Reductio and Wishful Thinking:

Enderton, in his excellent introduction "Elements of Set Theory" (1976) shows why, in his official opinion of expert of Set Theory, the set of all sets that do not belong to themselves cannot be a set and, most importantly, why the subset formed out of the elements of the suspicious set, is itself not a member of that set, and is therefore contradiction free.

Here the whole argumentation:

Theorem 2A There is no set to which every set belongs. 
Proof Let A be a set; we will construct a set not belonging to A. Let 
В = {x  ε A | x ф x}. 
We claim that В ф A. We have, by the construction of B, 
B ε В iff B ε А&ВфВ. 
If BεA, then this reduces to 
BεB iff ВфВ, 
which is impossible, since one side must be true and the other false. Hence 
ВфА. 

[I had to replace the gibberish character with iff (if and only if). ф  means the negation of the epsilon symbol, that is " not a member of" or "does not belong to".]

This could be a nice example of a reductio were it not for a fundamental difference. The reductio argument assumes a premise which is shown to lead to a contradiction. In other words, the contradiction is created by the assumption even though it needs not be. When Euclid, as (one of) the first used this form of proof to prove the non-existence of a highest prime, and therefore of the infinity of primes, he did not create a contradiction by affirming: 

1) There is a highest prime. 

There is nothing self-contradictory in this assertion. Only, the argumentation shows that we necessarily arrive at the negation of this assertion: 

2) There is no highest prime.

And that is the end of it. (1) is taken out of the running once and for all. It has been proven to be a false assertion.

This is certainly not the case with 

3) The set B (whose members, belonging to A, do not belong to themselves) is not a member of A.

This proposition Is not proven ad reductio by

4) The set B (whose members, belonging to A, do not belong to themselves) is a member of A.

The proposition (4) is not taken out of the running since it has not been proven to be a false assertion.

Both (3) and (4) form both horns of the dilemma. Asserting one is the same as asserting the other. Affirming that it "is impossible, since one side must be true and the other false" is a pure declaration of faith that does not make the antinomy go away.