Blurry HOD - a sketch of a landscape
Classically, a set is ordinal definable if it is the unique object satisfying a formula with ordinal parameters. Generalizing this concept, given a cardinal κ, I call a set <κ-blurrily definable if it is one of less than κ many objects satisfying a formula with ordinal parameters (called a <κ-blurry definition). By considering the hereditary versions of this notion, one arrives at a hierarchy of inner models, one for each cardinal κ: the collection of all hereditarily <κ-blurrily ordinal definable sets, which I call <κ-HOD. In a ZFC-model, this hierarchy spans the entire spectrum from HOD to V.
The special cases κ=ω and κ=ω1 have been previously considered, but no systematic study of the general setting has been done, it seems. One main aspect of the study is the notion of a leap, that is, a cardinal at which a new object becomes hereditarily blurrily definable. The talk splits into two parts: first, the ZFC-provable properties of blurry HOD, which are surprisingly rich, and second, the effects of forcing on the structure of blurry HOD and the achievable leap constellations.
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