Gunter Fuchs, CUNY
The blurry HOD hierarchy
Classically, an object is ordinal definable if it is the unique one satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal $\kappa$, I call an object $\lt\kappa$-blurrily ordinal definable if it belongs to an ordinal definable set with fewer than $\kappa$ elements (called a $\lt\kappa$-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, indexed by cardinals $\kappa$: the collection of all hereditarily $\lt\kappa$-blurrily ordinal definable sets, which I call $\lt\kappa$-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.
The special cases $\kappa=\omega$ and $\kappa=\omega_1$ have been previously considered, but no systematic study of the general setting has been carried out, it seems. One main aspect of the analysis is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable.
In this talk, I will focus on the ZFC-provable structural properties of the blurry HOD hierarchy, which turn out to be surprisingly plentiful. So for the most part, the talk will be forcing-free. Time permitting, I may hint at the result of the equiconsistency between the least leap being the successor of a singular strong limit cardinal and the existence of a measurable cardinal, for which, admittedly, forcing is used in one direction.