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

Question

I have recently run into this Wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set theory, which is founded on the idea of set membership, mereology is built upon what I consider conceptually more elementary, namely the relation between parts and the whole.

Personally, I have always found a little bit unsatisfactory (philosophically speaking) the fact that set theory postulates the existence of an empty set. But of course there is the technical aspect and current axiomatizations of set theory seem to be quite good regarding what it allows us to prove.

Now it seems there have been some attempts to relate mereology and set theory, and according to the article, some authors have recently tried to deduce ZFC axioms as theorems in certain axiomatizations of it. Yet, apparently only a couple of well trained mathematicians (one of them Tarski) have discussed mereology, since most people have shown indifference towards the whole subject.

So my questions are: how is it that mereology had no success as a possible foundation for mathematics? Are axiomatizations based on mereology not suitable for most developments or simply not worth the while? If so, which would be the technical reason behind?

Answer 1

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.

Answer 2

Lesniewski's idea was not only to replace set theory with mereology but to construct entirely new foundation for mathematics which consisted of three systems:

• prototethics - the counterpart of propositional logic
• ontology - which from contemporary point of view is a first-order theory of a binary predicate, this could be roughly described as a theory of "is" (but do not confound it with $\in$)
• mereology - nominalistically motivated theory of sets.

Lesniewski's motivations were first of all philosophical in spirit. He wrote explicitly that he could not accept either the notion of class of Russell's and Whitehead's or the notion of the extension of a concept of Frege's. Moreover he could not accept existence of the empty class. One of the most important, so to say, technical motivations was Russell's paradox.

As for mereology (I know very little about other systems) Lesniewski's original system of axioms (as well as the one introduced by Leonard and Goodman under the name calculus of individuals) is definitely too weak to reconstruct even a fragment of arithmetic, for example. It was proved by Tarski (in the 30's of the previous century) that Lesniewski's mereology determine structures which bear a very strong resemblance to complete Boolean algebras. Every mereological structure can be transformed into complete Boolean lattice by adding zero element (its non-existence is a consequence of axioms for mereology). And vice versa, every complete Boolean lattice can be turned into (mutatis mutandis) a mereological structure by deleting the zero element. Thus it is by far too little to think of rebuilding mathematics in this framework.

However, as it was said by Jeremy Shipley above there is some work towards building point-free geometrical and topological systems based on mereology enhanced with some additional relation which according to its intended interpretation is to model the situation in which regions are in contact (or are separated). Alfred Tarski himself was one of the first to do this in his Foundations of geometry of solids. One can then try to express separation axioms in the language of mereology plus connection, or require some other topological properties by means of axioms put upon connection. These all can be done, however usually with an application of ZF (ZFC) on metalevel, which is far from Lesniewski's intentions.

Answer 3

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).

Answer 4

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.

Answer 5

In algebraic set theory a la Joyal and Moerdijk, the subset relation is taken as fundamental, with membership only being a derived notion (specifically, the cumulative hierarchy is taken to be the free "ZF-algebra"*; i.e., partial order with small joins and an abstract "singleton" operator. The order corresponds to subsethood, and x is defined to be an element of y just in case the singleton operator applied to x yields a subset of y). I can never quite grasp what it is that mereology is supposed to be all about as a supposed contrast to set theory, but if it's just a matter of viewing subsethood as more elementary a concept than membership, well, there you go.

[*: ZF-algebra isn't a great name for the general concept of such structures, in my opinion, since they have very little to do with specifically Zermelo-Fraenkel set theory. Note that, while every object in the cumulative hierarchy is uniquely a join of singletons (and in this way can be viewed as a plain old bag of elements), in more general ZF-algebras, there may be objects which are not joins of singletons, thus carrying a more mereological flavor; in particular, these illustrate that subsethood is not definable in terms of membership, firmly establishing subsethood as the more primitive notion in this context]

Answer 6

I think one could argue that just as there are categorical versions of set theory, for example Lawvere's Elementary Theory of the Category of Sets, there are analogous categorical versions of mereology, even if the word 'mereology' is almost never invoked as far as I know.

