I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\beta \mapsto V_\beta \setminus X$ where $V_\beta$ is the $\beta$ 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?

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?