Say universe $U$ is "complete" iff every $U$-small set is a $U$-set. Does there exist a "complete" universe?
Definitions: A $U$-set is an element of $U$. A $U$-small set is isomorphic (i.e. bijective) to an element of $U$.
I'm reading about category theory and I thought of the above question. The reason is that there seems to be no "real difference" between $U$-small sets and $U$-sets. It seems like any such $U$ would be "too big to be a set," but I don't know how to prove this.
I figured out the answer.
If $U$ were "complete," then $\{U\} \in U$, a contradiction.