# 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.
[Bibtex]
@ARTICLE{HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD,
author = {Joel David Hamkins and Jonas Reitz},
title = {The set-theoretic universe {$V$} is not necessarily a class-forcing extension of {HOD}},
journal = {ArXiv e-prints},
year = {2017},
volume = {},
number = {},
pages = {},
month = {September},
note = {Manuscript under review},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
eprint = {1709.06062},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/the-universe-need-not-be-a-class-forcing-extension-of-hod},
}

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.
[Bibtex]
@ARTICLE{HamkinsReitz:The-set-theoretic-universe-is-not-necessarily-a-forcing-extension-of-HOD,
author = {Joel David Hamkins and Jonas Reitz},
title = {The set-theoretic universe {$V$} is not necessarily a class-forcing extension of {HOD}},
journal = {ArXiv e-prints},
year = {2017},
volume = {},
number = {},
pages = {},
month = {September},
note = {Manuscript under review},
abstract = {},
keywords = {under-review},
source = {},
doi = {},
eprint = {1709.06062},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://jdh.hamkins.org/the-universe-need-not-be-a-class-forcing-extension-of-hod},
}

# A model of the generic Vopěnka principle in which the ordinals are not $\Delta_2$-Mahlo

• V. Gitman and J. D. Hamkins, “A model of the generic Vopěnka principle in which the ordinals are not Mahlo,” Archive for Mathematical Logic, p. 1–21, 2018.
[Bibtex]
@ARTICLE{GitmanHamkins2018:A-model-of-the-generic-Vopenka-principle-in-which-the-ordinals-are-not-Mahlo,
author = {Gitman, Victoria and Hamkins, Joel David},
year = {2018},
title = {A model of the generic Vopěnka principle in which the ordinals are not Mahlo},
journal = {Archive for Mathematical Logic},
issn = {0933-5846},
doi = {10.1007/s00153-018-0632-5},
month = {5},
pages = {1--21},
eprint = {1706.00843},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://wp.me/p5M0LV-1xT},
abstract = {The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a Δ2-definable class containing no regular cardinals. In such a model, there can be no Σ2-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.},
}

Abstract. The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $\Delta_2$-definable class containing no regular cardinals. In such a model, there can be no $\Sigma_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

The Vopěnka principle is the assertion that for every proper class of first-order structures in a fixed language, one of the structures embeds elementarily into another. This principle can be formalized as a single second-order statement in Gödel-Bernays set-theory GBC, and it has a variety of useful equivalent characterizations. For example, the Vopěnka principle holds precisely when for every class $A$, the universe has an $A$-extendible cardinal, and it is also equivalent to the assertion that for every class $A$, there is a stationary proper class of $A$-extendible cardinals (see theorem 6 in my paper The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme) In particular, the Vopěnka principle implies that ORD is Mahlo: every class club contains a regular cardinal and indeed, an extendible cardinal and more.

To define these terms, recall that a cardinal $\kappa$ is extendible, if for every $\lambda>\kappa$, there is an ordinal $\theta$ and an elementary embedding $j:V_\lambda\to V_\theta$ with critical point $\kappa$. It turns out that, in light of the Kunen inconsistency, this weak form of extendibility is equivalent to a stronger form, where one insists also that $\lambda<j(\kappa)$; but there is a subtle issue about this that comes up with the virtual forms of these axioms, where the virtual weak and virtual strong forms are no longer equivalent. Relativizing to a class parameter, a cardinal $\kappa$ is $A$-extendible for a class $A$, if for every $\lambda>\kappa$, there is an elementary embedding
$$j:\langle V_\lambda, \in, A\cap V_\lambda\rangle\to \langle V_\theta,\in,A\cap V_\theta\rangle$$
with critical point $\kappa$, and again one may equivalently insist also that $\lambda<j(\kappa)$. Every such $A$-extendible cardinal is therefore extendible and hence inaccessible, measurable, supercompact and more. These are amongst the largest large cardinals.

