Is there a set that contains all sets?
This question was answered in our Set Theory course by providing Russell's Paradox. I understand the logic behind Russell's Paradox and that there exists no set whose condition is not being a member of itself.
However, how is this directly relevant to the question of whether there is a set that contains all sets?
How does it contradict the existence of such a universal set?
On
Well if you defined a "collection" $U$ like this, $U = \{ u \in U \ iff\ u\ a\ set\}$. Your statement is about $U$. In the definition I put, I defined the elements of $U$ but I did not qualify it as a set. Your statement asserts that the statement which asserts that $U$ defined above can not be a set.
The problem isnt that a set is in itself, it's that there is no bound or maximal set.
Russell's paradox is based on the assumption that if $A$ is a set and $P$ is a predicate then $\{ x \in A : P(x) \}$ is a set. It tells us that if $A$ is allowed to be a set of all sets, then $P$ can't be an arbitrary predicate; and if $P$ is allowed to be any predicate, then $A$ can't be allowed to be a set of all sets. Either one of these approaches can form an ostensibly consistent set theory. ZF set theory happens to choose the second one, but NF set theory chooses the first one instead.
In other words, there is no inherent contradiction in defining a universal set, but if you do, then you can't use comprehension by arbitrary predicates anymore.