Abstract: It is well known that certain natural statements of set theory, like Cantor's continuum hypothesis (CH), cannot be proved or disproved on the basis of the standard axioms (Zermelo–Fraenkel with Choice or ZFC). Some philosophers take this to be the end of the story on these questions, but set theorists continue to look for answers by investigating candidates for new axioms. One way to understand this work, an approach pioneered by Gödel, is to embrace some brand of realism (sometimes called ‘Platonism’) about sets: there is an objective world of sets in which the CH is either true or false; ZFC does not completely describe this world; our job is to find new true axioms that will give a fuller description detailed enough to decide the CH. Recent versions of realism in the philosophy of mathematics have rested on Quine's indispensability arguments: the world posited by our best scientific theories includes mathematical entities. In this book, I argue that attention to the actual details of scientific methodology substantially undermines Quine's argument, leaving realism without its best support. As an alternative, I develop a naturalistic approach (drawing on other themes from Quine, Gödel, and Wittgenstein) that finds the justification for mathematical methods in mathematics rather than extra-mathematical philosophy, and I apply this naturalism to the test case of a particular new axiom candidate (Gödel's Axiom of Constructibility).

Keywords: Axiom of Constructibility, Cantor, continuum hypothesis, Zermelo–Fraenkel, Gödel, Penelope Maddy, naturalism, philosophy of mathematics, Platonism, Quine, realism, scientific method, set theory, Wittgenstein