Difference between revisions of "Russell's Paradox"
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Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it. | Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it. | ||
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| + | ==External Links== | ||
| + | :[http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox at Stanford Encyclopedia of Philosophy] | ||
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[[Category: set theory]] | [[Category: set theory]] | ||
Revision as of 01:48, April 14, 2007
Bertrand Russell's Paradox revealed a flaw in 1901 in early or naive set theory. (It is likely others saw this flaw before Bertrand Russell.)
The defect is as follows. Let S be the set of all sets that do not contain themselves as members. In other words, set T is an element of S if, and only if, T is not an element of T:
Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it.