can someone explain the proof of russels paradox (barber)?

Question

So I understand Russels paradox (barber) but I do not understand the proof, I've looked everywhere online and youtube videos but it doesn't seem to make sense.

Please note, I have compensated dyslexic so I find it hard to interpret and understand text. I think this is the reason why I am finding it difficult to understand the proof.

This is as much as I understand: P(x) <----> x is not a member of itself. What I don't understand is if it means the element x is not a member of itself? And if so, I do not understand how a element can be a member of itself.

Answer 1

HINT:

Don't get hung up on the set-theoretic $\in$ relation. The proof works for every binary relation.

Consider the statement: $$\exists r: \forall x:[E(x,r) \iff \neg E(x,x)]$$ where $E$ is just some logical predicate having nothing to do with set membership.

Show how, from the above statement, you can obtain the contradiction: $$E(r,r) \iff \neg E(r,r)$$ What would this tell you about the original statement?

Answer 2

Don't use set-theoretic notation for the barber paradox. The proof is just:

Take any barber who cuts the hair of exactly those who don't cut their own hair.

Either the barber cuts his own hair or he does not.

If he does cut his own hair, then by his own rule he is not supposed to cut his hair.

If he does not cut his own hair, then by his own rule he is supposed to cut his hair.

In all cases we get a contradiction, and so such a barber does not exist.