“Russell’s paradox”的概念、定义、翻译、参考文献-科学参考

Russell’s paradox

Mathematics

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.

Computer

A contradiction originally formulated by Bertrand Russell and phrased in terms of set theory. Let T be the set of all sets that are not members of themselves, i.e.
$T = {S | S \notin S}$
Then it can be shown that T is a member of T if and only if T is not a member of T.
The paradox results from certain kinds of recursive definitions. It arises for example in the following situation: the barber in a certain town shaves everyone who does not shave himself; who shaves the barber?

Philosophy

The most famous of the paradoxes in the foundations of set theory, discovered by Russell in 1901. Some classes have themselves as members: the class of all abstract objects, for example, is an abstract object. Others do not: the class of donkeys is not itself a donkey. Now consider the class of all classes that are not members of themselves. Is this class a member of itself? If it is, then it is not, and if it is not, then it is.
The paradox is structurally similar to easier examples, such as the paradox of the barber. But it is not so easy to say why there is no such class as the one Russell defines. It seems that there must be some restriction on the kinds of definition that are allowed to define classes, and the difficulty is that of finding a wellmotivated principle behind any such restriction. The paradox also bears a resemblance to others such as the liar paradox, but since Ramsey insisted on the distinction it has been usual to distinguish Russell’s paradox, and others in the same family, from the semantic paradoxes of which the liar is a member. See also types, theory of, impredicative definitions.

单词	Russell’s paradox
释义	Russell’s paradox Mathematics 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. Computer A contradiction originally formulated by Bertrand Russell and phrased in terms of set theory. Let T be the set of all sets that are not members of themselves, i.e. $T = {S \| S \notin S}$ Then it can be shown that T is a member of T if and only if T is not a member of T. The paradox results from certain kinds of recursive definitions. It arises for example in the following situation: the barber in a certain town shaves everyone who does not shave himself; who shaves the barber? Philosophy The most famous of the paradoxes in the foundations of set theory, discovered by Russell in 1901. Some classes have themselves as members: the class of all abstract objects, for example, is an abstract object. Others do not: the class of donkeys is not itself a donkey. Now consider the class of all classes that are not members of themselves. Is this class a member of itself? If it is, then it is not, and if it is not, then it is. The paradox is structurally similar to easier examples, such as the paradox of the barber. But it is not so easy to say why there is no such class as the one Russell defines. It seems that there must be some restriction on the kinds of definition that are allowed to define classes, and the difficulty is that of finding a wellmotivated principle behind any such restriction. The paradox also bears a resemblance to others such as the liar paradox, but since Ramsey insisted on the distinction it has been usual to distinguish Russell’s paradox, and others in the same family, from the semantic paradoxes of which the liar is a member. See also types, theory of, impredicative definitions.
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