## Logic Seminar

Soon after forcing was developed, Solovay used it to construct a model of Zermelo-Frankel (ZF) set theory in which all sets of reals are Lebesgue measurable, all sets of reals have the Property of Baire, and all sets of reals have the perfect set property. While his construction seemed *ad hoc* at the time, we now know that Solovay's model plays an important foundational role in set theory and explains why independence results rarely arise outside of set theory and when they should be expected. The \(L(\mathrm{R})\) Absoluteness Theorem was proved by Shelah and Woodin and asserts that if there is a supercompact cardinal, the theory of Solovay's model can not be changed by the method of forcing. This talk will give an overview of this theorem and how it is proved. This result is the starting point for Woodin's \(\mathbb{P}_{\mathrm{max}}\)-extension, which will be developed over the course of the semester.