In the first-order ZFC context, set theorists commonly consider a first-order version of the Vopěnka principle, which we call the Vopěnka scheme, the scheme making the Vopěnka assertion of each definable class separately, allowing parameters. That is, the Vopěnka scheme asserts, of every formula $\varphi$, that for any parameter $p$, if $\{\,x\mid \varphi(x,p)\,\}$ is a proper class of first-order structures in a common language, then one of those structures elementarily embeds into another.

The Vopěnka scheme is naturally stratified by the assertions $\text{VP}(\Sigma_n)$, for the particular natural numbers $n$ in the meta-theory, where $\text{VP}(\Sigma_n)$ makes the Vopěnka assertion for all $\Sigma_n$-definable classes. Using the definable $\Sigma_n$-truth predicate, each assertion $\text{VP}(\Sigma_n)$ can be expressed as a single first-order statement in the language of set theory.

In my previous paper, The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme, I proved that the Vopěnka principle is not provably equivalent to the Vopěnka scheme, if consistent, although they are equiconsistent over GBC and furthermore, the Vopěnka principle is conservative over the Vopěnka scheme for first-order assertions. That is, over GBC the two versions of the Vopěnka principle have exactly the same consequences in the first-order language of set theory.

In this article, Gitman and I are concerned with the virtual forms of the Vopěnka principles. The main idea of virtualization, due to Schindler, is to weaken elementary-embedding existence assertions to the assertion that such embeddings can be found in a forcing extension of the universe. Gitman and Schindler had emphasized that the remarkable cardinals, for example, instantiate the virtualized form of supercompactness via the Magidor characterization of supercompactness. This virtualization program has now been undertaken with various large cardinals, leading to fruitful new insights.

Carrying out the virtualization idea with the Vopěnka principles, we define the generic Vopěnka principle to be the second-order assertion in GBC that for every proper class of first-order structures in a common language, one of the structures admits, in some forcing extension of the universe, an elementary embedding into another. That is, the structures themselves are in the class in the ground model, but you may have to go to the forcing extension in order to find the elementary embedding.

Similarly, the generic Vopěnka scheme, introduced by Bagaria, Gitman and Schindler, is the assertion (in ZFC or GBC) that for every first-order definable proper class of first-order structures in a common language, one of the structures admits, in some forcing extension, an elementary embedding into another.

On the basis of their work, Bagaria, Gitman and Schindler had asked the following question:

Question. If the generic Vopěnka scheme holds, then must there be a proper class of remarkable cardinals?

There seemed good reason to expect an affirmative answer, even assuming only $\text{gVP}(\Sigma_2)$, based on strong analogies with the non-generic case. Specifically, in the non-generic context Bagaria had proved that $\text{VP}(\Sigma_2)$ was equivalent to the existence of a proper class of supercompact cardinals, while in the virtual context, Bagaria, Gitman and Schindler proved that the generic form $\text{gVP}(\Sigma_2)$ was equiconsistent with a proper class of remarkable cardinals, the virtual form of supercompactness. Similarly, higher up, in the non-generic context Bagaria had proved that $\text{VP}(\Sigma_{n+2})$ is equivalent to the existence of a proper class of $C^{(n)}$-extendible cardinals, while in the virtual context, Bagaria, Gitman and Schindler proved that the generic form $\text{gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$-extendible cardinals.

