Russell’s Paradox

1. Russell’sParadox Emre ERDOĞAN 01.03.2011 CMPE 220

2. In the foundations of mathematics, Russell's paradox was discovered by Bertrand Arthur William Russell(18 May 1872 – 2 February 1970) in 1901. • The same paradox had been discovered a year before by Ernst Friedrich Ferdinand Zermelo(July 27, 1871 – May 21, 1953)but he did not publish the idea.

3. The Paradox 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.

4. Russell-like Paradoxes • Russell's paradox is not hard to extend. We can take a transitive verb <V>, that can be applied to its substantive form and thenwe can form the sentence: The <V>er that <V>s all (and only those) who don't <V> themselves.

5. An example would be “kiss": The kisser that kisses all (and only those) that don't kiss themselves. • or “punish" The punisher that punishes all that don't punish themselves.

6. Paradoxes that fall in this scheme The barber with "shave". The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves. The Grelling–Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves. Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves.

7. References • http://en.wikipedia.org/wiki/Russell's_paradox • http://en.wikipedia.org/wiki/Bertrand_Russell • http://en.wikipedia.org/wiki/Grelling–Nelson_paradox • http://en.wikipedia.org/wiki/Ernst_Zermelo • http://en.wikipedia.org/wiki/Punisher