On June 01, 2019 I was invited to give a talk at the Logic Fest in the Windy City in Chicago, USA.
The interplay between inner model theory and descriptive set theory in a nutshell
Abstract: The study of inner models was initiated by Gödel’s analysis of the constructible universe $L$. Later, it became necessary to study canonical inner models with large cardinals, e.g. measurable cardinals, strong cardinals or Woodin cardinals, which were introduced by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin’s proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel’s proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later Neeman improved the result in the projective hierarchy by showing that in fact the existence of a countable iterable model, a mouse, with Woodin cardinals and a top measure suffices to prove determinacy in the projective hierarchy.
This opened up the possibility for an optimal result stating the equivalence between local determinacy hypotheses and the existence of mice in the projective hierarchy, just like the equivalence of analytic determinacy and the existence of $x^\sharp$ for every real $x$ which was shown by Martin and Harrington in the 70’s. The existence of mice with Woodin cardinals and a top measure from levels of projective determinacy was shown by Woodin in the 90’s. Together with his earlier and Neeman’s results this estabilishes a tight connection between descriptive set theory in the projective hierarchy and inner model theory.
In this talk, we will outline the main concepts and results connecting determinacy hypotheses with the existence of mice with large cardinals. Neeman’s methods mentioned above extend to show determinacy of projective games of arbitrary countable length from the existence of inner models with many Woodin cardinals. We will discuss a number of more recent results, some of which are joint work with Juan Aguilera, showing that inner models with many Woodin cardinals can be obtained from the determinacy of countable projective games.