Separation: A Different Answer

Type theory asks which expressions a mathematical language should permit. Axiomatic set theory asks a different question: under what conditions may that language assert that a set exists? The distinction becomes sharp when we return to Russell's property, “does not belong to itself.”

Ernst Zermelo's 1908 axiomatization developed alongside the early work on types. It did not wait for Principia Mathematica to be completed. Its decisive move was to replace an unrestricted promise of new sets with rules for constructing sets from sets already available.

One kind of set variable

In the modern first-order presentation of ZF set theory, every variable ranges over sets. The basic nonlogical relation is membership, written $\in$. Equality is also available. Quantifiers such as $\forall x$ and $\exists x$ range over sets, while formulas may combine membership statements with logical connectives.

This language allows us to write $x\in x$. A formula being well formed does not make it true, and it does not ensure that every collection described by a formula is itself a set. That is the opening Zermelo's approach uses. It constrains the existence claims rather than assigning a new grammatical type to each level of collection.

There is a historical qualification. Zermelo spoke of “definite” properties in 1908. The exact modern first-order axiom schema below is a later sharpening of that idea. It is useful for seeing the mathematical mechanism without attributing today's formal syntax word for word to Zermelo.

Select from a set already given

Unrestricted comprehension would promise a set of all objects with a property $\varphi(x)$:

$$\exists S\,\forall x\bigl(x\in S\leftrightarrow\varphi(x)\bigr).$$

The schema of separation begins instead with a set $A$. For each suitable first-order formula $\varphi(x)$, possibly with parameters, it supplies the subset of elements of $A$ satisfying that formula:

$$\forall A\,\exists B\,\forall x \bigl(x\in B\leftrightarrow(x\in A\land\varphi(x))\bigr).$$

We abbreviate this as $B=\{x\in A\mid\varphi(x)\}$. The formula serves as a filter. It does not conjure a set from the whole universe of sets.

For example, the even numbers can be selected from $\mathbb N$. A circle can be selected from the plane once that plane has been constructed as a set. These familiar uses of set-builder notation survive. The unrestricted Russell collection is a different claim, because it names no set from which to select.

Russell's test, this time with a boundary

Choose any set $A$ and apply separation to the condition $x\notin x$:

$$R_A=\{x\in A\mid x\notin x\}.$$

Now ask whether $R_A$ belongs to $A$. Suppose that it does. By the defining condition, $R_A$ belongs to $R_A$ exactly when it does not belong to $R_A$:

$$R_A\in R_A \quad\leftrightarrow\quad (R_A\in A\land R_A\notin R_A).$$

Under the supposition $R_A\in A$, this is a contradiction. Hence

$$R_A\notin A.$$

The relative Russell set exists as a subset of $A$. The argument tells us that this particular subset cannot be one of the elements of $A$. The paradoxical step was the earlier demand for a single set covering every possible starting set.

Why there is no universal set

Suppose there were a set $V$ containing every set. Separation would form $R_V=\{x\in V\mid x\notin x\}$. The result just proved gives $R_V\notin V$. Yet $R_V$ is a set, so a universal $V$ would have to contain it. The assumption fails: in ZF, there is no set of all sets.

Mathematicians can still speak informally of the class of all sets. In some alternative foundations, classes are part of the formal language. A class that is too large to be a set is called a proper class. Calling the totality a class does not turn it into a universal set inside ZF.

Separation is powerful, but it only extracts subsets. By itself it cannot supply an infinite set, a power set, or the pair of two given sets. The next essay asks what further existence principles are needed to build a useful mathematical universe.

Sources

• Zermelo, “Untersuchungen über die Grundlagen der Mengenlehre I” (1908), the original axiomatization.
• Russell, letter to Frege, 16 June 1902, for the paradox that motivated much of the foundational debate.