# Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My understanding is that ZFC doesn't support ordinal-indexed sequences, so my question is, what is a good set theory in which to study this concept?

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Note that the class-length sequence $V_\alpha$ is definable, so given a set $X$, so is the sequence $V_\alpha\setminus X$ going to be definable (via the parameter $X$).
If you are going to talk about collections of classes, then perhaps it is easier to assume an inaccessible cardinal exists, and have two levels of universes: $V_\kappa$ as the world of sets, and its classes are also sets in the larger universe, allowing you to talk about "all complements" or so, if you'd like.
I was actually hoping to avoid classes, because I was hoping to model "large" sets (or "classes") as ordinal-indexed sequences as follows. Suppose one wishes to study $R = \{x | x \notin x\}$, for example. One would instead study $R_{\alpha} = \{x \in V_{\alpha} | x \notin x\}$. (I'm not sure of this idea works, though.) A "large set" or "proper class" would be an object such that this sequence never becomes constant. That being said, perhaps a class theory such as NBG would be a good context in which to study this idea. –  goblin Oct 6 '12 at 1:30
Yianni, note that $R$ is $V$, the universe, in ZF due to the axiom of regularity which implies $x\notin x$ for all $x$. It is not clear what you are planning to do while studying; it may be sufficient to use ZF after all. I suggest that you work on your set theory a bit first and learn how classes are dealt with within ZFC. –  Asaf Karagila Oct 6 '12 at 10:00