Definable sets and forcing (Z. Norwood, UCLA)

It is a central thesis of modern set theory that, under strong hypotheses called 'large cardinal' assumptions, definable sets of real numbers are well behaved; for example, they should be Lebesgue-measurable. The first result of this kind was proved in 1970 by Solovay, who used forcing and an inaccessible cardinal to construct a universe of set theory, called the Solovay model, in which all definable sets are Lebesgue-measurable. Later in the 1970s, Mathias extended Solovay's work and asked whether there are infinite mad families in the Solovay model. In 2015 T\"ornquist answered this question, showing that there are no infinite mad families in the Solovay model. (A mad family is a maximal family of subsets of the integers any two members of which have finite intersection.)

After an introduction to forcing and large cardinals, we will discuss a new proof of T\"ornquist's theorem and, time permitting, some recent related work on generic absoluteness. This is joint work with Itay Neeman.

No background in forcing or large cardinals will be assumed.