On the Elusive Notion of a Set: Sets as Products of Collectings

In an important article, Max Black writes:

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.   A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set  is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many.  A set is one in that it is a single item 'over and above' its members or elements as I have just explained.  It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.)  The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect.  So it seems there is a genuine puzzle here.  How remove it?

2. That sets are treated in set theory as single items 'over and above' their members can be seen from the fact that sets can have sets as members without having their members as members. (This is one respect in which sets differ from mereological sums.)  It follows that a set and its members are distinct.  Suppose that the pair of shoes I am wearing forms a set, the set {a, b}. Set theory implies that there exists a set of all this set's subsets, namely, the power set of the given set. This power set can be displayed as follows:

{{ }, {a, b}, {a}, {b}}.

You will note that, although {a} and {b} are members of the power set, a and b are not. Therefore, a and b are distinct from their respective singletons, and the plurality a, b is distinct from {a, b}.

3. So a set is distinct from its members, even though it is "wholly constituted by them," as Black puts it. The problem is to make sense of this. Given that a set is distinct from its members, what makes the set distinct from its members? How can a set be an item in addition to its members while remaining "wholly constituted by them"? Bill exists and Phil exists.  Set theory implies that there also exists {Bill, Phil}.  But if this third existent is permitted, so also is an infinity: {{Bill, Phil}}, {{{Bill, Phil}}} . . . .  I have no problem with infinity, not even actual infinity; nor do I have any problem with what lovers of desert landscapes will decry as a bloated ontology. My problem is to understand how {Bill, Phil} does not involve a contradiction.

4. One answer is that a set is a product of an act of collecting. This answer is suggested by Georg Cantor's talk of a Menge as a Zusammenfassung zu einem Ganzen "of definite and separate objects of our intuition or our thought." You are given three things, say, whether sets or nonsets, and you collect them into one thing. On this approach, a set is a mental construct. A mind operates on a mere plurality and creates out of that plurality a single item, the set.

How does this remove the contradiction?  Well, considered in itself, a set is both one and many: it is just its members and yet more than its members.  But if a set is a product of an act of mental collecting, then it is not one in itself, but one in relation to a mind that collects its members into a whole.  So it is not one in the same sense in which it is many: it is many in itself, but one only in relation to a mental collecting.  A set is many in itself but not one in itself.  So the contradiction disappears since the set is many and one in different respects.

5. Objection. "This is no answer at all. Your problem is to understand how a set of three things can be one thing without contradiction. So you ascribe the oneness to a mental act of collecting. But if you think of three things as one thing, then you think incorrectly. Three things are three things, not one thing, and if you think of them as one thing, then your thinking is just false."

6. Reply. "You are not getting my point. I grant that I think falsely if I think of three things as one thing. But why can't I think three things together, thereby creating in thought a new thing, a unity of the three things I started with? Granted, no many is one, and no one is many. But e pluribus unum must be possible, no? A many, a plurality, can be combined in thought to create a new object. So why not say that sets are created by thinking?  My left shoe is not created by thinking, nor is my right shoe; but it seems plausible to say that the set consisting of the two shoes is created by thinking.

7. Objection. "There are far more sets than there are mental constructions. For each natural number, there is that number's singleton. There are infinitely many of these singletons but only finitely many acts of thinking. And there are modal problems. Each of these singletons is a necessary being; but no object of a finite act of thinking is a necessary being. If {2} exists only as the accusative of my act of collecting 2, or my act of forming a set from 2 and 2 alone, then {2} is as contingent as my thinking.  Furthermore, there are uncountable infinities that could not possibly be collected by any finite mind. To put it picturesquely, no finite mind could 'wrap itself around' the real numbers in order to collect them into a single item, the set of reals. There is no way to accommodate Cantor's Paradise on your conception."

8. Reply. "Your objection does not show that sets cannot be mental constructions; it shows that sets cannot be mental constructions of a finite mind. If there were an infinite, necessarily existent mind, then sets could be the constructs of such a mind. If you maintain that there is no such mind, then you should also maintain that there are no sets. If, however, you hold that there are mathematical sets, and that nothing contradictory can exist, then you should hold that they are the mental constructs of an infinite mind. If you deny that there is an infinite mind, but hold that there are sets, then you owe us an alternative explanation of how a set can be both one and many.

9. Objection. "Isn't this all a bit quick?"

10. Reply. "What do you expect in the blogosphere?"

Comments

[David Brightly]

Rather than ask what has to be added to a mere multiplicity to arrive at a set---an approach to the idea of a set from below, as it were---we could make an approach from above. Presumably we have no difficulty thinking of a house as a unity over many bricks (plus other stuff which we can ignore for the sake of an example). The bricks have complex relations between them---one is above another or abuts another, and so on, but let's abstract these details away. What we are left with is a house, a unity, thought of as a set of bricks, also therefore a unity. A set is the abstraction arrived at when the relations between the parts of a one-over-many are forgotten, and set theory is an account of how such abstractions behave. (I'm using 'part' here in the engineering sense of 'component' rather than its mereological sense)

[Vlastimil Vohánka]

Bill,

You wrote:

"... This power set can be displayed as follows:

{{ }, {a, b}, {a}, {b}}.

... although {a} and {b} are members of the power set, a and b are not."

Let’s have the set S = {a}.
The power set *S (of the set S) = {{ }, {a}}.
And the power set **S (of the power set *S) = {{ }, {{ }, {a}}, {{ }}, {{a}}}.
Am I right?

