  It is sensible to approach the search for new axioms and the question of pluralism in a stepwise fashion, seeking first axioms that resolve certain low-level questions and making the case for non-pluralism at that level, and then proceeding upward to questions of greater complexity and the question of pluralism at higher levels. The present entry focuses on this debate as it takes place in the setting of classical descriptive set theory. There are three reasons for this choice. First, a popular view embraces non-pluralism for first-order arithmetic but rejects the search for new axioms for full second-order arithmetic and set theory and embraces pluralism at these   levels.[1]   The questions of descriptive set theory are important in this connection since many of them are statements of second-order arithmetic which are independent of the standard axioms, ZFC. It thus represents a critical juncture in the debate between the pluralist and non-pluralist. Second, here the mathematical landscape with regard to the implications of new axioms has largely stabilized and we now have the resources required to address the issue between the pluralist and the non-pluralist. (In the case of questions of third- and higher-order arithmetic the matter still awaits further mathematical developments. That topic is treated in the entry  “The Continuum Hypothesis”.)   Finally, the case for new axioms at this level involves so-called “extrinsic justifications” (something we shall describe below) and, moreover, provides the strongest current example of such justifications. 