[bibtex key=HamkinsKikuchi2016:Set-theoreticMereology]

**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.

In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, a mathematical philosopher naturally wonders whether one might find a similar success for mereology, based upon a mathematical or set-theoretic parthood relation rather than the element-of relation $\in$. Can set-theoretic mereology serve as a foundation of mathematics? And what should be the central axioms of set-theoretic mereology?

We should like therefore to consider a mereological perspective in set theory, analyzing how well it might serve as a foundation while identifying the central axioms. Although set theory and mereology, of course, are often seen as being in conflict, what we take as the project here is to develop and investigate, within set theory, a set-theoretic interpretation of mereological ideas. Mereology, by placing its focus on the parthood relation, seems naturally interpreted in set theory by means of the inclusion relation $\of$, so that one set $x$ is a *part* of another $y$, just in case $x$ is a subset of $y$, written $x\of y$. This interpretation agrees with David Lewis’s *Parts of Classes* (1991) interpretation of set-theoretic mereology in the context of sets and classes, but we restrict our attention to the universe of sets. So in this article we shall consider the formulation of set-theoretic mereology as the study of the structure $\langle V,\of\rangle$, which we shall take as the canonical fundamental structure of set-theoretic mereology, where $V$ is the universe of all sets; this is in contrast to the structure $\langle V,{\in}\rangle$, usually taken as central in set theory. The questions are: How well does this mereological structure serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics as taking place in $\langle V,\of\rangle$ to the same extent that set theorists have argued (with whatever degree of success) that one may find faithful representations in $\langle V,{\in}\rangle$? Can we get by with merely the subset relation $\of$ in place of the membership relation $\in$?

Ultimately, we shall identify grounds supporting generally negative answers to these questions. On the basis of various mathematical results, our main philosophical thesis will be that the particular understanding of set-theoretic mereology via the inclusion relation $\of$ cannot adequately serve by itself as a foundation of mathematics. Specifically, the following theorem and corollary show that $\in$ is not definable from $\of$, and we take this to show that one may not interpret membership-based set theory itself within set-theoretic mereology in any straightforward, direct manner.

**Theorem.** In any universe of set theory $\langle V,{\in}\rangle$, there is a definable relation $\in^*$, different from $\in$, such that $\langle V,{\in^*}\rangle$ is a model of set theory, in fact isomorphic to the original universe $\langle V,{\in}\rangle$, for which the corresponding inclusion relation $$u\subseteq^* v\quad\longleftrightarrow\quad \forall a\, (a\in^* u\to a\in^* v)$$ is identical to the usual inclusion relation $u\subseteq v$.

**Corollary.** One cannot define $\in$ from $\subseteq$ in any model of set theory, even allowing parameters in the definition.

A counterpoint to this is provided by the following theorem, however, which identifies a weak sense in which $\of$ may identify $\in$ up to definable automorphism of the universe.

**Theorem.** Assume ZFC in the universe $\langle V,\in\rangle$. Suppose that $\in^*$ is a definable class relation in $\langle V,{\in}\rangle$ for which $\langle V,\in^*\rangle$ is a model of set theory (a weak set theory suffices), such that the corresponding inclusion relation $$u\subseteq^* v\quad\iff\quad\forall a\,(a\in^* u\to a\in^* v)$$is the same as the usual inclusion relation $u\subseteq v$. Then the two membership relations are isomorphic $$\langle V,\in\rangle\cong\langle V,\in^*\rangle.$$

That counterpoint is not decisive, however, in light of the question whether we really need $\in^*$ to be a class with respect to $\in$, a question resolved by the following theorem, which shows that set-theoretic mereology does not actually determine the $\in$-isomorphism class or even the $\in$-theory of the $\in$-model in which it arises.

**Theorem.** For any two consistent theories extending ZFC, there are models $\langle W,{\in}\rangle$ and $\langle W,{\in^*}\rangle$ of those theories, respectively, with the same underlying set $W$ and the same induced inclusion relation $\of=\of^*$.

For example, we cannot determine in $\of$-based set-theoretic mereology whether the continuum hypothesis holds or fails, whether the axiom of choice holds or fails or whether there are large cardinals or not. Initially, the following central theorem may appear to be a positive result for mereology, since it identifies precisely what are the principles of set-theoretic mereology, considered as the theory of $\langle V,{\of}\rangle$. Namely, $\of$ is an atomic unbounded relatively complemented distributive lattice, and this is a finitely axiomatizable complete theory. So in a sense, this theory simply *is* the theory of $\of$-based set-theoretic mereology.

