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So it has come to my attention that David Lewis, David M. Armstrong and others tried a mereological account of sets.

James Franklin states it as:

Armstrong adopts David Lewis’s proposal that a set is the mereological sum of its singletons, and adds the idea that the singleton of x is simply the state of affairs of there being some unit-making universal that singles out x.

I’m wondering if someone could explain to me how that should work and what is gained by it.

The mereological sum, the fusion of widely different objects, is a pretty strange thing, too. Why is it less problematic than a set? And how does the procedure work for sets as encountered in mathematics, which don’t seem to consist of concrete objects, as {1, 2, 3, 4}, or, worse, sets like {{0,1}, {1,2}, {2,3}}??

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    From what I understand the motivation comes from the clash between pre-existing intuitions about parts and wholes and the account given by set theory, e.g. "even after the many have been collected together into a one, it is still discernible which many they were... Mereological fusion, by contrast, obliterates the separate identities of the fused: A single whole can be taken apart in many ways". So Lewis reconstructs Cantorian set theory from something more "structuralist", a Boolean algebra, see Burgess's Lewis on Set Theory. – Conifold Feb 21 at 18:43
  • I just want to state, the whole question is slightly skewed. Mereology and set-theory are two competing theories for foundations of mathematics. Mereological account of sets you are speaking of are not, technically, set-theory, but a different theory all-together. – Bertrand Wittgenstein's Ghost Feb 22 at 1:56
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The question is a little bit vague, but I'll take it to mean what it says literally. I'll try to explain David Lewis account on that matter, though somewhat using Randall Holmes (chapter 8: Philosophical Interlude) terminology about it, and I'll also add some of my account on it. Think of a class as a totality of many labeling atoms, i.e. an object whose material is exactly the sum material of all of those atoms, that's not difficult to think of, most important is to think that there is no excess material that the class contain over that sum material of those atoms, [this is the smallest kind of a totality of objects that one can think of really], now under this line of thought the totality of one atom is that atom itself, and also there cannot be an empty totality, since if there are no atoms, then there is no material available to speak of a totality of it. All of that line of thinking is simple and understandable. Notice that although Mereological fusions of phi objects can be difficult to comprehend for the general case of phi, but mereolgical fusion of atoms is very naive really. Now we come to the function of those atoms, each atom in that totality (i.e. the class) functions as a label of some object, think of each atom as a name, of course Lewis maintain that each labeling atom labels only one object and if an object is labeled by an atom then it is not labeled by a different atom. Now Lewis then go define Epsilon membership relation as being what is labeled by an atom in the class, let me clarify:

x is a member of y if and only if y is a totality of labeling atoms and there exists an atom that is a part of y such that this atom is the label of x.

So {x,y,z} for example is not the totality of objects x,y,z. No! Its rather the totality of the atom labeling x and the atom labeling y and the atom labeling z. So actually the object {x,y,z} can be disjoint of the totality of its members, much as the totality of non-self referring referents is disjoint from the totality of the objects they refer to.

Now we come to explain what is meant by the "singleton" function? It is a rule that sends each object to its labeling atom! so the singleton of x, written as {x} is actually the atom labeling x. So when its said that a set {x,y,z} is the totality (i.e., Mereological fusion) of all of singletons of its members, it is meant it is the totality of all atoms labeling its members.

This DOES provide some interpretation, actually a definition, of epsilon membership relation in terms of Part-hood and labeling (or if you want the singleton function) relation. Its a pretty much natural interpretation. Since part-hood has a natural understanding and also unique atomic labeling has a clear cut meaning.

As to how this procedure works for sets in mathematics, I don't see really where the exact difficulty is? the set {1,2,3,4} is the totality of all atoms labeling the objects 1,2,3,4. Of course the objects 1,2,3,4 themselves are defined according to the mathematical theory in question, so for example if you have a theory of arithmetic in which 1 is a primitive constant, and 2=1+1, 3=2+1 and 4=3+1, then you simply add Mereology on top of it and the singleton function (which is the atomic labeling function), then easily define {1,2,3,4} as the Mereological funsion (i.e. totality) of all atoms labeling objects 1,2,3,4. As for explaining the set {{0,1},{1,2},{2,3}}; to understand that you need to understand what each {x,y} mean, it means a pair of atoms one labeling the object x and the other labeling the object y. Now this pair of atoms, is itself a mereological funsion of two atoms, i.e. a totality of two atoms, of course this totality has x and y as its 'members' [according to definition of membership given above], now this totality itself would have an atom labeling it, and this atom would only have this totality as the sole member of it! (notice that every labeling atom is a singleton of what it labels). To make things easier lets denote the Mereolgical funsion of objects a,b,c,d.. as [a,b,c,d,..] and the atom labeling object x by Lx, then we have:

{{0,1},{1,2}{2,3}} = [ L[L0,L1], L[L1,L2], L[L2,L3] ]

Of course for that to work, you need a rule that tell's us which Mereological totality of labeling atoms, can itself have an atom labeling it?

The standard answer would be size related as Davis puts it. I myself have done some work on that and I can sum it into few principles:

  1. Axiom of existence of labels: There exists an object t such that there exists an atom Lt that labels t, and Lt is distinct of t.

