Set theory is the bedrock on which the rest of mathematics is built. Its objects are the simplest imaginable — a set is just a collection of things — yet from that single idea grow ordered pairs, relations, functions, the whole edifice of number, and, most spectacularly, a rigorous theory of the infinite. This branch begins where basic set notation leaves off and follows the thread all the way to the strange frontier where our very axioms run out of answers.

Building blocks

First, two constructions that turn sets into everything else. The power set gathers all subsets of a set — and is always strictly bigger than the set itself. Ordered pairs let us build the Cartesian product, and with it define relations and functions as sets — including the all-important bijection, a perfect pairing that will become our ruler for measuring size.

The sizes of infinity

Here is where set theory becomes astonishing. Using bijections, we can compare infinite sets. Some are countable — the integers and even the rationals can be listed in step with the counting numbers. But the real numbers are uncountable: Cantor's diagonal argument shows no list can ever contain them all. That is no fluke — Cantor's theorem proves the power set of any set is strictly larger, so there is an endless tower of infinities. The size of the real line, the continuum, raises a question — is there an infinity between the naturals and the reals? — whose answer, the Continuum Hypothesis, turns out to be undecidable.

Foundations and their price

Finally, the axiom that gives set theory its power and its paradoxes. The axiom of choice lets you pick one element from each of infinitely many sets, even with no rule to do so — indispensable across mathematics, but non-constructive. Its most notorious offspring is the Vitali set, a set of reals with no possible length, which is the very reason measure theory must confine itself to a well-behaved σ-algebra rather than every subset of the line.

Follow the pages in order and you will travel from "what is a set?" to the outer edge of what mathematics can decide.