# Ordinal definable set

A drawback to this informal definition is that requires quantification over all first order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating the definition that can be so formalized. In this approach, a set S is formally defined to be ordinal definable if there is some collection of ordinals α1...αn such that $S \isin V_{\alpha_1}$ and $S$ can be defined as an element of $V_{\alpha_1}$ by a first-order formula φ taking α2...αn 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, α2...αn) holds with its quantifiers ranging over $V_{\alpha_1}$.