But further, they achieved direct implications, with an interesting bifurcation feature that specifically suggested an affirmative answer to the question above. Namely, what they showed at the $\Sigma_2$-level is that if there is a proper class of remarkable cardinals, then $\text{gVP}(\Sigma_2)$ holds, and conversely if $\text{gVP}(\Sigma_2)$ holds, then there is either a proper class of remarkable cardinals or a proper class of virtually rank-into-rank cardinals. And similarly, higher up, if there is a proper class of virtually $C^{(n)}$-extendible cardinals, then $\text{gVP}(\Sigma_{n+2})$ holds, and conversely, if $\text{gVP}(\Sigma_{n+2})$ holds, then either there is a proper class of virtually $C^{(n)}$-extendible cardinals or there is a proper class of virtually rank-into-rank cardinals. So in each case, the converse direction achieves a disjunction with the target cardinal and the virtually rank-into-rank cardinals. But since the consistency strength of the virtually rank-into-rank cardinals is strictly stronger than the generic Vopěnka principle itself, one can conclude on consistency-strength grounds that it isn’t always relevant, and for this reason, it seemed natural to inquire whether this second possibility in the bifurcation could simply be removed. That is, it seemed natural to expect an affirmative answer to the question, even assuming only $\text{gVP}(\Sigma_2)$, since such an answer would resolve the bifurcation issue and make a tighter analogy with the corresponding results in the non-generic/non-virtual case.

In this article, however, we shall answer the question negatively. The details of our argument seem to suggest that a robust analogy with the non-generic/non-virtual principles is achieved not with the virtual $C^{(n)}$-cardinals, but with a weakening of that property that drops the requirement that $\lambda<j(\kappa)$. Indeed, our results seems to offer an illuminating resolution of the bifurcation aspect of the results we mentioned from Bagaria, Gitmand and Schindler, because it provides outright virtual large-cardinal equivalents of the stratified generic Vopěnka principles. Because the resulting virtual large cardinals are not necessarily remarkable, however, our main theorem shows that it is relatively consistent with even the full generic Vopěnka principle that there are no $\Sigma_2$-reflecting cardinals and therefore no remarkable cardinals.

Main Theorem.

1. It is relatively consistent that GBC and the generic Vopěnka principle holds, yet ORD is not Mahlo.
2. It is relatively consistent that ZFC and the generic Vopěnka scheme holds, yet ORD is not definably Mahlo, and not even $\Delta_2$-Mahlo. In such a model, there can be no $\Sigma_2$-reflecting cardinals and therefore also no remarkable cardinals.

For more, go to the arcticle:

• V. Gitman and J. D. Hamkins, “A model of the generic Vopěnka principle in which the ordinals are not Mahlo,” Archive for Mathematical Logic, p. 1–21, 2018.
[Bibtex]
@ARTICLE{GitmanHamkins2018:A-model-of-the-generic-Vopenka-principle-in-which-the-ordinals-are-not-Mahlo,
author = {Gitman, Victoria and Hamkins, Joel David},
year = {2018},
title = {A model of the generic Vopěnka principle in which the ordinals are not Mahlo},
journal = {Archive for Mathematical Logic},
issn = {0933-5846},
doi = {10.1007/s00153-018-0632-5},
month = {5},
pages = {1--21},
eprint = {1706.00843},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://wp.me/p5M0LV-1xT},
abstract = {The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a Δ2-definable class containing no regular cardinals. In such a model, there can be no Σ2-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.},
}

# The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme

• J. D. Hamkins, “The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme,” ArXiv e-prints, 2016.
[Bibtex]
@ARTICLE{Hamkins:The-Vopenka-principle-is-inequivalent-to-but-conservative-over-the-Vopenka-scheme,
author = {Joel David Hamkins},
title = {The {Vop\v{e}nka} principle is inequivalent to but conservative over the {Vop\v{e}nka} scheme},
journal = {ArXiv e-prints},
year = {2016},
volume = {},
number = {},
pages = {},
month = {},
note = {Under review},
abstract = {},
keywords = {under-review},
source = {},
eprint = {1606.03778},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://wp.me/p5M0LV-1lV},
}

Abstract. The Vopěnka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vopěnka scheme, which makes the Vopěnka assertion only for the first-order definable classes of structures. Nevertheless, the two Vopěnka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for first-order assertions in the language of set theory.

The Vopěnka principle is the assertion that for every proper class $\mathcal{M}$ of first-order $\mathcal{L}$-structures, for a set-sized language $\mathcal{L}$, there are distinct members of the class $M,N\in\mathcal{M}$ with an elementary embedding $j:M\to N$ between them. In quantifying over classes, this principle is a single assertion in the language of second-order set theory, and it makes sense to consider the Vopěnka principle in the context of a second-order set theory, such as Godel-Bernays set theory GBC, whose language allows one to quantify over classes. In this article, GBC includes the global axiom of choice.

