Russell Paradox

AKA WhoShavesTheBarber?. Related to, but not identical to, the CretanParadox.

The paradox associated with BertrandRussell's name shows that the naive idea of identifying a set with a predicate that selects its members from the set of all existing things (the all-set, or universe) leads to inconsistencies. Namely: A given set may or may not contain itself as a member. Now consider the set consisting exactly of those sets that do not contain themselves as a member. Then this set contains itself as a member if and only if it does not contain itself as a member. See ImpossibleSet.

At a more fundamental level: there are "not-quite sets" called ProperClass?es that you cannot think of ever being finally specified because their definition permits for a process to enlarge them whenever you think of them as finished entities.

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This could be some more verbose. I'll give it a try.

The RussellParadox was given in the beginning of the 20th century by BertrandRussell. It shows that careless handling of sets can lead to difficulties, or, actually, to severe paradoxes. Namely, he said the following: we all know what sets are. Now, a set may or may not have the property P which is that it contains itself.

Most sets we know of don't contain themselves, obviously. For example, the set {red,blue,green} does not contain itself, and the set of natural numbers does not contain itself. Now, we could imagine a set S which contains only itself, or say, a set T which contains S, 1 and itself. This is something surprising at first, but quite ok.

Now, consider all sets which do not contain itself (these are 'most' sets, as we saw). Let's call the set which contains all these sets A. The question now is: is A contained in A? Unfortunately, this already leads to problems:

• if A is in A, it certainly contains itself, thus A should not be in A.
• if A is not in A, it obviously does not contain itself, thus it should be in A.

Only after a second thought, one sees that there is not so much spectacular here. Obviously, we did something which we should not have done; something that sets do not allow. People then started on a more strict level, and namely assumed (the somewhat hidden assumption) that there exists no set which contains all sets; or more explicitly, that the set A does not exist. Actually, what is usually done is just to give specific axioms, the ZermeloAxioms?, which say what sets exist - and then one may not take any set for granted, unless it is given by these axioms. One of these axioms is the AxiomOfChoice, which has lead to many controversies in mathematics.

See SetOfAllSets. Also see the book NaiveSetTheory.

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