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!
Best Answer
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