# Blog Archives

# Topic Archive: HOD dichotomy

# Deciphering the HOD Dichotomy: Part II

I will continue to attempt to present my version of the notes of Woodinâ€™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.

# Does the HOD dichotomy preclude a model of a proper class of supercompact cardinals which are not HOD-supercompact?

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.