October 23
Joel David Hamkins,
Notre Dame University
The contingent HOD dichotomy
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 $V\neq HOD$; 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).