  It was also observed by Gödel (1964) (and, independently, by Myhill and Scott 1971, Takeuti 1963 and Post 1951) that a simpler proof of the relative consistency of AC can be formulated in terms of ordinal definability. If we write \(\rD(X)\) for the set of all subsets of \(X\) which are first-order definable in the structure \((X, \in)\), then the class OD of ordinal definable sets is defined to be the union \(\bigcup_{\alpha\in \Ord}  D(V_{\alpha})\).  The class HOD of hereditarily ordinal definable sets consists of all sets \(a\) for which \(a\), the members of \(a\), the members of members of \(a\), … etc., are all ordinal definable. It can then be shown that the structure (HOD, \(\in\)) is a model of ZF + AC , from which the relative consistency of AC  with ZF again  follows.[8]