Russell’s Paradox

I can’t be the only one who looked up five Youtube videos and seven websites to fully understand what Russell’s Paradox is all about. Before I explain, to help you save time and effort, here are three key points I learned:

(1) “a is not in a” is the basic premise of the paradox. It defines a set that cannot exist.

(2) Barber paradox is an easy illustration of the paradox. If we imagine a small town where everyone needs to be shaved, and define a barber as someone “who shaves all those, and those only, who do not shave themselves”, who shaves the barber?

The barber cannot shave himself as he only shaves those who do not shave themselves. As such, if he shaves himself he ceases to be the barber.

If the barber does not shave himself, he needs to be shaved by a barber, and thus, as the barber, he must shave himself — but we defined a barber to be someone who only shaves those who do not shave themselves. See the paradox?

(3) This was meant to show the logical flaw of the naive set theory, so if you’re wondering if the paradox has an answer — there is no answer.

Naive set theory: Given any property there exists a set containing all objects that have that property.

example: S = {x: T(x)}

The paradox arises within naïve 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 (Stanford.edu)

Russell’s paradox becomes: let y = {x: x is not in x}, is y in y (Scientific American)

Conclusion: Russell’s paradox was used to question a particular logic. There is no answer because the question is flawed to begin with. The moral of the paradox is to set better definitions, a set of axioms that clarify the case.

P.S. (In case you want to “solve” the paradox, read about Zermelo-Fraenkel set theory).