Russell's Paradox

Bertrand Russell's Paradox revealed a flaw in 1901 in early or naive set theory. (It is likely others saw this flaw before Bertrand Russell.) It is a form of the liar's paradox expressed in the terms of set theory.

The defect is as follows. Let S be the set of all sets that do not contain themselves as members. In other words, set T is an element of S if, and only if, T is not an element of T:

$S=\{T\mid T\not \in T\}.$

Here is the flaw. Is Set S a member of itself? If S is a member of itself, then it cannot be a member of itself by the very definition of S. But if S is not a member of itself, then it must be a member of itself, again by its very definition. Hence there is a fundamental logical contradiction in this type of set, and in any theory that allows it.

Russel's paradox and others like it can be avoided only at the expense of giving up the idea that any criteria can be used to construct sets. For example in Zermelo-Frankel set theory the construction $S=\{T\mid T\not \in T\}$ would not qualify as well-founded. Some formulations of set theory would allow such a construction but then S would be a proper class rather than a set.

External Links

Russell's Paradox at Stanford Encyclopedia of Philosophy