# What is the name of $V_\alpha$?

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 $$V_\alpha:={...}"$$

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The $\alpha$th level of the cumulative hierarchy?... The family of all sets of rank less than $\alpha$?... I don't recall hearing a zippy name applied to these families. – Arthur Fischer Aug 26 '12 at 14:15
Well "the $\alpha$th level ..." is more creative than anything I was able to come up with just now. – Travis Bemrose Aug 26 '12 at 14:16

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.

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Welcome Trevor! – Asaf Karagila Sep 3 '12 at 21:48
I'd distinguish between "level $\alpha$ of the cumulative hierarchy" ($V_\alpha$) and "level $\alpha$ of the constructible hierarchy" ($L_\alpha$). – Carl Mummert Sep 3 '12 at 21:51
Thanks, Asaf! (Also, I will now edit my answer to reflect Carl's comment.) – Trevor Wilson Sep 3 '12 at 21:58

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).

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