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

• J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
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Abstract. In light of the celebrated theorem of Vopěnka, proving in ZFC that every set is generic over $\newcommand\HOD{\text{HOD}}\HOD$, 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 $\langle\HOD,\in,A\rangle$. 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 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 $0^\sharp$.

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 $\HOD$.

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 $\HOD$. In other words, for every set $a$ there is a forcing notion $\mathbb{B}\in\HOD$ and a $\HOD$-generic filter $G\subseteq\mathbb{B}$ for which $a\in\HOD[G]\subseteq V$. The full set-theoretic universe $V$ is therefore the union of all these various set-forcing generic extensions $\HOD[G]$.

It is natural to wonder whether these various forcing extensions $\HOD[G]$ can be unified or amalgamated to realize $V$ as a single class-forcing extension of $\HOD$ by a possibly proper class forcing notion in $\HOD$. 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 $\HOD$?

We intend the question to be asking more specifically whether the universe $V$ arises as a bona-fide class-forcing extension of $\HOD$, in the sense that there is a class forcing notion $\mathbb{P}$, possibly a proper class, which is definable in $\HOD$ and which has definable forcing relation $p\Vdash\varphi(\tau)$ there for any desired first-order formula $\varphi$, such that $V$ arises as a forcing extension $V=\HOD[G]$ for some $\HOD$-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 $\HOD$.

Main Theorem. If ZFC is consistent, then there is a model of ZFC which is not a forcing extension of its $\HOD$ by any class forcing notion definable in that $\HOD$ 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 $\HOD$, such that the set-theoretic universe $V$ is a generic extension of $\langle \HOD,\in,A\rangle$.

This is a postive answer to the main question, if one is willing to augment $\HOD$ with a class $A$ that may not be definable in $\HOD$. 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).

• J. D. Hamkins and J. Reitz, “The set-theoretic universe $V$ is not necessarily a class-forcing extension of HOD,” ArXiv e-prints, 2017. (manuscript under review)
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# Inner-model reflection principles

• N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, and J. Reitz, “Inner-model reflection principles,” ArXiv e-prints, 2017. (manuscript under review)
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Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH.

Every set theorist is familiar with the classical Lévy-Montague reflection principle, which explains how truth in the full set-theoretic universe $V$ reflects down to truth in various rank-initial segments $V_\theta$ of the cumulative hierarchy. Thus, the Lévy-Montague reflection principle is a form of height-reflection, in that truth in $V$ is reflected vertically downwards to truth in some $V_\theta$.

In this brief article, in contrast, we should like to introduce and consider a form of width-reflection, namely, reflection to nontrivial inner models. Specifically, we shall consider the following reflection principles.

Definition.

1. The inner-model reflection principle asserts that if a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then there is a proper inner model $W$, a transitive class model of ZF containing all ordinals, with $a\in W\subsetneq V$ in which $\varphi(a)$ is true.
2. The ground-model reflection principle asserts that if $\varphi(a)$ is true in $V$, then there is a nontrivial ground model $W\subsetneq V$ with $a\in W$ and $W\models\varphi(a)$.
3. Variations of the principles arise by insisting on inner models of a particular type, such as ground models for a particular type of forcing, or by restricting the class of parameters or formulas that enter into the scheme.
4. The lightface forms of the principles, in particular, make their assertion only for sentences, so that if $\sigma$ is a sentence true in $V$, then $\sigma$ is true in some proper inner model or ground $W$, respectively.

We explain how to force the principles, how to separate them, how they are consequences of various large cardinal assumptions, consequences of the maximality principle and of the inner model hypothesis. Kindly proceed to the article (pdf available at the arxiv) for more.

• N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, and J. Reitz, “Inner-model reflection principles,” ArXiv e-prints, 2017. (manuscript under review)
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This article grew out of an exchange held by the authors on math.stackexchange
in response to an inquiry posted by the first author concerning the nature of width-reflection in comparison to height-reflection:  What is the consistency strength of width reflection?

