Russell's Paradox
First published Fri Dec 8, 1995; substantive revision Wed May 27, 2009
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.
Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. 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.
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Russell does not offer a solution to the paradox, rather his theory of types avoids the paradox. The function (x e x) cannot be expressed within his type theory.
The answer to Russell's question, "Is the class of those classes that are not members of themselves a member of itself or not?" is No, because it does not exist.
There is a theorem of first order predicate logic, 1. ~EyAx(xRy <-> ~(xRx)),
that resolves the Russell Paradox and the Barber Paradox.
Because of 1, there is no y such that Ax(x e y <-> ~(x e x)),
and there is no y such that Ax(y shaves x <-> ~(x shaves x)).
1. ~EyAx(xRy <-> ~(xRx))
Proof:
1a. Ax(xRy <-> ~(xRx)) -> (yRy <-> ~(yRy)) ..when x has the value y.
2a. (yRy <-> ~(yRy)) is a contradiction for all y. (p <-> ~p) is contradictory.
3a. ~Ax(xRy <-> ~(xRx))
4a. Ay(~Ax(xRy <-> ~(xRx))).
5a. ~EyAx(xRy <-> ~(xRx))
QED.
That is, ~EyAx(x e y <-> ~(x e x)) is true. (it is an instance of 1.)
That is, ~EyAx(y shaves x <-> (x shaves x)) is true. (it is also an instance of 1.)
The Russell class is not a member of any class, including proper classes if such there be. And it has no members, nor does it have any properties.
Both Frege (axiom V of
Grundlagen der Arithmetik) and Cantor, were wrong to assume EyAx(x e y <-> Fx) as an axiom.
That there is a class for every predicate, is false in their systems.