Set-theoretic mereology

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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.