Russell's paradox: the proof that Cantor's classic set theory is selfcontradictory. ►Georg Cantor defined a ►set as an aggregation of elements. These elements could, of course, also be sets themselves. But in the early 20th century, the philosopher and mathematician Bertrand Russell discovered the following paradox: "Let S be the set of all sets that do not contain themselves. Now, does S contain itself?" This definition of a set leads to a contradiction, since S can contain itself only if it does not contain itself, and vice versa. Russell also came up with a more tangible version of this paradox, namely the barber paradox: "The barber of Sevilla shaves all those men in Sevilla who do not shave themselves. Who shaves the barber?" Russell's own solution to the paradox of set theory consisted in a systematic arrangement of all sets into different types. Thus, sets of the lowest Type I could have only simple elements but not sets as elements. Sets of Type II could have simple elements and sets of Type I as elements, sets of type III could include Type II sets among their elements, and so forth. Thus, the set of all sets that do not contain themselves would be a set of Type ∞ and is not definable in this system. Russell published this theory of types in his famous work of 1910, Principia Mathematica, which he cowrote together with A.N. Whitehead. Contemporary mathematicians use the axiomatic set theoretical system redefined in 1922 by Ernst Zermelo and Abraham Fraenkel, for which Russell's paradox no longer arises.
