The theory of projective sets of complexity greater than co-analytic is completely undetermined by ZFC. For example, in \(L\) there is a \(\mathbf{\Sigma}^1_2\) set that is not Lebesgue measurable and does not have the Baire property, whereas if Martin’s axiom holds (see Section 11), every such set has those regularity properties. There is, however, an axiom, called the axiom of Projective Determinacy, or PD, that is consistent with ZFC, modulo the consistency of some large cardinals (in fact, it follows from the existence of some large cardinals), and implies that all projective sets are regular. Moreover, PD settles essentially all questions about the projective sets. See the entry on  large cardinals and determinacy  for further details.