A drawback to the above informal definition is that it requires quantification over all first-order formulas, which cannot be formalized in the standard language of set theory. However, there is a different, formal such characterization: :A set
S is
ordinal definable if there is some collection of ordinals
α1, ...,
αn and a first-order formula
φ taking
α2, ...,
αn as parameters that uniquely defines S as an element of V_{\alpha_1}, i.e., such that
S is the unique object validating
φ(
S,
α2,...,
αn), with its quantifiers ranging over V_{\alpha_1}. The latter denotes the set in the
von Neumann hierarchy indexed by the ordinal
α1. The
class of all ordinal definable sets is denoted OD; it is not necessarily
transitive, and need not be a model of
ZFC because it might not satisfy the
axiom of extensionality. A set further 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 is equivalent to the existence of a (definable)
well-ordering of the universe. Note however that the formula expressing V = HOD need not hold true within HOD, as it is not
absolute for models of set theory: within HOD, the interpretation of the formula for HOD may yield an even smaller inner model. HOD has been found to be useful in that it is an
inner model that can accommodate essentially all known
large cardinals. This is in contrast with the situation for
core models, as core models have not yet been constructed that can accommodate
supercompact cardinals, for example. == References ==