November 19
Corey Switzer,
University of Vienna
Definable Well Orders and Other Beautiful Pathologies
Many sets of reals - well orders of the reals, MAD families, ultrafilters on omega etc - only necessarily exist under the axiom of choice. As such, it has been a perennial topic in descriptive set theory to try to understand when, if ever, such sets can be of low definitional complexity. Large cardinals rule out such the existence of projective well orders, MAD families etc while it's known that if $V=L$ (or even just 'every real is constructible') then there is a $\Delta^1_2$ well order of the reals and $\Pi^1_1$ witnesses to many other extremal sets of reals such as MAD families and ultrafilter bases. Recently a lot of work on the border of combinatorial and descriptive set theory has focused on considering what happens to the definitional complexity of such sets in models in which the reals have a richer structure - for instance when $\mathsf{CH}$ fails and various inequalities between cardinal characteristics is achieved. In this talk I will present a recent advance in this area by exhibiting a model where the continuum is $\aleph_2$, there is a $\Delta^1_3$ well order of the reals, and a $\Pi^1_1$ MAD family, a $\Pi^1_1$ ultrafilter base for a P-point, and a $\Pi^1_1$ maximal independent family, all of size $\aleph_1$. These complexities are best possible for both the type of object and the cardinality hence this may be seen as a maximal model of 'minimal complexity witnesses'. This is joint work with Jeffrey Bergfalk and Vera Fischer.