Historical Note

Set theory as we use it today is largely the work of one mathematician: the German mathematician Georg Cantor (1845-1918). His papers on the subject appeared between 1874 and 1897, and — perhaps surprisingly — they did not start out as an investigation of sets at all. Cantor's route into set theory began with trigonometric series, expressions of the form $a_1 \sin x + a_2 \sin 2x + a_3 \sin 3x + \dots$, and the question of how uniquely such a series could represent a function led him to think carefully about collections of points where a series might fail to converge.

That line of thinking produced one of the most striking results in the history of mathematics. In a paper published in 1874, Cantor proved that the set of real numbers cannot be placed in a one-to-one correspondence with the set of integers — in other words, there are "more" real numbers than integers, even though both sets are infinite. This was a genuinely new idea: infinite sets could be compared in size, and not all infinities were the same size. From 1879 onward, Cantor published a series of further papers establishing properties of abstract sets in general.

How the idea was received

Cantor's work did not arrive to universal applause. Richard Dedekind (1831-1916), another major mathematician of the period, received it warmly and built on it. But Leopold Kronecker (1810-1893) was sharply critical, objecting to the way Cantor treated infinite sets with the same rigor as finite ones — Kronecker's own view of mathematics had little room for actual infinities. Separately, the German logician Gottlob Frege attempted, around the turn of the century, to present set theory as a branch of logic itself, grounding numbers and arithmetic in purely logical set-theoretic terms.

Russell's Paradox

Up to that point, set theory rested on a very permissive assumption: that for any well-defined property, there exists a set of all things having that property — including, in principle, a "set of all sets." In 1902, the English philosopher Bertrand Russell (1872-1970) showed that this assumption is self-contradictory. Consider the set of all sets that do not contain themselves as a member — does this set contain itself? Either answer leads to a contradiction. This became known as Russell's Paradox, and it exposed a real crack in the informal foundations set theory had been built on. The mathematician Paul R. Halmos later summarized the lesson memorably in his book Naive Set Theory: "nothing contains everything."

Rebuilding on firmer ground …