In the presence of a supercompact cardinal, the theory of Solovay's model \(L(\mathbf{R})\) cannot be changed by forcing. Given that \(L(\mathbf{R})\) fails to satisfy the Axiom of Choice, it is natural to wonder if this inner model can be enlarged in a controlled way to obtain a model of ZFC. Woodin's \(\mathbb{P}_{\mathrm{max}}\)-extension of \(L(\mathbf{R})\) is such an enlargement. It has a theory which, in many ways, mirrors the theory of Martin's Maximum, the forcing axiom for stationary set preserving forcings. This talk will give an introduction to the \(\mathbb{P}_{\mathrm{max}}\)-extension and its properties. This topic will be further developed in the Wednesday lectures in this semester (fall 2019).