Why every small set is a moderate set, but not conversely (as universe itself)? I read here > a subset of the universe, which may be small or large. (every small set is moderate, but not conversely; again, the universe itself is the standard counterexample.) But I don't understand well because the universe itself is the standard counterexample, i.e why every small set is moderate, but not conversely ? This because universe is a pure set ? This situation is for me cryptic. **Why can't the universe belong to itself ?**

The article you link is about foundations in general. In each foundation the precise reason that a universe cannot contain itself may be different.

For example, in ZFC, the axiom of regularity forbids that any set contains itself. So in that system, if we have a set that functions as a universe, it cannot contain itself.

The more general answer is that having a universe contain itself would give us Russell's paradox:

Most foundations will have some form of separation. That is, given a set $X$ in our universe, and some first-order formula $\varphi$, we can form a set $$ Y = \\{x \in X : \varphi(x)\\}, $$ such that $Y$ is again in our universe. If our universe $U$ contains itself, we could build the set $$ R = \\{ x \in U : x \
ot \in x\\}, $$ which would be in $U$. But now we have that $R \in R$ if and only if $R \
ot \in R$, which is problematic.