
related topics 
{math, number, function} 
{theory, work, human} 
{work, book, publish} 
{law, state, case} 
{group, member, jewish} 
{government, party, election} 

Russell in 1907
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.
Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the nowcanonical Zermelo–Fraenkel set theory (ZF).
Contents
Informal presentation
Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all nonsquares, that set is itself not a square and so should be one of its own members. It is "abnormal".
Full article ▸


related documents 
Series (mathematics) 
Nonstandard analysis 
Boolean satisfiability problem 
Johnston diagram 
Riemann zeta function 
Functor 
Direct sum of modules 
Integration by parts 
Newton's method 
Principal components analysis 
List of trigonometric identities 
Forcing (mathematics) 
Stone–Čech compactification 
Pascal's triangle 
Infinity 
Cauchy sequence 
Groupoid 
Complete lattice 
Ruby (programming language) 
Braket notation 
Denotational semantics 
Mathematical induction 
Numerical analysis 
Logic programming 
Kernel (algebra) 
Cardinal number 
Sequence alignment 
Gaussian elimination 
Addition 
Entropy (information theory) 