In the usual forms of set theory such as ZF, the primary notion is a membership relation, and this is frequently coupled with an intuitive conception of what sets 'are' internally, such as the cumulative hierarchy picture. In categorical (or other structural) set theories, the primary notion is functional composition, and this is coupled with the intuition that mathematicians are not so interested in sets for their internal structure, but in terms of how they interact through universal structures which can be formulated categorically in terms of composition. In this framework, elements of a set $X$ are reckoned as special types of functions $1 \to X$ where $1$ is a terminal object.

In set theory, we think of sets as consisting of elements (categorically this is expressed by saying that functions $f: X \to Y$ are uniquely determined by their action on elements $1 \to X$). In 'mereology' we want to think of a set as consisting of its parts (subsets). The categorical form of mereology would complete the analogy

elements: functions :: parts: ??

and it's almost obvious here that the '??' should be relations. That is, just as elements are considered special types of functions $1 \to X$, so subsets of $X$ are considered special types of relations, namely morphisms of the form $1 \to X$ in a suitable category of relations. So the primary notion of such a categorical mereology would be that of relations from the point of view of how they compose, so that we are studying categories (or perhaps better, bicategories) of sets and relations on their own terms.

Now this is quite well developed in the categorical literature, and goes by various names such as allegories or bicategories of relations. It seems clear from their book Categories, Allegories that Freyd and Scedrov consider allegories to be just as viable (at the level of foundations) as their more functional counterparts.

I wouldn't view a category $Rel$ (of sets and relations) as 'competing' with a category $Set$ of sets and functions for foundational attention, but would see the two woven together into a whole:

• Categorically, functions as special types of relations are characterized as being precisely the left adjoints in the bicategory $Rel$. That is to say: a relation $R: X \to Y$ is a function iff it has a right adjoint $S: Y \to X$, meaning that there is a counit $R \circ S \leq 1_Y$ (translation: $R$ is well-defined) and a unit $1_X \leq S \circ R$ (translation: the domain of $R$ is all of $X$). We thus get a (locally full) inclusion $i: Set = Ladj(Rel) \hookrightarrow Rel$ of bicategories.

• Going the other way, the inclusion $i: Set \hookrightarrow Rel$ of categories has a right adjoint $P: Rel \to Set$. This says there is a natural bijection $$Rel(iX, Y) \cong Func(X, PY)$$ and in fact this property is practically the very definition of topos: we can define a topos to be a (regular) category $C$ for which the inclusion $i: C \to Rel(C)$ has a right adjoint $p: Rel(C) \to C$.

In this way we see categorical set theory and 'categorical mereology' (in the sense I'm giving it here) as fitting together in a structurally harmonious whole.

Answer 7

I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP's question, Andreas Blass's answer and Carl Mummert's comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory.

In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: $\sqsubseteq$) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar): $$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge z\mathrel{\bot} y) $$ where $z\mathrel{\bot} y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq y)$ ($z$ and $y$ are incompatible, otherwise they are compatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence:

an object $x$ is a mereological sum of the group of $S$-es if and only if every $S$ is part of $x$ and every part of $x$ is compatible with some $S$.

Notice that it is a consequence of the definition that there cannot be a mereological set of an empty group of objects. Using sets and set theoretical notation we may define the sum of a set $X$ as binary relation in the following way: $$ x\mathrel{\mathrm{Sum}} X\iff \forall y(y\in X\longrightarrow y\sqsubseteq x)\wedge\forall y(y\sqsubseteq x\longrightarrow\exists z(z\in X\wedge\neg z \mathrel{\bot} y). $$ What is usually called classical mereology is a second order system which is obtain by adding the following axiom: $$ \forall X(X\neq\emptyset\longrightarrow\exists x(x\mathrel{\mathrm{Sum}} X). $$ Building a first-order system is a little bit more painstaking. To simplify things a bit we may introduce some auxiliary notation: $$ x\mathrel{\mathbf{sum}_y}\varphi(y) $$ as an abbreviation of the following formula: $$ \forall y(\varphi(y)\longrightarrow y\sqsubseteq x)\wedge\forall u(u\sqsubseteq x\longrightarrow\exists z(\varphi(z)\wedge \neg z\mathrel{\bot} u)). $$ "$x\mathrel{\mathbf{sum}_y}\varphi(y)$" may be read as $x$ is a mereological sum of all $\varphi$-ers. From this we can prove for example that:

• $\forall z(z\mathrel{\mathbf{sum}_y}\text‘z=y\text')$
• $\forall z(z\mathrel{\mathbf{sum}_y}\text‘z\sqsubseteq y\text')$.

In this setting, mereological sum existence axiom schema can be expressed as: $$ \exists x\varphi(x)\longrightarrow\exists y(y\mathrel{\mathbf{sum}_x}\varphi(x)). $$ Since the consequence of the axioms presented is that there can only be one mereological sum of $\varphi$-ers we can introduce notation (analogous to the set-theoretical abstraction operator): $$ \bigl[x\mid\varphi(x)\bigr], $$ for those formulas, which are satisfied by at least one object. Now, important thing is that: $$ x=\bigl[x\bigr] $$ so we cannot distinguish between any given object and its mereological singleton (so to say), which is the first problem to interpret ZF(C).

Defining proper part as $x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y$ we may define mereological atoms (or points, if you prefer the name) as objects minimal w.r.t. $\sqsubset$: $$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$ Now, in case $a_1,\ldots,a_n$ are atoms we can indeed treat $\bigl[a_1,\ldots,a_n\bigr]$ as a counterpart of $\{a_1,\ldots,a_n\}$ (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass: $$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$ works fine. But it does not work for example for: $$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$ since under the interpretation in question for every $a_i$: $$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$ Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.)

To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set): $$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$ while its mereological counterpart is usually not true. That is it is the case that: $$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$ but is NOT the case that: $$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x). $$

EDIT: Originally I suggested that it might be interesting to consider a system of mereology with the implication above taken as an axiom. However, in the comment below Andreas pointed to the fact that this entails linearity of $\sqsubseteq$. The consequence is that the class of models of the theory which consists of poset axioms+separation+existence of mereological sums narrows down to one-element (up to isomorphism) class, the only model being degenerate one-element structure.

As Jeremy Shipley wrote above (in comments) part of is a decent interpretation of subsethood, but not membership. There are still some other points worth mentioning, but this post has already got out of control.

Answer 8

Considering the nature of your question, you might be interested in the following paper by Geoffrey Hellman and Stewart Shapiro:

"The Classical Continuum without Points", The Review of Symbolic Logic, vol. 6, No. 3 (2013). I will be referring to http://philsci-archive.pitt.edu/9409/

Let me quote their Abstract (from the 13'th draft) verbatim (my comments will be in square brackets):

We develop a point-free construction of the classical continuum, with an interval based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts [called "intervals" which, if one wishes, may be deemed as 'points'], thereby demonstrating the independence of "indecomposability" from a non-punctiform conception [for the intervals 'cover' the gunky line]. It is surprising that such simple axioms as ours already imply the Archimedian property [for the interval structure covering the gunky line] and that they [the intervals forming that interval structure satisfying these "simple axioms"] determine an isomorphism with the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

I believe this paper answers your title question in the following manner (this completely my own comment):

The reason why mereology has not succeeded as an alternative to set theory is that classical analysis was, in its conception, implicitly based on the notion of 'point set'. This was realized by Dedekind, Cantor, and Frege, although the naive set theory that they used assumed that every property had an extension that was a set (engendering the Russell, Burali-Forti, and Curry paradoxes). As we know, modern set theory was developed in order to rid us of such paradoxes, but had no reason (since it was originally an analysis of the notion of 'point set') to replace the notion of 'point' (or more generally, the notion of 'element') in its axiomatization, since most "ordinary mathematicians" have not found problems with the classical conception of analysis on which modern analysis is based.

As for your second question, I believe the Hellman-Shapiro axiomatization of the gunky real line is adequate (as they show in their paper) for the development of real analysis.

Finally, a conjecture (which, hopefully, is not out of place here). Since the Hellman-Shapiro axioms derive an interval structure which is isomorphic to the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field, it seems reasonable to infer that the cardinality of the class of these intervals is $2^{\aleph_0}$. Is it possible to define such isomorphic interval structures on the gunky real line (in their notation, $G$) that are also of cardinality $\aleph_1$, $\aleph_2$,... respectively? If so, then the mereology of the Hellman-Shapiro variety will have done much to explain why there can exist such a plethora of continua, and how such can come to be. But this is, of course, mere speculation....

Answer 9

The following remarks reflect personal research that may be relevant to the idea of a mereological foundation.

I devised a set of sentences intended to admit a universal class to Zermelo-Fraenkel set theory. The strategy involved a primitive part relation and a primitive membership relation with additional axioms to deal with identity and recharacterizing the part relation as a subset relation.

The proper part relation can be expressed as a self-defining predicate with a circular syntax. For this reason, I view the system as related to mereology.

The membership relation depends on the part relation, but is also introduced with a circular syntax.

The sense of these sentences is that to be a subset cannot exclude being a basic open set for a topology. To be an element cannot exclude being an element of a basic open set for a topology.

No functions or constants have such definition. A grammatical equivalence with relation to the primitive relations is defined. A first-order identity is defined after certain axioms establish familiar relations with respect to class equivalence. Second-order extensionality holds, but it is not the criterion of identity. Functions and constants may be introduced only with non-circular syntax in relation to the first-order identity predicate.

Although mereology is generally thought of in terms of the proper part relation, if one reads Lesniewski, there is a great deal of effort involved with investigation of logical equivalence. This work is done in response to Tarski's paper on primitive logistic. Tarski's analysis is done in second-order logic, as is Lesniewski's.

So, the manipulations to obtain an identity relation are consistent with Lesniewski's work, even though it does not seem that way because the usual feature discussed is the part relation.

All objects are classes, with exactly one class as a proper class. The proper part relation is essential to establish this distinction. The first-order identity relation is also essential since the single class that is not an element of any class is unique by virtue of first-order identity. Second-order extensionality does not permit this distinction. The sole proper class is the set universe.

Again, this is consistent with Lesniewski's work. In objecting to Russell's paradox, Lesniewski develops this notion of a full class. This becomes the general mereological principle that a class and its parts are uniform.

The membership relation could be stratified using the proper part relation. But, to establish singletons relative to the modified axiom of pairing, an empty set had to be assumed. This is not a typical mereological assumption. This stratification is comparable to what Quine found necessary in order to have a universal class for his New Foundations. If compared with Euclid, the empty set is "that which has no parts". It is the ground for units which are "that by which what exists is one".

There is a power set axiom. However, a similar axiom only collecting proper parts is included as needed to form the first-order identity. This, too, is comparable to Quine whose system has Cantorian and non-Cantorian classes. In order for the set universe to be differentiated from its elements, proper parts had to be associated with the membership relation in the sense of a power axiom. Once a first-order identity is described, the usual power set axiom can be defined for the Cantorian "finished classes".

If these things do not sound bad enough, the model theory would necessarily be unacceptable to those committed to a predicative model construction strategy. The mereological or topological emphasis is viewed as a second-order structure in spite of the manipulations to obtain a first-order identity relation. This is consistent with the Tarskian analysis and the Lesniewskian program of research. But, it is non-standard with respect to modern foundational thinking.