Now compare your claim I cited with G. Oppy (books.google.com/books?id=FPU9tzW-2HAC, pp. 23-24):
"A set x is transitive if every member of x is a subset of x, that is, every member of a member of x is a member of x." (Oppy, probably following Drake's 1974 Set Theory, employs this definition in a recursive definition of ordinal sets.) But you, Bill, seem to assume, contrary to Oppy, that for every set x, every member --which is not empty set -- of x is NOT a subset of x, and that every member -- which is not empty set -- of a member of x is NOT a member of x. For instance, empty set is both a member of and a subset of *S, but {a} is not among its subsets , though {a} is its member. *S is a member of **S, empty set is a member of both, but {a} is a member of *S and not of **S. Am I right? Is Oppy wrong? Is the way he proceeds standard?

[Bill Vallicella]

Vlastimil,

Yes, you are right about the power set **S. But then I have trouble following you. In paragraph #1 above I repeat the well known point that math. sets are entities over and above their members. Then, in para. #2 I give an argument why this must be the case. The argument was:

1. Sets can have sets as members without having their members as members; ergo
2. A set and its members are distinct.

To put that more clearly:

1*. Some sets have sets as members without having their members as members; ergo
2*. Some sets are such that they are distinct from their members.

You write, "your claim." What claim do you take me to be making? And how are transitive sets relevant to it?

[Bill Vallicella]

David,

I hope you are not suggesting that a brick house is a set. That can't be right for two reasons. First, a house can gain or lose parts while retaining its identity; a set by contrast cannot gain or lose members. If it does it ceases to exist. Second, a modal point. A house that actually has such-and-such parts might have had different parts (e.g. one of the red bricks might have had a yellow replacement). But no set might have had a different membership than the one it actually has.

But I may not have understood you.

Happy New Year.

[Vlastimil Vohánka]

Bill,

I do not challenge your argument (1*)-(2*).

By "your claim" I mean the last line of my quotation from your post ("although {a} and {b} are members of the power set, a and b are not"). It seems to me that this line implicitly states the principle I stated (for every set x, every member --which is not empty set -- of x is NOT a subset of x, and that every member -- which is not empty set -- of a member of x is NOT a member of x). Maybe I'm wrong.

And Oppy's transitive sets seem to be in conflict with the principle. Again, maybe I'm wrong; maybe I don’t understand the concept of transitive set.

Let's consider the following.

Your set = {a, b}.
Your power set = {{ }, {a, b}, {a}, {b}}.

My set S = {a}.
The power set *S = {{ }, {a}}.
The power set **S = {{ }, {{ }, {a}}, {{ }}, {{a}}}.

In this list, no set which have members which have members (i.e., all the power sets) is transitive. For instance, your power set has your set as a member, but your power set does not have any member of your set as a member. **S has *S as a member, but **S doesn't have all the members of *S as members. *S has S as a member, but *S does not have all the members of S as members. A layman like me naturally asks whether there are any transitive sets at all, or, rather, how do they look like.

[Vlastimil Vohánka]

Bill and David,

Could you please supply some example(s) of clearly transitive set(s) (by enumeration of their members)?

[David Brightly]

Hello Bill, and Happy New Year in return.

Can I refer you to my further comment on your empty set post? I am suggesting that a brick house is a set, but the 'is' here is not the 'is' of identity. Rather, it's the 'is' of abstraction, or perhaps of metaphor. What I mean by this is that a house, in some aspects, 'behaves like' a set of bricks. The problem is that the idea of 'set' is such a thin extension to the idea of multiplicity that it's hard to pin down quite what it adds. Suppose we want to know the weight of a house. Common sense tells us that we just have to add up the weights of the individual bricks. But this is a special case of a general rule: if a material thing consists of a multiplicity of disjoint sub-things then its weight is the sum of the weights of the individual sub-things. This can be formalised in the language of sets of objects and functions from objects to real numbers representing weights. We have a metaphor for a house taken from the world of mathematical objects. Like any metaphor, it claims to express only a limited aspect of the thing in question. So, Yes, the house, for the purpose of calculating its weight at a particular instant, is the set of bricks (with their own particular weights) that it's actually made of at that moment. How else would the engineering disciplines, and applied maths in general, succeed?

[Vlastimil Vohánka]

Addendum

Bill,

You wrote in your reply to me:

"I repeat the well known point that math. sets are entities over and above their members. Then, in para. #2 I give an argument why this must be the case."

Is the well known point supposed to be that ALL math. sets are entities over and above their members? If so, note that your conclusin (2*) is only that SOME math. sets are such that they are distinct from their members.

Secondly, what are the logical relations btw the words "over and above" and "distinct from" as you are using them? Do they mean the same? If not, what does "over and above" mean?

[David Brightly]

Hi Bill,

May I take another stab at explaining my original comment here? The problem is to understand how a set can be a unity over and above its members, even though it appears to be wholly constituted by its members. Let's return to the house made exclusively of bricks. We can think of the house as a unity of the bricks themselves plus the way they are arranged to make the house. I think you have said something very close to this in past posts. The 'way they are arranged to make the house' comes down to relations between the bricks, or alternatively, between the individual bricks and the unity that is the house. It will include such facts as that a certain brick belongs to the house, and that another brick lies in the topmost course of the chimney, and so on. Let's now imagine paring away these relations by a process of gradual forgetting or ignoring. At first we will retain the fact that a brick belongs to the house but forget where it sits in relation to the house. If we carry this abstraction process to the very end we arrive at a mere plurality of bricks with no relations between them or the house. But (and this is partly inspired by Alex's comment on the empty set posting) we can stop just short of this when what remains is just the plurality of bricks and the is-part-of relation with the house. At this point what we have left looks just like a mathematical set---a one-over-many consisting of a plurality plus a membership relation. This appears to be the 'thinnest' extension possible that turns a mere plurality into a unity. To summarise, we have a sorites-like argument: If we can understand the house as a unity over and above its elements and also understand that there is a continuum from the house to a set, then we ought to be able to accept a set as a unity over and above its members.