  Here is an overview of the present entry: Section 1 provides further philosophical motivation and sharpens the question of pluralism by briefly discussing the interpretability hierarchy, the incompleteness phenomenon, and the nature of axioms.  Section 2  describes the central notions and results of classical descriptive set theory along with the independence results concerning the questions that remained open and were ultimately shown to be unresolvable on the basis of the standard axioms, ZFC. Section 3 describes the two main approaches to new axioms—axioms of definable determinacy and large cardinal axioms—and discusses the implications of these axioms for the undecided questions of classical descriptive set theory. Section 4 treats of the intimate relationship between the two approaches and outlines the case that has been made for axioms of definable determinacy and large cardinal axioms. Section 5 revisits the debate between the pluralist and the non-pluralist in light of the foregoing mathematical developments. 