Kurt Gödel Research Center

2020 seminar talk: Definability of maximal families of reals in forcing extensions

Talk held by Jonathan Schilhan (KGRC) at the KGRC seminar on 2020-10-01.

Abstract

The definability of combinatorial families of reals, such as mad families, has a long history. The constructible universe $L$ is a good model for definability, for its nice structural properties. On the other hand, as a rule of thumb, the universe can't be too far from $L$ if it allows for low projective witnesses of such families. Thus it makes sense to look at forcing extensions of $L$.

We show that after a countable support iteration of Sacks forcing or splitting forcing (or many others) over $L$, every analytic hypergraph on a Polish space has a $\mathbf\Delta^1_2$ maximal independent set. This means that in the models obtained by these iterations, most types of interesting “maximal families” have $\mathbf\Delta^1_2$ witnesses. In particular, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a $\Pi^1_1$ mif (maximal independent family) while the independence number $\mathfrak{i}$ is bigger than $\aleph_1$.

The slides for this talk are available.

Time and Place

Talk at 3:00pm via Zoom