Russell's paradox wrecked Frege's attempt to build arithmetic from pure logic
Gottlob Frege wanted to ground all of mathematics in logic, assuming any well-defined property picks out a set. Bertrand Russell then pictured a set gathering every set that fails to contain itself. If it belongs to itself, it cannot; if it does not, it must. Frege's system collapsed under that contradiction.
Frege's approach explained numbers through counting: to say a barn holds four horses is to say that four things fit the concept of a horse in that barn. It was elegant, but the rule Russell exploited, that every property defines a set, proved fatal. The paradox arrived amid other unsettling discoveries, among them that the parallel postulate cannot be proved and that some arithmetic truths cannot be proved in Peano arithmetic. Together they produced a foundational crisis, answered in the early 20th century by axiomatic systems. The best known and most studied is Zermelo–Fraenkel set theory, used both with and without an extra rule called the axiom of choice.
The subject itself began in the 1870s with Richard Dedekind and Georg Cantor, and Cantor is usually credited as founder. His 1874 paper showed that the real numbers are uncountable: no list, however long, can contain them all. He later proved that the collection of all subsets of any set is strictly larger than the set, even when it is infinite, and developed transfinite numbers, labelling infinite sizes with the Hebrew letter aleph.
Many found this shocking. Leopold Kronecker and Henri Poincaré resisted, later joined by Hermann Weyl and L. E. J. Brouwer, and Ludwig Wittgenstein objected on philosophical grounds. Carl Friedrich Gauss had earlier insisted that completed infinity had no place in mathematics. Yet by around 1900 Cantor's ideas were winning ground, helped by Dedekind's construction of the real numbers in 1872 and by Giuseppe Peano, whose axioms for arithmetic also gave us the epsilon symbol for membership.
The roots run deeper. Grouping objects is as old as numbers, and Bernard Bolzano's 1851 Paradoxes of the Infinite is often seen as the first rigorous use of sets, though it made little impact at the time. Today set theory underpins most of mathematics and reaches into computer science, philosophy and linguistics.
Source: Set theory
