One letter in 1902 wrecked a logician's plan to build arithmetic from logic.
Consider the set of all sets that are not members of themselves. Is it a member of itself? If yes, then no; if no, then yes. Bertrand Russell found this contradiction in 1901 and wrote to Gottlob Frege, just as Frege was finishing his grand attempt to build arithmetic from pure logic.
In the late 19th century, mathematicians treated a set as any collection you could describe. The principle, now called unrestricted comprehension, says that for any clear property there is a set of exactly the things that have it. It seemed obviously true, and it underpinned the German logician Gottlob Frege's project to show that all of arithmetic could be derived from logic.
Bertrand Russell found the flaw in May 1901. Most sets are not members of themselves: the set of all squares is not a square. Call such sets normal. Now form the set of all normal sets. If it is normal, it belongs in itself, which makes it not normal. If it is not normal, it doesn't belong in itself, which makes it normal. Either answer contradicts itself. The barber paradox gives a homely version: a barber shaves all and only the men who do not shave themselves; who shaves the barber?
In 1902 Russell wrote to Frege, who was preparing the second volume of his Basic Laws of Arithmetic. Frege replied almost at once, in a letter dated 22 June 1902, and added an appendix acknowledging the problem and proposing a fix, one later judged unsatisfactory by some. Russell was not quite first: Ernst Zermelo had found the same paradox by 1902, possibly as early as 1899, but shared it only with colleagues at Göttingen, and Georg Cantor had already noticed contradictions lurking in set theory in the late 1890s.
The repair work shaped modern mathematics. In 1908 Russell proposed his theory of types, which changes the logical language so that a set cannot be asked about itself in the problematic way. The same year Zermelo instead restricted which sets can be formed, and his axioms, extended with help from Abraham Fraenkel, became Zermelo–Fraenkel set theory, the standard foundation of mathematics today. A single self-referential question forced mathematicians to decide what a set really is.
Source: Wikipedia — Russell's paradox · Text summarised from Wikipedia (CC BY-SA 4.0)
