This will be a talk for the CUNY Logic Workshop, 23 October 2026, 2pm, at the CUNY Graduate Center in midtown Manhattan.

The Contingent HOD Dichotomy

Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame

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 maximality principles, and with Woodin’s HOD dichotomy.

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