This will be a talk for the Logic Seminar at the University of Notre Dame on 15 September 2026, 2pm in the Hayes-Healy building room 125.

The contingent HOD dichotomy

Joel David Hamkins

Abstract: We shall discuss the contingently contingent nature of the class HOD of hereditarily ordinal-definable sets. In some models of set theory the axiom V=HOD is contingent by set forcing and in others it is not. After discussing some philosophical and historical puzzles concerning the nature of ordinal definability, I shall introduce and investigate what we call the contingent HOD dichotomy. Namely, it is provable in ZFC that either (1) the set-theoretic universe V is close to HOD in several respects: HOD is a ground model of V; the axiom V=HOD is forceable by set forcing; every object is ordinal-definable with a single additional parameter; and furthermore all these things are forcing invariant and hold throughout the generic multiverse; or (2) the universe is far from HOD; in particular V ≠ HOD;  more generally, HOD is not a ground; the axiom V=HOD is not forceable; the universe is not ordinal-definable from a parameter; and these things hold invariantly throughout the generic multiverse. We shall explore how the dichotomy engages with large cardinals, with set-theoretic geology, with the iterated HOD^alpha models, with the maximality principles, and with Woodin’s HOD dichotomy. 

This is new joint work in progress with Bokai Yao (Peking University).