The existence of large cardinals has dramatic consequences, even for simply-definable small sets, like the projective sets of real numbers. For example, Solovay (1970) proved, assuming that there exists a measurable cardinal, that all \(\mathbf{\Sigma}^1_2\) sets of reals are Lebesgue measurable and have the Baire property, which cannot be proved in ZFC alone. And Shelah and Woodin (1990) showed that the existence of a proper class of Woodin cardinals implies that the theory of \(L(\mathbb{R})\), even with real numbers as parameters, cannot be changed by forcing, which implies that all sets of real numbers that belong to \(L(\mathbb{R})\) are regular. Further, under a weaker large-cardinal hypothesis, namely the existence of infinitely many Woodin cardinals, Martin and Steel (1989) proved that every projective set of real numbers is determined, i.e., the axiom of PD holds, hence all projective sets are regular. Moreover, Woodin showed that the existence of infinitely many Woodin cardinals, plus a measurable cardinal above all of them, implies that every set of reals in \(L(\mathbb{R})\) is determined, i.e., the axiom \(AD^{L(\mathbb{R})}\) holds, hence all sets of real numbers that belong to \(L(\mathbb{R})\), and therefore all projective sets, are regular. He also showed that Woodin cardinals provide the optimal large cardinal assumptions by proving that the following two statements: