Good set theory in which to study ordinal-indexed sequences? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:11:17Z http://mathoverflow.net/feeds/question/108971 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108971/good-set-theory-in-which-to-study-ordinal-indexed-sequences Good set theory in which to study ordinal-indexed sequences? user1837 2012-10-06T00:52:14Z 2012-10-06T01:31:54Z <p>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?</p> http://mathoverflow.net/questions/108971/good-set-theory-in-which-to-study-ordinal-indexed-sequences/108972#108972 Answer by Asaf Karagila for Good set theory in which to study ordinal-indexed sequences? Asaf Karagila 2012-10-06T00:56:51Z 2012-10-06T00:56:51Z <p>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 $X$).</p> <p>If you intend to use more, perhaps a theory like ZFC+Global choice; or NBG which is more suitable for handling proper classes.</p> <p>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.</p>