Emre ERDOĞAN CMPE 220

In the foundations of mathematics, Russell's paradox was discovered by Bertrand Arthur William Russell (18 May 1872 – 2 February 1970) in 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.

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.

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

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.

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.

References Nelson_paradox Nelson_paradox