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.
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