What is the difference between the notation $\exists v(v=v)$ and $V:=\{x:x=x\}$

Question

This comes from Kunen, "Set Theory," p. 21:

He says: the statement there is some $v$ (our universe is non-empty or $\exists v(v=v)$) is a logical fact.

But a set that doesn't exist is our universe, $V:=\{x:x=x\}$.

What is the difference, in view of the seeming similarity $v=v$ and $x:x=x$?

I am convinced that the $V$ as mentioned does not exist as a set - e.g., Russell's Paradox, etc. It's just the similarity of notation that I'm wondering about.

Thanks

Answer 1

There exist sets which are equal to themselves, but the collection of all such sets is not a set.

In other words, let $\phi(x)$ be the formula $x=x$. Then there exists sets such that $\phi(x)$, but $\{x\mid \phi(x)\}$ is not a set.