Difference between revisions of "Russell's Paradox"
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| − | + | The '''Russell's Paradox''', credited to Bertrand Russell, was one of those which forced the axiomatization of set theory. | |
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| + | ==Paradox== | ||
| + | We start with the property <math>P</math>: (<math>x</math> does not belong to <math>x</math>). We define <math>C</math> to be the collection of all <math>x</math> with the property <math>P</math>. Now comes the question: does <math>C</math> have the property <math>P</math>? Assuming it does, it cannot be in itself, in spite of satisfying its own membership criterion, a contradiction. Assuming it doesn't, it must be in itself, in spite of not satisfying its own membership criterion. This is the paradox. | ||
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| + | ==See Also== | ||
| + | *[[Set]] | ||
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| + | [[Category:Set theory]] | ||
Revision as of 07:32, 25 November 2007
The Russell's Paradox, credited to Bertrand Russell, was one of those which forced the axiomatization of set theory.
Paradox
We start with the property 
: (
does not belong to ). We define 
to be the collection of all with the property . Now comes the question: does have the property ? Assuming it does, it cannot be in itself, in spite of satisfying its own membership criterion, a contradiction. Assuming it doesn't, it must be in itself, in spite of not satisfying its own membership criterion. This is the paradox.