  2. Axiom schema of Accessibility: If phi(x,y) is 1-1 relation between atomically labeled objects or atoms, and the fusion of all y's (i.e., the totality of all atoms of y) is a part of an object that has an atomic label, then the fusion of all x's has an atomic label.

  3. Axiom of Infinity: An atomic labeled totality of infinitely many atoms, exists.

Those would interpret all rules of ZFC. However the axioms are not purely mereological, it does incorporate the concept of "size", so it enroots ZFC in the contemplation about Part, Label, and Size. IF you understand labeling as a singleton function, then you'll enroot ZFC in conception about Part and Size.

Another approach is to enroot SET THEORY in the concept of "Definiteness", following Ackermann's approach. Here every class has a label if and only if it is a class having all of its parts being definite totalities of atoms labeling likewise definite totalities of atoms. So we are speaking of a hereditary concept of extreme form of definiteness, so not only every part of a definite class is a definite totality of atoms; no! even all of what is labeled by an atom in that class is also a likewise definite totality (where likewise also includes that labeling also!), and so on.. Here given that concept of extreme definiteness then we'd have only ONE comprehension rule from which we can derive all rules of set theory, and this concept is "Reflection":

for all x_1,..,x_n definite classes for all y (phi(y) -> y is a definite class) -> Exists a definite class x for all y (y in x <-> phi)

provided that phi doesn't mention definiteness, i.e. phi is a purely mereological +labeling formula. [you can think of such phi as being inherently about definiteness].

So here the rules of ZFC can be viewed to be enrooted in reflective inherent definite labeling of Mereological totalities.

So although Mereology alone cannot explain mathematics, yet, intuitive imports into it from labeling, definiteness or size, can indeed provide an almost natural interpretation of ZFC. Although the result is in reality technical, but at least some natural outlook, or at least some relation to a natural background can be visualized.

One of the benefits of this method is that it can visualize intuitively how we can have non-well founded membership, and also can supply us with an intuitive explanation for Russell's paradox, and other paradoxes, and also motivate existence of proper classes. Now a class that has its label being a part of it is a class that is a member of itself! which is even not that strange thing since we have examples in life where a representative of a group of people is one of them. Now the class of all atoms that are not parts of what they label would be the class of ALL classes that are elements of some classes, and that are not elements of themselves, this class does EXIST by the atomic Mereological composition principle, which is provable from General composition principle. Now Russell's paradox disappear because simply this class itself doesn't have a label! i.e. its a proper class. However, proper classes are expected to occur since the singleton function of Lewis (i.e. the function that sends classes to atoms) which is the labeling function here, this function is not expected to be total in the first place! in other words there is no clear apriori principle that binds us to think that all classes must have atomic labels! This shows that the common expectation that every class ought to be a member of some class, is incorrect; which in some sense underlays the development of Russell's paradox in the face of unrestricted comprehension principles.

In addition to that, this method can easily define Ur-elements, that are actually not that far away from the material of classes themselves. So it does provide a visualization into the multiple versions of Ontology that you find in set\class theories, as well as different kinds of membership relations, well founded or not.

All in all, this method do possess useful implications at least at the conceptual level, and it might prove useful in motivating further technical developments in class\set theory.

  • Hi, this seems very murky to me, being trained in set theory but not philosophy. What are the atoms of the empty set? What are the atoms of the number 3? – user4894 Feb 22 at 20:26
  • the empty set is just a technical fix. You can of course interpret it as Lewis did by the totality of all non-labeling atoms (should we work in atomic mereology), David Lewis doesn't consider the empty set as a class. Now about number 3 it depends on your theory in question if you mean Zermelo 3 for example, then it would be the atom labeling the atom labeling the atom labeling the empty set i.e. it is L[L[L0]]. If you mean Von Neumann 3, then you can also define it along similar lines as [L0, L[L0], L[L0,L[L0]] ] – Zuhair Feb 22 at 20:42
  • Can you explain to me what is the benefit of all this, as opposed to what we do in math? "Set is an undefined term. A set is whatever obeys the axioms of set theory." Do philosophers consider that simple viewpoint unsatisfactory? I have to say that although I certainly believe that philosophers talk this way about sets, to me it really seems ... well murky is as polite as I can go here. Can someone put this in context for me please? – user4894 Feb 22 at 23:44
  • Also: I clicked on your link above. The title page is missing so I can't actually know who wrote it or what it is, a book of set theory or a book on the philosophy of set theory. But right there on the first page is says: What is a set? This is not a question that we will answer directly. “Set” is an undefined notion. The reader will have to count on her intuitive notion of what a “set” or “collection” is at the start. She may find that some of the properties of the set concept that we develop here will be somewhat unfamiliar. This certainly does not support your thesis. Context please? – user4894 Feb 22 at 23:48
  • The author is Randal Holmes, see math.boisestate.edu/~holmes, see section "New: my book to be available on-line in a new version". He is a mathematician, a set theorist! With special interest and achievements in New Foundations. And the book is of set theory, but the referred section in it is about philosophy of sets. Anyhow about using axioms, of course this would be a rigorous way, but still you need to investigate the informal background, here this notion together with axioms of Mereology would interpret ALL of ZFC axioms. It adds to one's intuitive arsenal about that subject. – Zuhair Feb 23 at 6:29

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