3The axiom of extensionality

This, then, is arguably the most fundamental fact about sets: they are fully specified by their elements in the sense that if two sets have exactly the same elements, then they are not two sets but one set. (That is, those two sets are numerically identical.)

This is known as the axiom of extensionality:

Axiom 1 (Extensionality). If two sets have exactly the same elements, then they are identical.

Mathematicians, because they like scary notation, sometimes write it down like this:¹

∀ x ∀ y (∀ z (z ∈ x ↔ z ∈ y) → x = y)

But what this means is just what I said in English: if two sets have exactly the same elements, they are identical. The lesson here, by the way, is that mathematics often looks much scarier than it really is.

Notice, by the way, that I’ve been talking a lot about “sets” and “belonging to” and “membership”, but I haven’t given you a clean definition of what these things are. That’s not an accident, because these things don’t really have a full, rigorous definition in the usual sense. And there’s a good reason for that: a definition is generally a kind of reduction of advanced concepts to simpler concepts. We might, for example, define a bachelor as an unmarried man; this is a definition, but what it does is define one complex concept, that of a bachelor, in terms of two simpler concepts, that of a man and an unmarried person. Definitions in mathematics work exactly the same way: we define one thing in terms of other things. But we cannot define all of our concepts like this, else our definitions would be circular. We have to bottom out somewhere, and the concepts of “set” and “membership” actually turn out to be plausible candidates for where all of mathematics might bottom out.²

What can be said is that the axiom of extensionality does a lot to tacitly define what sets are. Sets are the kinds of things which are defined by their elements: once you have fully specified which elements belong to a set, you have fully specified the set. Philosophers of mathematics call this the idea of implicit definition. (There are other axioms of set theory which put further flesh on the bones, but we won’t need to concern ourselves with those axioms in this primer.)

This is written in the language of set theory, where the quantifiers are tacitly taken to range over only the pure sets. If you don’t know what that means, don’t worry about it. ↩
A logician might put it like this: the belonging to symbol, which by the way is ∈, is a primitive predicate—indeed, the only non-logical primitive predicate—in the language of set theory. ↩