# Approximation and cover properties propagate upward

I should like to record here the proof of the following fact, which Jonas Reitz and I first observed years ago, when he was my graduate student, and I recall him making the critical observation.

It concerns the upward propagation of the approximation and cover properties, some technical concepts that lie at the center of my paper, Extensions with he approximation and cover properties have no new large cardinals, and which are also used in my proof of Laver’s theorem on the definability of the ground model, and which figure in Jonas’s work on the ground axiom.

The fact has a curious and rather embarrassing history, in that Jonas and I have seen an unfortunate cycle, in which we first proved the theorem, and then subsequently lost and forgot our own proof, and then lost confidence in the fact, until we rediscovered the proof again. This cycle has now repeated several times, in absurd mathematical comedy, and each time the proof was lost, various people with whom we discussed the issue sincerely doubted that it could be true.  But we are on the upswing now, for in response to some recently expressed doubts about the fact, although I too was beginning to doubt it again, I spent some time thinking about it and rediscovered our old proof! Hurrah!  In order to break this absurd cycle, however, I am now recording the proof here in order that we may have a place to point in the future, to give the theorem a home.

Although the fact has not yet been used in any application to my knowledge, it strikes me as inevitable that this fundamental fact about the approximation and cover properties will eventually find an important use.

Definition. Assume $\delta$ is a cardinal in $V$ and $W\subset V$ is a transitive inner model of set theory.

• The extension $W\subset V$ satisfies the $\delta$-approximation property if whenever $A\subset W$ is a set in $V$ and $A\cap a\in W$ for any $a\in W$ of size less than $\delta$ in $W$, then $A\in W$.
• The extension $W\subset V$ satisfies the $\delta$-cover property if whenever $A\subset W$ is a set of size less than $\delta$ in $V$, then there is a covering set $B\in W$ with $A\subset B$ and $|B|^W\lt\delta$.

Theorem. If $W\subset V$ has the $\delta$-approximation and $\delta$-cover properties and $\delta\lt\gamma$ are both infinite cardinals in $V$, then it also has the $\gamma$-approximation and $\gamma$-cover properties.

Proof. First, notice that the $\delta$-approximation property trivially implies the $\gamma$-approximation property for any larger cardinal $\gamma$. So we need only verify the $\gamma$-cover property, and this we do by induction. Note that the limit case is trivial, since if the cover property holds at every cardinal below a limit cardinal, then it trivially holds at that limit cardinal, since there are no additional instances of covering to be treated. Thus, we reduce to the case $\gamma=\delta^+$, meaning $(\delta^+)^V$, but we must allow that $\delta$ may be singular here.

If $\delta$ is singular, then we claim that the $\delta$-cover property alone implies the $\delta^+$-cover property: if $A\subset W$ has size $\delta$ in $V$, then by the singularity of $\delta$ we may write it as $A=\bigcup _{\alpha\in I}A_\alpha$, where each $A_\alpha$ and $I$ have size less than $\delta$. By the $\delta$-cover property, there are covers $A_\alpha\subset B_\alpha\in W$ with $B_\alpha$ of size less than $\delta$ in $W$.  Furthermore, the set $\{B_\alpha\mid\alpha\in I\}$ itself is covered by some set $\mathcal{B}\in W$ of size less than $\delta$ in $W$. That is, we cover the small set of small covers. We may assume that every set in $\mathcal{B}$ has size less than $\delta$, by discarding those that aren’t, and so $B=\bigcup\mathcal{B}$ is a set in $W$ that covers $A$ and has size at most $\delta$ there, since it is small union of small sets, thereby verifying this instance of the $\gamma$-cover property.

