What is the name of $V_\alpha$?

Question

In the Von Neumann cumulative hierarchy, $V:=\bigcup_\alpha(V_\alpha)$ is called the universe. Is there a name for the individual levels $V_\alpha$?

Just as one can say "The closure of $A$ is defined as $$cl(A):={...}"$$

I would like to be able to say "The _______ of $\alpha$ is defined as $$V_\alpha:={...}"$$

Answer 1

The sets $V_\alpha$ are usually referred to either as levels of the cumulative hierarchy (as mentioned in the comments) or as rank initial segments of V. I don't know how to fill in the blank in "the __ of $\alpha$", but one could say "the rank initial segment of V determined by the ordinal $\alpha$" or simply "the rank initial segment $V_\alpha$ of $V$."

It's not good to call it just the $\alpha$th level of $V$ without specifying which hierarchy; that could cause confusion when $V=L$, because $V_\alpha \ne L_\alpha$ in general.

Answer 2

To give a minor modification to Trevor's answer,

You can say that $V_\alpha$ is "the set of set with von Neumann rank $<\alpha$" (or $\leq\alpha$, depending on your definition of rank).

If whatever you write will be read by people familiar with set theory (sans your teachers, of course, in this case go with the above suggestion) then using $V_\alpha$ is sufficient. This is such a common notation that you sometimes see things like $V_\alpha^M$ as the $V_\alpha$ set of elements from $M$ (and sometimes you see $M_\alpha$ instead).