1. The empty or null set is a strange animal. It is a set, but it has no members. This is of course not a contingent fact about it, but one bound up with its very identity: the null set is *essentially* null. Intuitively, however, one might have thought that a set is a group of two or more things. Indeed, Georg Cantor famously defines a set (*Menge*) as "any collection into a whole (*Zusammenfassung zu einem Ganzen*) of definite and separate objects of our intuition and thought." (**Contributions to the Founding of the Theory of Transfinite Numbers**, Dover 1955, p. 85) In the case of the null set, however, there are no definite objects that it collects. So in what sense is the null set a set? One might ask a similar question about singletons, sets having exactly one member. But I leave this for later.

2. A related puzzle concerns the *existence* of the null set, a puzzle that arises even if we don't raise any questions about the existence of mathematical (as opposed to commonsense) sets in general. Set theory can be done either naively or axiomatically. In the standard axiomatic approach to the subject, that of Zermelo-Fraenkel, the existence of the null set is posited in a special axiom. In Zermelo's 1908 formulation, both the null set and singleton sets are posited in his *Axiom der Elementarmengen*. About the null set Zermelo writes, "There exists a (fictitious) set, the *null set*, 0, that contains no element at all." (van Heijenoort, p. 202) One curious feature of this Zermelian formulation is that 'fictitious' appears to cancel out 'exists.' To *exist*, if it means anything, is to exist in reality, in splendid independence of language and mind. Something that exists as a fiction precisely does not exist. But let's not quibble over this infelicity of formulation. There is a more serious problem.

Intuitively, the existence of a set depends on the existence of its members. The set consisting of me and my cat cannot exist unless both man and cat exist: if either of us should cease to exist, the set would cease to exist. It exists *because* we exist, not vice versa. (This is of course not a causal use of 'because.') In the case of the null set, however, there is nothing on which the null set can depend for its existence. Bertrand Russell refers to the difficulty in his early **Principles of Mathematics** (1903):

. . . with the strictly extensional view of classes [sets] propounded above, a class which has no terms [members] fails to be anything at all: what is merely and solely a collection of terms cannot subsist when all the terms are removed. Thus we must either find a different interpretation of classes, or else find a method of dispensing with the null-class. (p. 74)

In Whitehead and Russell's **Principia Mathematica**, we learn that "to say that a class exists is equivalent to saying that the class is not equal to the null-class." (24.495) It seems to follow from this that the null set does not exist!

3. No working mathematician is likely to lose any sleep over this, however. He will tell us that the null set is convenient, computationally useful and ought to be judged by its practical fruits. Here is an argument for admission of the null set:

Two sets A and B are said to be disjoint if they have no members in common. What then is the intersection of two disjoint sets? One wants to be able to say that the intersection of any two sets is a set, just as the subtraction of any integer from any other is an integer, whether positive or negative. Thus one needs to posit a null set just as one needs to posit negative integers. We also want the intersection of A' and B' (also disjoint) to be the same as the intersection of A and B. Thus we speak of *the* null set, where 'the' connotes uniqueness.

4. Here is a second argument for admission of the null set. The Union Axiom states that, given any set x, there exists a set Ux the members of which are exactly the members of the members of x. Now suppose we apply the Union Axiom to the set {Socrates, Plato}. Since the members of this set do not have members, U{Socrates, Plato} = the null set. In general, the application of the Union Axiom to any set the members of which are nonsets yields the null set.

5. The uniqueness of the null set can be proven by *reductio ad absurdum*. In such a mode of proof one attempts to show that a certain assumption, in the presence of propositions antecedently accepted, implies a contradiction. So assume that the null set is not unique: assume that there are two null sets, N and N'. Then, by the Axiom of Extensionality (two sets are the same iff they have all the same members), N has a member that N' does not have, or vice versa. But this issues in a contradiction inasmuch as neither N nor N' has a member. Therefore, the null set is unique.

6. So on the one hand, the null set is useful and well-motivated from within the circle of set-theoretical ideas, but on the other hand, it appears philosophically to be a creature of darkness. Is there a way to get rid of the darkness?

By my lights, the philosopher aims at a degree and a type of clarity that the mathematician qua mathematician -- not to mention other nonphilosophers -- does not care about. Of course, I am not saying that he *should* care about it. He is within his rights in simply dismissing concerns like the one raised in this post as irrelevant to his concerns or unimportant given his goals and priorities. What is intolerable, though, is the mathematician who gives a lousy philosophical answer to a philosophical question, especially if he is only half-aware that it is a philosophical question. He is then like the neuroscientist who, refusing to stick to his subject-matter, says silly things about mind and consciousness, all the while oblivious to the philosophical problems to which he gives silly answers.

