I am reading Charles Pinter's Introduction to Set Theory
Every proper class is in one-to-one correspondence with the universal class $\mathscr{U}$, that is, the class of all sets [emph. added].
He uses the term in the beginning of the book, and then makes a sudden transition to the term "universe of sets," denoted by $V$. What's the difference between $\mathscr{U}$ and $V$? If they are the same,why use different terms?
Any clarification is appreciated.
There are plenty of proper classes which are not the universe of sets, just like how there are many singletons which are not $\{\varnothing\}$.
For example the class of all singletons, or the class of all pairs, or the class of power sets, or the class of ordinals, or the class of successor ordinals, or the class of ... you get the point.
It should be noted, perhaps, that the assumption that every two classes have a bijection between them is equivalent to the axiom of global choice, stating that given a proper class of non-empty sets there is a class-function which chooses one from each of the sets.