seminar on 2015-06-18.

Abstract

Given a graph (or just a binary relation) $G$ on a set $X$, we call $D \subseteq X$ $G$-discrete iff any two of its elements are $G$-unrelated. A maximal discrete set is one which has no proper discrete superset. If $X$ is an effectively presented Polish space and $G$ is say Borel, we can measure how hard it is to construct a maximal $G$-discrete set is in terms of the (effective) projective hierarchy.

A good example is the space of Borel probability measures together with the relation of being orthogonal. It is known that there can be no analytic maximal orthogonal family of measures (m.o.f), but in the constructible universe, there is an (effectively) $\Pi^1_1$ m.o.f. It is also known that there can be no such m.o.f in the Cohen or Random extensions.

In recent joint work with Asger Törnquist, we investigate the complexity of maximal discrete sets and in particular, of m.o.fs in forcing extensions by Sacks, Miller and Mathias forcing. We show that not only in the constructible universe is it possible to have a $\Pi^1_1$ m.o.f.