  The above results show that axioms of definable determinacy extend the results on regularity properties that can be proved in ZFC. ZFC can establish that all Σ̰11 sets of reals have the regularity properties and Δ̰11-determinacy (a theorem of ZFC) lies at the heart of this result. The extension of Δ̰11-determinacy to stronger forms of definable determinacy (PD, ADL(ℝ), etc.) generalizes these results by extending the regularity properties to the corresponding pointclasses (projective, in L(ℝ), etc.) 