Kurt Gödel Research Center

2017 seminar talk: Projective sets and inner models

Talk held by Yizheng Zhu (Universität Münster, Germany) at the KGRC seminar on 2017-03-09.

Abstract

The collection of projective sets of reals is the smallest one containing all the Borel sets and closed under complements and continuous images. The Axiom of Projective Determinacy (PD) is the correct axiom that settles the regularity properties of projective sets. Inner model theory provides a systematic way of studying the projective sets under PD. In this talk, we describe some recent progress in this direction. A key theorem is the following inner-model-theoretic characterization of the canonical model associated to $\Sigma^1_3$:

Let $\mathcal{O}_{\Sigma^1_{3}}$ be the universal $\Sigma^1_3$ subset of $u_\omega$ in the sharp codes for ordinals in $u_\omega$. Let $M_{1,\infty}$ be the direct limit of iterates of $M_1$ via countable trees and let $\delta_{1,\infty}$ be the Woodin cardinal of $M_{1,\infty}$. Then $M_{1,\infty}|\delta_{1,\infty} = L_{u_\omega}[\mathcal{O}_{\Sigma^1_{3}}]$.

This theorem paves the way for further study of $\Sigma^1_3$ sets using inner model theory. It also generalizes to arbitrary $\Sigma^1_{2n+1}$ and $M_{2n-1,\infty}$.