universal set paradox
 

"Try to take the power set of the universal set. You will find that the universal set must be a member of itself and not a member of itself

Russell does use the Russell set and hierarchy to solve this problem."

I understand the contradiction of russell set, but not of the universal set. Why must the universal set not be a member of itself. How does russell use the russell set to solve the problem. I understnd the theory of types, the hierarchy, but not why russell's set is needed to solve the problem. I also understand the contradiction that the power set must be bigger and the same size as universal set but not the contradiction noted above. Thanks.

It's been a while since I've studied set theory, but in a nutshell, every set has to have a domain, so there is no such thing as the universal set.

What if there were a set whose members are all the members of the power set of this set, so x is a member of S if x is a member of P(S). Doesn't this violate Cantor's theorem, and also is not able to be resolved by Russell's hierarchy?

I came up with that myself.

Do you mean x is a member of S if and only if x is a member of P(S)? I think you could use Cantor's theorem to show that such a set is impossible -- take f(x) -> x. Since this function is one-to-one (forgive me if I'm forgetting terminology here), S = P(S). But this is impossible, so your definition of a set defines a different set.

I think that's just an impossible set, and so a bad definition of a set. But I'd really feel more confident if someone who's had more set theory could confirm this. Maybe if I have time today or tonight, I'll try to refresh my memory a bit better.