[bibtex key=”HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD”]

Abstract. In light of the celebrated theorem of Vopěnka, proving in ZFC that every set is generic over $\text{HOD}$, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of $\text{HOD}$ by some possibly proper-class forcing notion in $\text{HOD}$. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its $\text{HOD}$ for any class forcing notion definable in $\text{HOD}$ and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments $\text{HOD}$ 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 $⟨\text{HOD},\in ,A⟩$. 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 a counterexample to the intermediate model property, namely, a class-forcing extension $V\subseteq V[G]$ by a certain definable tame forcing and a transitive intermediate inner model $V\subseteq W\subseteq V[G]$ with $W\models \text{ZFC}$, such that $W$ is not a class-forcing extension of $V$ by any class forcing notion with definable forcing relations in $V$. This improves upon a previous example of Friedman (1999) by omitting the need for $0^{\mathrm{♯}}$.

In 1972, Vopěnka proved the following celebrated result.

Theorem. (Vopěnka) If $V=L[A]$ where $A$ is a set of ordinals, then $V$ is a forcing extension of the inner model $\text{HOD}$.

The result is now standard, appearing in Jech (Set Theory 2003, p. 249) and elsewhere, and the usual proof establishes a stronger result, stated in ZFC simply as the assertion: every set is generic over $\text{HOD}$. In other words, for every set $a$ there is a forcing notion $\mathbb{B}\in \text{HOD}$ and a $\text{HOD}$-generic filter $G\subseteq \mathbb{B}$ for which $a\in \text{HOD}[G]\subseteq V$. The full set-theoretic universe $V$ is therefore the union of all these various set-forcing generic extensions $\text{HOD}[G]$.

It is natural to wonder whether these various forcing extensions $\text{HOD}[G]$ can be unified or amalgamated to realize $V$ as a single class-forcing extension of $\text{HOD}$ by a possibly proper class forcing notion in $\text{HOD}$. We expect that it must be a very high proportion of set theorists and set-theory graduate students, who upon first learning of Vopěnka’s theorem, immediately ask this question.

Main Question. Must the set-theoretic universe $V$ be a class-forcing extension of $\text{HOD}$?

We intend the question to be asking more specifically whether the universe $V$ arises as a bona-fide class-forcing extension of $\text{HOD}$, in the sense that there is a class forcing notion $\mathbb{P}$, possibly a proper class, which is definable in $\text{HOD}$ and which has definable forcing relation $p⊩\phi (\tau )$ there for any desired first-order formula $\phi$, such that $V$ arises as a forcing extension $V=\text{HOD}[G]$ for some $\text{HOD}$-generic filter $G\subseteq \mathbb{P}$, not necessarily definable.

In this article, we shall answer the question negatively, by providing a model of ZFC that cannot be realized as such a class-forcing extension of its $\text{HOD}$.

Main Theorem. If ZFC is consistent, then there is a model of ZFC which is not a forcing extension of its $\text{HOD}$ by any class forcing notion definable in that $\text{HOD}$ and having a definable forcing relation there.

Throughout this article, when we say that a class is definable, we mean that it is definable in the first-order language of set theory allowing set parameters.

The main theorem should be placed in contrast to the following result of Sy Friedman.

Theorem. (Friedman 2012) There is a definable class $A$, which is strongly amenable to $\text{HOD}$, such that the set-theoretic universe $V$ is a generic extension of $⟨\text{HOD},\in ,A⟩$.

This is a postive answer to the main question, if one is willing to augment $\text{HOD}$ with a class $A$ that may not be definable in $\text{HOD}$. Our main theorem shows that in general, this kind of augmentation process is necessary.

It is natural to ask a variant of the main question in the context of set-theoretic geology.

Question. Must the set-theoretic universe $V$ be a class-forcing extension of its mantle?

The mantle is the intersection of all set-forcing grounds, and so the universe is close in a sense to the mantle, perhaps one might hope that it is close enough to be realized as a class-forcing extension of it. Nevertheless, the answer is negative.

Theorem. If ZFC is consistent, then there is a model of ZFC that does not arise as a class-forcing extension of its mantle $M$ by any class forcing notion with definable forcing relations in $M$.

We also use our results to provide some counterexamples to the intermediate-model property for forcing. In the case of set forcing, it is well known that every transitive model $W$ of ZFC set theory that is intermediate $V\subseteq W\subseteq V[G]$ a ground model $V$ and a forcing extension $V[G]$, arises itself as a forcing extension $W=V[G_0]$.

In the case of class forcing, however, this can fail.

Theorem. If ZFC is consistent, then there are models of ZFC set theory $V\subseteq W\subseteq V[G]$, where $V[G]$ is a class-forcing extension of $V$ and $W$ is a transitive inner model of $V[G]$, but $W$ is not a forcing extension of $V$ by any class forcing notion with definable forcing relations in $V$.

Theorem. If ZFC + Ord is Mahlo is consistent, then one can form such a counterexample to the class-forcing intermediate model property $V\subseteq W\subseteq V[G]$, where $G\subset \mathbb{B}$ is $V$-generic for an Ord-c.c. tame definable complete class Boolean algebra $\mathbb{B}$, but nevertheless $W$ does not arise by class forcing over $V$ by any definable forcing notion with a definable forcing relation.

More complete details, please go to the paper (click through to the arxiv for a pdf). [bibtex key=”HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD”]