TITLE: How is Russell’s Paradox equal to the universal set?
QUESTION [7 upvotes]: How is Russell’s Paradox, which is defined as the collection
$$
\{ x \mid x \notin x \},
$$
equal to the universal set $ \{ x \mid x = x \} $? Do we assume that if $ x = x $, then $ x \notin x $? My question is specifically what is written at the end of Page 21 of this document.
Thanks for any help!

REPLY [8 votes]: You seem confused about the relationship between Russell's Paradox and the non-existence of the so-called universal set. I will try to clarify without referring to the notion of a class.
If you assume the existence of a universal set $U$ such that $\forall x:x\in U$ and your set theory allows for arbitrary subsets and does not disallow $x\in x$, then you can define set $R$ such that $\forall x:[x\in R\iff x\in U \land x\notin x]$. Note the similarities to the standard presentation of Russell's Paradox.
Applying the definition of $R$ to itself, we have $R\in R \iff R\in U \land R\notin R$. Since, by definition, we must have $R\in U$, this is a contradiction. Thus, the existence of $U$, as defined above, results in a contradiction. Thus, $U$ cannot exist. 
Thus, the non-existence of the universal set can be proven (in the set theories described here) by using the same kind of contradiction that arises in Russell's Paradox