This is a talk for the Workshop on Mereology at Shandong University in Jinan, China, a part of the week-long conference Week of Fusion Philosophy, 22-26 June 2026. The mereology talks are on 22 June 2026.
Title: Set-theoretic mereology as a foundation of mathematics?
Speaker: Joel David Hamkins, University of Notre Dame, Peking University
Abstract. Mereology, the study of the relation of part to whole, is often contrasted with set theory and its membership relation, the relation of element to set. Whereas set theory has found success in the foundation of mathematics, since the time of Cantor, Zermelo and Hilbert, nevertheless mereology has been strangely absent. Why is this? In this talk, I shall introduce and discuss set-theoretic mereology, a form of mereology based upon the set-theoretic inclusion relation ⊆ rather than the element-of relation ∈. In particular, we shall investigate the role to be played by set-theoretic mereology in the foundations of mathematics, and come perhaps to an explanation of why it has been absent.
You can define the empty set as that which is a subset of every set, and singletons as those nonempty sets whose only subsets are themselves and the empty set, and you can define the membership of A in B as the singleton of A being a subset of B. So the only obstacle is defining an operation taking A to the singleton of A or vice versa. There are other operations that would also work, but I don’t see how to get there if the only symbols I can start with are the subset symbol and the equality symbol. If I had a unary union operation I can do it (the singleton of A is the unique singleton whose union is A), can we define that?
The binary union of A U B is definable but that’s not the same thing, I can’t take A U A as the unary union because there’s no type lowering, A U B is the unary union of the pair {A, B}.
So do you add a singleton operation or do you build things up some other way?
Yes, with the singleton operator, one can easily recover the whole of set theory. The more difficult part is to get there with just the pure mereological theory having only ⊆ or otherwise to analyze what is possible in this theory. I will post my slides later so you can see what I have to say about it.