Set theory, philosophy of

The various attitudes that have been taken to mathematics can be split into two camps according to whether they take mathematical theorems to be true or not. Mathematicians themselves often label the former camp realist and the latter formalist. (Philosophers, on the other hand, use both these labels for more specific positions within the two camps.) Formalists have no special difficulty with set theory as opposed to any other branch of mathematics; for that reason we shall not consider their view further here. For realists, on the other hand, set theory is peculiarly intractable: it is very difficult to give an unproblematic explanation of its subject matter.

The reason this difficulty is not of purely local interest is an after effect of logicism. Logicism, in the form in which Frege and Russell tried to implement it, was a two-stage project. The first stage was to embed arithmetic (Frege) or, more ambitiously, the whole of mathematics (Russell) in the theory of sets; the second was to embed this in turn in logic. The hope was that this would palm off all the philosophical problems of mathematics onto logic. The second stage is generally agreed to have failed: set theory is not part of logic. But the first stage succeeded: almost all of mathematics can be embedded in set theory. So the logicist aim of explaining mathematics in terms of logic metamorphoses into one of explaining it in terms of set theory.

Various systems of set theory are available, and for most of mathematics the method of embedding is fairly insensitive to the exact system that we choose. The main exceptions to this are category theory, whose embedding is awkward if the theory chosen does not distinguish between sets and proper classes; and the theory of sets of real numbers, where there are a few arguments that depend on very strong axioms of infinity (also known as large cardinal axioms) not present in some of the standard axiomatizations of set theory.

All the systems agree that sets are extensional entities, so that they satisfy the axiom of extensionality: ∀x(xЄa ≡ xЄb) → a=b. What differs between the systems is which sets they take to exist. A property F is said to be set-forming if {x:Fx} exists: the issue to be settled is which properties are set-forming and which are not. What the philosophy of set theory has to do is to provide an illuminating explanation for the various cases of existence.

The most popular explanation nowadays is the so-called iterative conception of set. This conceives of sets as arranged in a hierarchy of stages (sometimes known as levels). The bottom level is a set whose members are the non-set-theoretic entities (sometimes known as Urelemente) to which the theory is intended to be applicable. (This set is often taken by mathematicians to be empty, thus restricting attention to what are known as pure sets, although this runs the danger of cutting set theory off from its intended application.) Each succeeding level is then obtained by forming the power set of the preceding one. For this conception three questions are salient:

Why should there not be any sets other than these? How rich is the power-set operation? How many levels are there?

An alternative explanation which was for a time popular among mathematicians is limitation of size. This is the idea that a property is set-forming provided that there are not too many objects satisfying it. How many is too many is open to debate. In order to prevent the system from being contradictory, we need only insist that the universe is too large to form a set, but this is not very informative in itself: we also need to be told how large the universe is.