If $\delta$ is regular, consider a set $A\subset W$ with $A\in V$ of size $\delta$ in $V$, so that $A=\{a_\xi\mid\xi\lt\delta\}$. For each $\alpha\lt\delta$, the initial segment $\{a_\xi\mid\xi\lt\alpha\}$ has size less than $\delta$ and is therefore covered by some $B_\alpha\in W$ of size less than $\delta$ in $W$.  By adding each $B_\alpha$ to what we are covering at later stages, we may assume that they form an increasing tower: $\alpha\lt\beta\to B_\alpha\subset B_\beta$. The choices $\alpha\mapsto B_\alpha$ are made in $V$.  Let $B=\bigcup_\alpha B_\alpha$, which certainly covers $A$. Observe that for any set $a\in W$ of size less than $\delta$, it follows by the regularity of $\delta$ that $B\cap a=B_\alpha\cap a$ for all sufficiently large $\alpha$.  Thus, all $\delta$-approximations to $B$ are in $W$ and so $B$ itself is in $W$ by the $\delta$-approximation property, as desired. Note that $B$ has size less than $\gamma$ in $W$, because it has size $\delta$ in $V$, and so we have verified this instance of the $\gamma$-cover property for $W\subset V$.

Thus, in either case we’ve established the $\gamma$-cover property for $W\subset V$, and the proof is complete. QED

(Thanks to Thomas Johnstone for some comments and for pointing out a simplification in the proof:  previously, I had reduced without loss of generality to the case where $A$ is a set of ordinals of order type $\delta$; but Tom pointed out that the general case is not actually any harder.   And indeed, Jonas dug up some old notes to find the 2008 version of the argument, which is essentially the same as what now appears here.)

Note that without the $\delta$-approximation property, it is not true that the $\delta$-cover property transfers upward. For example, every extension has the $\aleph_0$-cover property.

# Jonas Reitz

Jonas Reitz earned his Ph.D under my supervision in June, 2006 at the CUNY Graduate Center.  He was truly a pleasure to supervise. From the earliest days of his dissertation research, he had his own plan for the topic of the work: he wanted to “undo” forcing, to somehow force backwards, from the extension to the ground model. At first I was skeptical, but in time, ideas crystalized around the ground axiom (now with its own Wikipedia entry), formulated using a recent-at-the-time result of Richard Laver.  Along with Laver’s theorem, Jonas’s dissertation was the beginning of the body of work now known as set-theoretic geology.  Jonas holds a tenured position at the New York City College of Technology of CUNY.

Jonas Reitz

web page | math genealogy | MathSciNet | ar$\chi$iv | google scholar | related posts

Jonas Reitz, “The ground axiom,” Ph.D. dissertation, CUNY Graduate Center, June, 2006.  ar$\chi$iv

Abstract.  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.

# Set-theoretic geology

• G. Fuchs, J. D. Hamkins, and J. Reitz, “Set-theoretic geology,” Annals of Pure and Applied Logic, vol. 166, iss. 4, pp. 464-501, 2015.
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A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.

# Pointwise definable models of set theory

• J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal Symbolic Logic, vol. 78, iss. 1, pp. 139-156, 2013.
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One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is heard at a good math tea—that there must be real numbers that we cannot describe or deﬁne, because there are are only countably many deﬁnitions, but uncountably many reals.  Does it withstand scrutiny?

This article provides an answer.  The article has a dual nature, with the first part aimed at a more general audience, and the second part providing a proof of the main theorem:  every countable model of set theory has an extension in which every set and class is definable without parameters.  The existence of these models therefore exhibit the difficulties in formalizing the math tea argument, and show that robust violations of the math tea argument can occur in virtually any set-theoretic context.

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

# The ground axiom is consistent with $V\ne{\rm HOD}$

• J. D. Hamkins, J. Reitz, and W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc.~Amer.~Math.~Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.
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Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of $V=\text{HOD}$. In this article, we show that the Ground Axiom is relatively consistent with $V\neq\text{HOD}$. In fact, every model of ZFC has a class-forcing extension that is a model of $\text{ZFC}+\text{GA}+V\neq\text{HOD}$. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with $\text{ZFC}+\text{GA}+V\neq\text{HOD}$ in which this supercompact cardinal is preserved.

# The Ground Axiom

• J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, pp. 3160-3162, 2005.
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This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.