**Theorem.** Set-theoretic mereology, considered as the theory of $\langle V,\of\rangle$, is precisely the theory of an atomic unbounded relatively complemented distributive lattice, and furthermore, this theory is finitely axiomatizable, complete and decidable.

But upon reflection, since every finitely axiomatizable complete theory is decidable, the result actually appears to be devastating for set-theoretic mereology as a foundation of mathematics, because a decidable theory is much too simple to serve as a foundational theory for all mathematics. The full spectrum and complexity of mathematics naturally includes all the instances of many undecidable decision problems and so cannot be founded upon a decidable theory. Finally, it follows as a corollary that the structure consisting of the hereditarily finite sets under inclusion forms an elementary substructure of the full set-theoretic mereological universe $$\langle \text{HF},\of\rangle\prec\langle V,\of\rangle.$$ Consequently set-theoretic mereology cannot properly treat or even express the various concepts of infinity that arise in mathematics.

Mereology on MathOverflow | Mereology on Stanford Encyclopedia of Philosophy | Mereology on Wikipedia

Some previous posts on this blog:

Different models of set theory with same $\of$ | $\of$ is decidable

Pingback: Different models of set theory with the same subset relation | Joel David Hamkins

Pingback: Set-theoretic mereology, Logic and Metaphysics Workshop, CUNY, October 2016 | Joel David Hamkins

Here are some possibly related points.

From a historic point of view it may be interesting to note that Mac Lane spoke about a set theory based on a system of inclusions at the ASL 2000 meeting in Urbana, see this note by Steve Awodey: http://www.andrew.cmu.edu/user/awodey/preprints/CT2000/MLsetsNewAbstract.ps

There is a wider project, algebraic set theory, which (often) axiomatizes set theory based on a system of inclusions. But instead of axiomatizing sets and inclusions they tend to axiomatize classes and inclusions, together with distinguished *small* inclusions, which correspond to the inclusion of a set into a class. See the project page at http://www.phil.cmu.edu/projects/ast/

Thanks for your comment and the links!

Regarding your second point, there is also the 1991 book Parts of Classes by David Lewis, which is explicitly about set-theoretic mereology using classes in Godel-Bernays set theory. Let me say, however, that much of the analysis of my paper with Makoto also applies in this class case. In particular, the pure mereology for classes gives rise to a finitely axiomatizable complete decidable theory, which is therefore unsuitable as a foundation of mathematics, because it is decidable. If one adopts something like the singleton operator, as Lewis does, then $\in$ is definable and the foundation is bi-interpretable with $\in$-based set theory.

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Your paper is great. I wonder if the overall “finitistic” result of set-theoretic mereology couldn’t end up in something important anyway. For example, abstract objects are often shaped by means of what they are not, e.g. causally defective or spatiotemporally unextended, rather than what they are, e.g. necessary perhaps. David Lewis ‘ megethology is ambitious and pretty interesting. Maybe it fails as a foundation of the whole of mathematics, but couldn’t be relevant to our knwoledge of set-sized objects? Moreover, isn’t lewisian original concern about the status of singleton left untouched? It seems that your paper is a great formal analysis which would need a metaphysical or, if you don’t like the term, a substantive analysis. What are your insights of the metaphysical moral of your set-theoretic mereology? Is supposed to work with the not-so-innocent theses of composition as (or analogous to) identity and Universal fusion axiom?

Thanks, I’m glad you liked the paper. (We have a follow-up paper at http://jdh.hamkins.org/inclusion-relations-are-all-isomorphic/, but it is less explicitly philosophical.)

Lewis makes a fundamental use of the singleton operator, and as we explain in the paper, to my way of thinking this makes his theory basically bi-interpretable with membership-based set theory. So it can be a perfectly fine foundation of mathematics. I view the analysis of our mereology paper in part as explaining why Lewis needed to add the singleton operator to have a satisfactory theory. If you don’t have it, then the theory is decidable. With it, the theory is basically equivalent to set theory.

Pingback: On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018 | Joel David Hamkins

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Pingback: The model theory of set-theoretic mereology, Notre Dame Math Logic Seminar, January 2022 | Joel David Hamkins