SetsIntroduction

In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but is the oldest and the youngest.

— G.H. HARDY

Every school has its clubs — the quiz club, the eco club, the chess club. A club works smoothly only when everyone can tell, without an argument, who is a member and who is not. Below, four clubs have lined up for their photo. In each line one person has sneaked in who does not fit the club's rule.

Spot the intruder in each club. Tap the one member that breaks the pattern.

How did you know? In every club the rule was sharp: hand you any object, and you could say for certain whether it belongs. A collection with a sharp rule like this is what mathematicians call a set. Sets are the language of almost all of modern mathematics. Relations and functions — your very next chapter — are defined using sets, and so are sequences, geometry and probability. In this chapter you will learn the basic words and operations of that language.

The Man Who Counted Infinity

In 1874 Cantor proved something startling: the real numbers cannot be paired off one-to-one with the whole numbers. Some infinite sets are bigger than others! Richard Dedekind (1831–1916) welcomed the idea, but Leopold Kronecker (1823–1891) attacked Cantor for treating infinite sets just like finite ones. Around the turn of the century Gottlob Frege tried to build all of logic on sets — until Bertrand Russell (1872–1970) showed in 1902 that the idea of a "set of all sets" leads to a contradiction, now called Russell's paradox.

To escape such paradoxes, mathematicians wrote careful rules (axioms) for sets: Ernst Zermelo in 1908, Abraham Fraenkel in 1922, John von Neumann in 1925 and Paul Bernays in 1937. Kurt Gödel completed one such system in 1940, now called von Neumann–Bernays–Gödel set theory.

Drag these milestones into the order in which they happened, earliest at the top.

Paul Halmos summed up Russell's discovery in his book Naive Set Theory in three words: nothing contains everything. Despite all the storms, Cantor's sets survived, and today almost every idea in mathematics is written in their language. Let's learn it.