Logic and math
Russell's paradox
Bertrand Russell, 1901; sent to Gottlob Frege in a letter of 16 June 1902.
A set is any collection of things. Some sets contain themselves as members: the set of everything that is not a teacup is not a teacup, so it belongs to itself. Most sets do not: the set of all teacups is not a teacup. Now consider R, the set of every set that is not a member of itself.
Is R a member of itself? If it is, it must meet the condition for membership, so it is not a member of itself. If it is not, it meets the condition, so it is. A homelier version: a village barber shaves exactly those villagers who do not shave themselves. Who shaves the barber?
Bertrand Russell found this in 1901 while reading Gottlob Frege, who was building arithmetic out of pure logic on the assumption that any condition you can state defines a set. Set aside whether the puzzle is clever. What does the contradiction show about R?