| 释义 |
Russell's paradox By using the notation of set theory, a set can be defined as the set of all x that satisfy some property. Now it is clearly possible for a set not to belong to itself: any set of numbers, say, does not belong to itself because to belong to itself the set would have to be a number. But it is also possible to have a set that does belong to itself: for example, the set of all sets belongs to itself. In 1901, Bertrand Russell drew attention to the following paradox, by considering the set R={x | x ∉ x}. If R ∈ R, then R fails the condition for being an element of R and so R ∉ R; and if R ∉ R, then R ∈ R by definition. The paradox points out the necessity of defining mathematical objects carefully (compare Perron's paradox). For example, such a set R cannot be defined using the Zermelo-Frankel axioms, so no paradox arises; likewise the ‘set of all sets' does not exist within ZF theory.
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