This will be a one-off talk at the University of California, Davis. Slides available here.
The iterative conception of set is often thought to provide a (if not “the”) mainstream resolution of the set-theoretic paradoxes. Whilst many philosophical questions remain (e.g. about the underlying axioms and metaphysics of the iterative conception), it is taken to largely be clear what the conception states—the universe of sets is formed by forming all possible subsets at additional stages and continuing this into the transfinite. In this talk, I’ll present some work suggesting that there are different kinds of iterative conception. The “default” version just mentioned is the “strong” version, but there is a “weaker” version too—we might not insist that we get all possible subsets of a given set at additional stages. Rather, we could instead have operations of set formation that are more fine-grained, and allow us to slowly “feed in” more and more subsets. Taking this approach, I argue, allows us more flexibility with the foundational theory we adopt, and reveals some interesting possibilities for the advancement of the philosophy of mathematics. Towards the end of the talk, I’ll mention some relationships with other areas of philosophy, including negative epistemology, conceptual engineering, and relationships with pluralism and perspectivism.