The set of all sets that are not members of themselves is a member of itself only if it is not a member of itself.
Russell's paradox is the most famous of the logical or set-theoretical paradoxes.
The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself.
Similarly, if R is not a member of itself, then by definition it must be a member of itself.
Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics.
The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P ∨ Q by the rule of Addition; then from P ∨ Q and ~P we obtain Q by the rule of Disjunctive Syllogism.
Because of this, and because set theory underlies all branches of mathematics, many people began to worry that, if set theory was inconsistent, no mathematical proof could be trusted completely.
Four responses to Russell's paradox have helped logicians develop an explicit awareness of the nature of formal systems and of the kinds of metalogical and metamathematical results commonly associated with them today.
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