The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD

TL;DR

This paper demonstrates that the universe V need not be a class-forcing extension of HOD, providing counterexamples and exploring the necessity of augmenting HOD with certain classes to achieve such extensions.

Contribution

It proves that there exist models of ZFC where V is not a class-forcing extension of HOD, even with definable class forcing notions, highlighting limitations of the class-forcing approach.

Findings

Counterexamples to V being a class-forcing extension of HOD

Augmentation of HOD with a definable class can make V a class-forcing extension

Existence of intermediate models not obtainable via class forcing

Abstract

In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amenable class $A$, definable in $V$, then the set-theoretic universe $V$ is a class-forcing extension of the expanded structure $\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentation process can be necessary. The same example shows that $V$ is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, namely, a class-forcing extension $V\subseteq W\subseteq V[G]$ with an intermediate transitive inner model $W$ that is neither a class-forcing extension of $V$ nor a ground model of $V[G]$ by any definable class forcing notion with definable forcing relations.