Logic Workshop

February 2
Gunter Fuchs, CUNY
Blurry HOD and the structure of leaps

For a cardinal $\kappa\ge 2$, one can weaken the concept 'x is ordinal definable' (i.e., x is the unique object satisfying some condition involving ordinal parameters) to 'x is $\lt\kappa$-blurrily ordinal definable,' meaning that x is one of fewer than $\kappa$ many objects satisfying some condition involving ordinal parameters. By considering the hereditary version of this, one naturally arrives at the inner model $\lt\kappa$-HOD, the class of all hereditarily $\lt\kappa$-blurrily ordinal definable sets. In ZFC, by varying $\kappa$, one obtains a hierarchy of inner models spanning all the way from HOD to V. The leaps are those stages in the hierarchy where something new is added. I have previously given a logic workshop talk about the basic theory of the blurry HOD hierarchy, and in this talk, after reviewing the basics, I want to focus on the consistency strengths of certain leap constellations, ranging from outright consistent, to equiconsistent with a measurable cardinal, to inconsistent.