The Oxford Biblical Studies Online and Oxford Islamic Studies Online have retired. Content you previously purchased on Oxford Biblical Studies Online or Oxford Islamic Studies Online has now moved to Oxford Reference, Oxford Handbooks Online, Oxford Scholarship Online, or What Everyone Needs to Know®. For information on how to continue to view articles visit the subscriber services page.
Show Summary Details

Page of

PRINTED FROM OXFORD REFERENCE (www.oxfordreference.com). (c) Copyright Oxford University Press, 2023. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single entry from a reference work in OR for personal use (for details see Privacy Policy and Legal Notice).

date: 07 December 2023

Russell's paradox

The Oxford Companion to Philosophy
Mark SainsburyMark Sainsbury

Russell's paradox. 

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.

Prof. Mark Sainsbury


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: