RationalWiki's 2018 Fundraiser

There is no RationalWiki without you. We are a small non-profit with no staff — we are hundreds of volunteers who document pseudoscience and crankery around the world every day. We will never allow ads because we must remain independent. We cannot rely on big donors with corresponding big agendas. We are not the largest website around, but we believe we play an important role in defending truth and objectivity.

If everyone seeing this today donates $5, we will meet our goal for 2018.

Fighting pseudoscience isn't free.
We are 100% user-supported! Help and donate $5, $20 or whatever you can today with PayPal Logo.png!

Donations so far: $3383Goal: $5000

Russell's paradox

From RationalWiki
Jump to: navigation, search
Part of a
convergent series on


Russell's paradox is as follows: Consider the set of all sets which are not members of themselves. Is this set a member of itself? If it is, then it is not. If it is not, then it is. A contradiction ensues. In an interesting application, this can be exploited to preclude the existence of a bijection between any set and its power set. The result is an elegant proof that the cardinality of the latter is always larger.

Bertrand Russell's paradox is a consequence of the axiom of comprehension and the existence of a universal set. Hence, in ZF set theory, the universal set does not exist. Alternatively, one could avoid Russell's paradox and retain a universal set by rejecting the axiom of comprehension.

Another version of this paradox goes: Does a barber, who shaves all those and only those who do not shave themselves, shave himself? If he shaves himself, then he is not one of those who do not shave themselves, and since he only shaves those who do not shave themselves, then he does not shave himself. If he doesn't shave himself, then he is one of the those who do not shave themselves, and since he shaves all those who do not shave themselves, then he shaves himself.

See also[edit]