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.

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.

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, manuscript under review, arxiv.org/abs/1601.06593, blog.

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.

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.