Talk: Work in progress on the iterative conceptions of set

This will be a talk in the Oslo Logic Seminar. Handout here.


In this talk, I will present some early-stage work in progress on the iterative conceptions of set. In a recent booklet (itself entitled Iterative Conceptions of Set) I’ve argued that we can think more generally about the iterative conception as a way of generating sets, without committing to the Powerset operation. The booklet is rather “programmatic”, however, and leaves a lot of mathematical questions open. In this talk I’ll try to accomplish the following things:

(1.) I’ll present the rough idea I gave in the booklet.

(2.) I’ll outline one application I’m working on (a version of Steel’s multiverse programme), and the challenges I’m facing there.

(3.) I’ll present some thoughts about how one might try to extract a general notion of an “iterative conception of set”, at least model-theoretically.

Talk: Make It So: Imperatival Foundations for Mathematics

This will be a talk at the workshop Generative Metaphysics and the Philosophy of Mathematics. Slides here.

Abstract: This article articulates and assesses an imperatival approach to the foundations of mathematics.  After Fine, we call the program `procedural postulationism’. The core idea for the program is that mathematical  domains of interest can fruitfully be viewed  as the outputs of construction procedures.  Fine argues that this viewpoint has various potential epistemic and ontological benefits in philosophy of mathematics; and in any case we think the guiding idea is interesting enough to warrant close inspection. We offer a logic for `creative’ imperatives—imperatives that command the introduction of new objects into the domain—and show how to treat the concept of indefinite iteration in that logic. We then give consistency proofs for  arithmetic and set theory starting from the formalized hypothesis that certain commands statable in the  resulting logic are executable. Using this framework, we will assess whether the view can claim to have epistemic and ontological benefits over standard `declarative’ approaches.

Talk: Philosophy and the Iterative Conceptions of Set

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.