Tag Archives: V=HOD

The contingent HOD dichotomy, Notre Dame Logic Seminar, September 2026

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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).

Slides – The Contingent HOD Dichotomy – Notre Dame 2026Download

The Contingent HOD Dichotomy, CUNY Logic Workshop, October 2026

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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).

On the contingent contingency of V = HOD and independence over the maximality principles, Fudan University, Shanghai, July 2026

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This will be a talk for the Fudan University logic group on 16 July 4:00pm in Shanghai.

Abstract: The axiom V = HOD is contingently contingent—more precisely, it is class-forcing contingent that V = HOD is contingent with respect to set forcing. It follows that V = HOD and □(V ≠ HOD) are logically independent of the maximality principles MP and MP(ℝ) over ZFC. I will give a full review of the relevant notions. This will be a chalkboard talk concerning very new joint work in progress with Bokai Yao (Peking University).

The paper is in progress and will be available I expect in a few weeks.