In contrast, the first-order Vopěnka scheme makes the Vopěnka assertion only for the first-order definable classes $\mathcal{M}$ (allowing parameters). This theory can be expressed as a scheme of first-order statements, one for each possible definition of a class, and it makes sense to consider the Vopěnka scheme in Zermelo-Frankael ZFC set theory with the axiom of choice.

Because the Vopěnka principle is a second-order assertion, it does not make sense to refer to it in the context of ZFC set theory, whose first-order language does not allow quantification over classes; one typically retreats to the Vopěnka scheme in that context. The theme of this article is to investigate the precise meta-mathematical interactions between these two treatments of Vopěnka’s idea.

Main Theorems.

1. If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka scheme holds, but the Vopěnka principle fails.
2. If ZFC and the Vopěnka scheme holds, then there is a class forcing extension, adding classes but no sets, in which GBC and the Vopěnka principle holds.

It follows that the Vopěnka principle VP and the Vopěnka scheme VS are not equivalent, but they are equiconsistent and indeed, they have the same first-order consequences.

Corollaries.

1. Over GBC, the Vopěnka principle and the Vopěnka scheme, if consistent, are not equivalent.
2. Nevertheless, the two Vopěnka axioms are equiconsistent over GBC.
3. Indeed, the two Vopěnka axioms have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme for assertions in the first-order language of set theory. $$\text{GBC}+\text{VP}\vdash\phi\qquad\text{if and only if}\qquad\text{ZFC}+\text{VS}\vdash\phi$$

These results grew out of my my answer to a MathOverflow question of Mike Shulman, Can Vopěnka’s principle be violated definably?, inquiring whether there would always be a definable counterexample to the Vopěnka principle, whenever it should happen to fail. I interpret the question as asking whether the Vopěnka scheme is necessarily equivalent to the Vopěnka principle, and the answer is negative.

The proof of the main theorem involves the concept of a stretchable set $g\subset\kappa$ for an $A$-extendible cardinal, which has the property that for every cardinal $\lambda>\kappa$ and every extension $h\subset\lambda$ with $h\cap\kappa=g$, there is an elementary embedding $j:\langle V_\lambda,\in,A\cap V_\lambda\rangle\to\langle V_\theta,\in,A\cap V_\theta\rangle$ such that $j(g)\cap\lambda=h$. Thus, the set $g$ can be stretched by an $A$-extendibility embedding so as to agree with any given $h$.

# Norman Lewis Perlmutter

Norman Lewis Perlmutter successfully defended his dissertation under my supervision and will earn his Ph.D. at the CUNY Graduate Center in May, 2013.  His dissertation consists of two parts.  The first chapter arose from the observation that while direct limits of large cardinal embeddings and other embeddings between models of set theory are pervasive in the subject, there is comparatively little study of inverse limits of systems of such embeddings.  After such an inverse system had arisen in Norman’s joint work on Generalizations of the Kunen inconsistency, he mounted a thorough investigation of the fundamental theory of these inverse limits. In chapter two, he investigated the large cardinal hierarchy in the vicinity of the high-jump cardinals.  During this investigation, he ended up refuting the existence of what are now called the excessively hypercompact cardinals, which had appeared in several published articles.  Previous applications of that notion can be made with a weaker notion, what is now called a hypercompact cardinal.

Norman Lewis Perlmutter, “Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2013.

Abstract.  This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other.

The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards fully characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. If the inverse limit exists, then it is given by either the entire thread class or a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved in both directions by forcing under fairly general assumptions. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter.

The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing.  A high-jump cardinal is the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\{\ j(f)(\kappa) \mid f: \kappa \to \kappa\ \}$.  Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to an Woodin-for-supercompactness cardinal. The existence of an excessively hypercompact cardinal is inconsistent.