# Question concerning the universe of sets.

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.

• I have the book in front of me. My guess is they are the same. Another possibility is that V is any subclass of the universal class. Could you give an example of when he uses V? – Rachmaninoff Dec 21 '14 at 1:50
• @Rachmaninoff The first usage of $V$ is definition 11.1; I cannot give a page number; for I have the book as an E-book. – Conor O'Brien Dec 21 '14 at 2:45
• You must have a later edition. My Pinter Set Theory ends with chapter 10 with a discussion on inacessible ordinals – Rachmaninoff Dec 27 '14 at 1:38

There are plenty of proper classes which are not the universe of sets, just like how there are many singletons which are not $\{\varnothing\}$.
• So… how do $\mathscr U$ and $V$ fit in? – Conor O'Brien Dec 21 '14 at 4:11
• $\mathscr U$ is a subclass of $V$. It's a collection of some sets, not necessarily all of them. But the working assumption is that it is just too darn big to be a set itself. – Asaf Karagila Dec 21 '14 at 4:12
• Then, the difference is that $\mathscr U\subset V$ (because $\mathscr U$ is the set of all elements being considered?)? – Conor O'Brien Dec 21 '14 at 4:18
• The class $\scr U$ is some class being considered. It need not be equal to $V$ itself. – Asaf Karagila Dec 21 '14 at 4:20