Ordinal definable set

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In mathematical set theory, a set S is said to be ordinal definable if there is some collection of ordinals α₁...α_n such that $S \isin V_{\alpha_1}$ and can be defined there by a first-order formula φ taking α₁...α_n as parameters. Here $V_{\alpha_1}$ denotes the set indexed by the ordinal α₁ in the von Neumann hierarchy of sets. In other words, S is the unique object such that φ(S, α₁...α_n) holds with its quantifiers ranging over $V_{\alpha_1}$ .

The class of all ordinal definable sets is denoted OD; it is not necessarily transitive. A set is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable. The class of hereditarily ordinal definable sets is denoted by HOD, and is a transitive model of ZFC, with a definable well ordering. It is consistent with the axioms of set theory that all sets are ordinal definable, and so hereditarily ordinal definable. The assertion that this situation holds is referred to as V = OD or V = HOD. It follows from V = L, and implies that the universe can be well-ordered.