Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
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. – arjafi 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
up vote 3 down vote accepted

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.

share|cite|improve this answer
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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.