I think we have discussed this before. The arguments are valid, but depend on assumptions that are not questioned in the arguments (e.g. the Axiom of union depends on the questionable assumption that absolutely everything is a set).

It is not a creature of darkness if its existence is required to explain basic mathematical truths. The question is whether basic mathematical truths (those requiring induction, in particular) do require it.

Posted by: ocham | Tuesday, December 30, 2008 at 04:49 AM

I once tried to put it this way: you would think that what a set is, essentially, is something to which other objects bear the "element of" relation. But the empty set is not such an object, since by definition it has no elements. And you can't say it could have elements, because if it doesn't, it couldn't.

Posted by: Alex Leibowitz | Friday, January 02, 2009 at 01:49 AM

Alex,

Nice and pithy and exactly right.

O writes, "e.g. the Axiom of union depends on the questionable assumption that absolutely everything is a set." I guess I don't see this.

Posted by: Bill Vallicella | Friday, January 02, 2009 at 05:09 AM

But if we make the is-an-element-of relation primitive we can think of it as a table with objects labelling the rows and sets labelling the columns and with a tick in the cell at row r and column c iff object r is an element of set c. Suppose the table contains a column with no ticks---after all, why should this possibility be excluded?---then the relation is telling us that there is an empty set.

Posted by: David Brightly | Friday, January 02, 2009 at 04:10 PM

David Brightly -- hmmm...that's an interesting image. It makes me think of Cantor's diagonal argument -- but I can't quite see why it should.

Posted by: Alex Leibowitz | Tuesday, January 06, 2009 at 10:05 PM

Bill,

How can we dispel the darkness surrounding the empty set? The main charge against it in your post is that

My suggestion is that it's this realist (and presentist?) view that absorbs the light. For if we take Cantor's idea outlined in the subsequent post that a set is the result of a (possible) mental act of collecting then the difficulties fade away. If I'm interested in my genealogy it makes sense to talk about the set of my ancestors. Granted, there is a presupposition here that there are such things as people and a relation of descent between them. Your point, perhaps, is that there first must be (or must have been?) things in existence before we can find collections of things. But that seems to be digging deeper for foundations than we need. For if Cantor is right then sets are a way of thinking about multiplicity when a universe of objects is already given. So I can think of my ancestors as a subset of the universe of all the people who have ever existed. Again it makes sense to partition my ancestors into two further sets, those living and those dead. Many people have living ancestors. I don't. If you deny us the empty set then we cannot talk in general about our living ancestors using the language of sets, for I am an exception, apparently havingnoset of living ancestors. But this, of course, is just another mathematician's motivation for admitting the empty set---we want to be able freely to take complements, and the complement of the universe is the empty set.To summarise, we differ in that your view appears to be that sets must be in the world, external to ourselves, and somehow dependent on the objects in the world. My view is that they are mental constructs, abstracted from a world in which multiplicity is a given. I don't know how sustainable my view is---it may well collapse under further probing---but it doesn't find the empty set a creature of darkness. And it's this (and the effectiveness of the mathematical system erected upon it) that suggests to me that your realist view is mistaken conceptualisation of sets. But then I'm a mathematician, not a philosopher, and can only hope this isn't a silly answer to a philosopher's question.

Posted by: David Brightly | Wednesday, January 07, 2009 at 02:45 AM

Alex,

It's the mathematician's standard view of a relation.

Posted by: David Brightly | Wednesday, January 07, 2009 at 02:47 AM

David

On some consideration I wonder if your objection does not beg the question. For that is exactly what is at issue -- whether the table *should* contain a column with no ticks.

Posted by: Alex Leibowitz | Saturday, January 10, 2009 at 09:41 AM

Alex,

Yes, I can see that I might be seen as begging the question. But the point I'm trying to make is that there is a perfectly good way of thinking about a relation that does allow for tickless columns (or rows). Consider the relation is-a-child-of. If we think of it defined between children on the left (labelling the rows) and parents on the right (labelling the columns) then there can be no tickless rows or columns. But this introduces an undesirable asymmetry. If we think of the relation as defined between people we will find tickless columns (childless people) and if we restrict to living people we will also find tickless rows (children whose parents have died). If we allow flexibility over the domain on which the relation is defined we gain substantially. For example, if we think of the child relation R as specified symmetrically between people on the left and people on the right, then we can define the product relation of R with itself, and this is exactly the is-a-grandchild-of relation between people. Further, we can

calculatethis relation by multiplication of boolean matrices. We can't do this if we insist on distinct domains on the left and right. The lesson I take from this is that there is no right or wrong way of thinking about relations, but therearemore and less useful and effective ways. With my apologies if this stuff is old hat.Posted by: David Brightly | Monday, January 12, 2009 at 12:53 AM