Inmathematicalset theory, asetS is said to beordinal definable if, informally, it can be defined in terms of a finite number ofordinals by afirst-order formula. Ordinal definable sets were introduced byGödel (1965).
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:
The latter denotes the set in thevon Neumann hierarchy indexed by the ordinalα1. Theclass of all ordinal definable sets is denoted OD; it is not necessarilytransitive, and need not be a model ofZFC because it might not satisfy theaxiom of extensionality.
A set further ishereditarily ordinal definable if it is ordinal definable and all elements of itstransitive closure are ordinal definable. The class of hereditarily ordinal definable sets is denoted by HOD, and is atransitive 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 fromV = 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 notabsolute 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 aninner model that can accommodate essentially all knownlarge cardinals. This is in contrast with the situation forcore models, as core models have not yet been constructed that can accommodatesupercompact cardinals, for example.