I will continue to attempt to present my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy, including the HOD conjecture. Principally, I hope to present how weak extender models relate to the HOD conjecture. I will also present the initial part of new results that were inspired by the HOD conjecture, in particular, by failures of the cover property.

I will be presenting my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy. In particular I will be discussing Woodin’s version and definition of a model of set theory being “far from HOD”. I will also be discussing how this relates to some possible future results (see title) as well as a recent result of Cheng,Friedman and Hamkins, which seem on their face to contradict Woodin’s premise.