There are two key points: First, the theory of L(ℝ)ℙmax is “effectively complete” in the sense that it is invariant under set forcing. Second, the model L(ℝ)ℙmax is “maximal” (or “saturated”) in the sense that it satisfies all Π2-sentences (about the relevant structure) that can possibly hold (in the sense that they can be shown to be consistent by set forcing over the model). 