Central paradox in the theory of classes. Most classes are not members of themselves, but some are; for example, the class of non-men, being itself not a man, is a member of itself. Let R be the class of all classes which are not members of themselves. If it exists, it is a member of itself if and only if it is not a member of itself: a contradiction. So it does not exist. This is paradoxical, because it conflicts with the seemingly inescapable view that any coherent condition determines a class. (Even a contradictory condition, like being round and square, determines a class: the class with no members.) Standard responses, like Russell's theory of types, aim to find some limitation on what classes there are which is (a) intuitively satisfactory, (b) excludes R, and (c) includes all classes needed by mathematicians.
B. Russell, ‘Mathematical Logic as Based upon the Theory of Types’, American Journal of Mathematics (1908); repr. in Bertrand Russell: Logic and Knowledge. Essays 1901–1950, ed. R. C. Marsh (London, 1965).Find this resource:
R. M. Sainsbury, Paradoxes (Cambridge, 1988), ch. 5.Find this resource: