Blurry Definability

Abstract

I begin the study of a hierarchy of (hereditarily) <κ-blurrily ordinal definable sets. Here for a cardinal κ, a set is <κ-blurrily ordinal definable if it belongs to an OD set of cardinality less than κ, and it is hereditarily so if it and each member of its transitive closure is. I show that the class of hereditarily <κ-blurrily ordinal definable sets is an inner model of ZF. It satisfies the axiom of choice iff it is a κ-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 λ such that some set is hereditarily <λ-blurrily ordinal definable but not hereditarily <κ-blurrily ordinal definable for any cardinal κ<λ. Such cardinals I call leaps. The main results concern the structure of leaps. For example, I show that if λ is a limit of leaps, then the collection of all hereditarily <λ-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.