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