Why hasn't mereology succeeded as an alternative to set theory?

I have long found this question interesting, and in some recent joint work with Makoto Kikuchi, now available, we consider various aspects of the question of whether a set-theoretic version of mereology can form a foundation of mathematics. In particular, for our main thesis we argue that the particular understanding of mereology by means of the inclusion relation $\subseteq$ cannot, by itself, form a foundation of mathematics.

Joel David Hamkins and Makoto Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, vol. 25, iss. 3, pp. 285-308, 2016. arxiv.org/abs/1601.06593, (blog post).

Abstract. We consider a set-theoretic version of mereology based on the inclusion relation $\newcommand\of{\subseteq}\of$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\of$, we identify the natural axioms for $\of$-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.

Please follow through to the arxiv for a pdf version of the article.

Update. Here is a link to a follow-up article:

Joel David Hamkins and Makoto Kikuchi, The inclusion relations of the countable models of set theory are all isomorphic, manuscript under review. arxiv.org/abs/1704.04480, (blog post).

Abstract. The structures $\langle M,\newcommand\of{\subseteq}\of^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle M,\of^M\rangle$ are exactly the countable saturated models of the theory of set-theoretic mereology: an unbounded atomic relatively complemented distributive lattice. A very weak set theory suffices, even finite set theory, provided that one excludes the $\omega$-standard models with no infinite sets and the $\omega$-standard models of set theory with an amorphous set. Analogous results hold also for class theories such as Gödel-Bernays set theory and Kelley-Morse set theory.

And see the related question, Do all countable models of ZF with an amorphous set have the same inclusion relation up to isomorphism? That question remains an open question in the paper.

It seems worthwhile to point out that Steve’s answer also essentially answers Carl Mummert’s question (in a comment) about why one can’t get set theory as a definitional extension of mereology by defining points (as things with no proper parts) and then using “point $x$ is a part of object $y$” as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there’s no good way to handle sets of sets. Mereology (at least in Leśniewski’s version — I’m not familiar with other versions) would make no distinction between a collection of sets and the union of those sets. I think you can get somewhat closer to set theory by combining (as Leśniewski did) mereology with ontology, but even then I don’t think you get anywhere near ZF. To really handle something like the cumulative hierarchy of ZF (or even the shorter hierarchy of Russell-style type theory, I believe), mereology would have to be supplemented with some way to treat sets as (new) points, something like Frege’s notion of Wertverlauf (which would probably be anathema to Leśniewski).

Unlike category theory which is in many ways a freer framework in which to do mathematics and which very nicely captures universal objects and constructions (e.g., limits and colimits), mereology is a more restrictive framework than set theory. The whole/part relation can be captured by set/subset, but set/member cannot simply be recaptured in mereology. For instance, in mereotopology a space is comprised entirely of extended parts, no points. Try reformulating the separation axioms and deriving Urysohn's theorem, for example. (Maybe it can be done. I think so. But it's not immediately clear how.) For these reasons, mereology will remain of interest to nominalistically inclined mathematical philosophers (like Tarski, not to mention Russell and Whitehead in whose work I find mereological inclinations) but is not likely to spark a major mathematical research program, in my opinion.