One can prove in ZFC—and the use of the AC is necessary—that there are non-determined sets. Thus, the Axiom of Determinacy (AD), which asserts that all subsets of \(\mathcal{N}\) are determined, is incompatible with the AC. But Donald Martin proved, in ZFC, that every Borel set is determined. Further, he showed that if there exists a large cardinal called measurable (see  Section 10), then even the analytic sets are determined. The axiom of Projective Determinacy (PD) asserts that every projective set is determined. It turns out that PD implies that all projective sets of reals are regular, and Woodin has shown that, in a certain sense, PD settles essentially all questions about the projective sets. Moreover, PD seems to be necessary for this. Another axiom, \(AD^{L(\Bbb R )}\), asserts that the AD holds in \(L(\Bbb R)\), which is the least transitive class that contains all the ordinals and all the real numbers, and satisfies the ZF axioms (see  Section 7). So, \(AD^{L(\Bbb R )}\) implies that every set of reals that belongs to \(L(\Bbb R)\) is regular. Also, since \(L(\Bbb R)\) contains all projective sets, \(AD^{L(\Bbb R )}\) implies PD.