We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-deﬁnability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a ﬁnitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.
Hamkins, Joel David and Kikuchi, Makoto, "Set-Theoretic Mereology" (2016). CUNY Academic Works.