Blurry Definability

I begin the study of a hierarchy of (hereditarily) <kappa-blurrily ordinal definable sets. Here for a cardinal kappa, a set is <kappa-blurrily ordinal definable if it belongs to an OD set of cardinality less than kappa, and it is hereditarily so if it and each member of its transitive closure is. I show that the class of hereditarily <kappa-blurrily ordinal definable sets is an inner model of ZF. It satisfies the axiom of choice iff it is a kappa-c.c. forcing extension of HOD, and HOD is definable inside it (even if it fails to satisfy the axiom of choice). Of particular interest are cardinals lambda such that some set is hereditarily <lambda-blurrily ordinal definable but not hereditarily <kappa-blurrily ordinal definable for any cardinal kappa <lambda. Such cardinals I call leaps. The main results concern the structure of leaps. For example, I show that if lambda is a limit of leaps, then the collection of all hereditarily <lambda-blurrily ordinal definable sets is a model of ZF in which the axiom of choice fails. Using forcing, I produce models exhibiting various leap constellations, for example models in which there is a (regular/singular) limit leap whose cardinal successor is a leap. Many open questions remain.