Any ideas on how to do this one?
Let A be a set. Show that a complement of A does not exist. So I need to show that there isn't a set of all x not in A.
Hi,
$\displaystyle A$ is a set and $\displaystyle B=\{x:\, x \not \in A\}$ ?
Assuming axioms of ZF set theory, $\displaystyle B$ can't be a set: If $\displaystyle B$ were a set, then since $\displaystyle A$ is a set, axiom of pairing gives that $\displaystyle \{A,B\}$ is a set. Axiom of union then tells us that that there exist a set $\displaystyle C$ whose elements are elements of $\displaystyle A$ and elements of $\displaystyle B$. But from the definition of $\displaystyle B$ we get that $\displaystyle C$ is the universal class of all sets, which is a proper class, i.e. is not a set. This is a contradiction, thus $\displaystyle B$ can't be a set.
Can you see why the class of all sets is not a set?