In this sense, the system is Brouwerian. Logicism and logical atomism reduce the notion of object to presupposed denotations and treat the universe as Ax(x=x) with respect to ontology. When Leibniz introduced the principle of identity of indiscernibles, he did so while invoking geometric principles. The system interprets the Cantorian theory of ones in relation to his topological ideas as reflecting Leibniz' original statement. This is actually the source of the stratified membership relation. I compare it to Brouwerian ideals in that a focus on geometry is a rejection of the logicist interpretation of Leibniz principle of identity of indiscernibles.

In general, it would be best to view the structure as a closure algebra. The set universe would be the intersection over the empty set. So, the system is closed under arbitrary intersection in the same sense that an axiom of union may be interpreted as arbitrary union. With regard to statements in Aristotle, a choice has been made with regard to what "exists". In naive set theory and set theories such as New Foundations, no distinction is made with respect to partitions in relation to negation. Aristotle remarks that one should not attempt to negate substance. A closure algebra interpretation makes a distinguishing choice of closed sets over open sets. This actually derives from the model-theoretic axiom of foundation. The transitive closures satisfy the closure axioms.

It is a very strong system. It is as least as strong as Tarski's axiom. So, it would be modeled by an inaccessible cardinal or stronger.

Although this system will never be published, it was developed carefully. I hope that these remarks help anyone who might wonder what would be involved in a mathematics based on a part relation. But, if you read Lesniewski, and the paper by Tarski, you will see that much of a Lesniewskian system has nothing to do with the part relation. The part relation had merely been an outcome of his analysis of Russell's paradox, and, he insisted that the paradox should be ignored in the development of foundations because it was the result of a mistaken analysis concerning classes.

Answer 10

The most sophisticated philosophical treatment of the relation between set theory and mereology is David Lewis' book Parts of Classes. For each non-empty set S, Lewis views the singleton set of each member of S as a part of S in the classical mereological sense of part. This results in a reconception of set theory for which the only distinctively set-theoretical notion is the singleton operation. Everything else is mereology.

An application on this basis to issues of granularity -- whereby parts can hold between entities on different levels (for instance cells as parts of organs, organs as parts of organisms) -- is provided in Smith and Brogaard's paper "Quantum Mereotopology" (Annals of Mathematics and Artificial Intelligence, September 2002, Volume 36).

A good technical overview of theories of mereology is Peter Simons book, Parts (Clarendon, 1984).

Answer 11

Although Mereology cannot by itself manage to formalize most of ordinary mathematics, yet still it is not too far from it! An addition of a single primitive with rather trivial axioms about it can result in a system strong enough to interpret full second order arithmetic and thus most of ordinary mathematics! For example add to Atomic General Extensional Mereology (AGEM) a primitive one place coding injective partial function $F$ that uniquely sends any pair of atoms (objects having no more than two atoms as parts of) to an atom, provided that there is an atom outside of the range of $F$. Now this coding function $F$ is very trivial, yet still its addition to AGEM would interpret full second order arithmetic $Z_2$. One cannot do that for example with Identity theory, we cannot top it with alike trivial primitive function to result in a theory that can interpret most of mathematics, yet still, one hears of first order logic with equality (or identity), actually one sees many expositions of axiomatic set theory as formal extensions of first order logic with identity, see for example this Wikipedia exposition of ZFC. This appears in some sense unfair to Mereology. One would have expected to see many generally known axiomatic systems topping Mereology in which a sizable amount of mathematics can be encoded. Actually as far as formalizing "ordinary mathematics" is concerned, the set concept seems to be too excessive, the standard formal capture of the set concept is ZFC which is far much stronger than a formal theory needed to capture what most people know of as mathematics, one would think that what is needed for that strength is actually Mereology plus theories, i.e. Mereological theories topped with some trivial mathematical concepts, like the system spoken about above, in which most of ordinary mathematics is encodable! rather than a sky-high formal system (like ZFC) far beyond what is really needed.