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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 intend to use more, perhaps a theory like ZFC+Global choice; or NBG which is more suitable for handling proper classes.

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.

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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
"I suggest that you work on your set theory a bit first and learn how classes are dealt with within ZFC." Yes, I would like to do that, but I thought classes were NOT dealt with within ZFC? I would like to study a set theory that actually allows these sorts of sequences. – goblin Oct 6 '12 at 21:09
Yes, classes are not real elements in ZFC but it doesn't mean we cannot say things about them, and prove things about them. There are a lot of delicate points, but it is mathematically valid to do that. This is like saying that you cannot prove anything in PA about infinite sets (e.g. all even numbers can be divided by two). I wrote some math.SE answers about this topic, – Asaf Karagila Oct 6 '12 at 21:53

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