Sets

A set is simply a well defined collection of distinct objects, considered together, but the idea, developed rigorously, turned out able to serve as a common foundation underneath nearly the whole of mathematics; a number, a function, a geometric point, can each be reconstructed as a particular kind of set. Georg Cantor created set theory from 1874 onward while investigating the size of infinite collections, and his own naive version of the theory was shown in the early twentieth century to permit genuine logical paradoxes (Bertrand Russell's paradox, concerning the set of all sets that do not contain themselves, is the most famous); the axiomatic version developed afterward, chiefly by Ernst Zermelo and Abraham Fraenkel, repairs this and remains the standard foundation used today.