# The weakly compact embedding property, Apter-Gitik celebration, CMU 2015

This will be a talk at the Conference in honor of Arthur W. Apter and Moti Gitik at Carnegie Mellon University, May 30-31, 2015.  I am pleased to be a part of this conference in honor of the 60th birthdays of two mathematicians whom I admire very much.

Abstract. The weakly compact embedding property for a cardinal $\kappa$ is the assertion that for every transitive set $M$ of size $\kappa$ with $\kappa\in M$, there is a transitive set $N$ and an elementary embedding $j:M\to N$ with critical point $\kappa$. When $\kappa$ is inaccessible, this property is one of many equivalent characterizations of $\kappa$ being weakly compact, along with the weakly compact extension property, the tree property, the weakly compact filter property and many others. When $\kappa$ is not inaccessible, however, these various properties are no longer equivalent to each other, and it is interesting to sort out the relations between them. In particular, I shall consider the embedding property and these other properties in the case when $\kappa$ is not necessarily inaccessible, including interesting instances of the embedding property at cardinals below the continuum, with relations to cardinal characteristics of the continuum.

This is joint work with Brent Cody, Sean Cox, myself and Thomas Johnstone.

Slides | Article | Conference web site

# Embeddings of the universe into the constructible universe, current state of knowledge, CUNY Set Theory Seminar, March 2015

This will be a talk for the CUNY Set Theory Seminar, March 6, 2015.

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe.

Question.(Hamkins) Can there be an embedding $j:V\to L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $V\neq L$?

The notion of embedding here is merely that $$x\in y\iff j(x)\in j(y),$$ and such a map need not be elementary nor even $\Delta_0$-elementary. It is not difficult to see that there can generally be no $\Delta_0$-elementary embedding $j:V\to L$, when $V\neq L$.

Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, Every countable model of set theory embeds into its own constructible universe, where the title theorem is the following.

Theorem.(Hamkins) Every countable model of set theory $\langle M,\in^M\rangle$, including every countable transitive model of set theory, has an embedding $j:\langle M,\in^M\rangle\to\langle L^M,\in^M\rangle$ into its own constructible universe.

The methods of proof also established that the countable models of set theory are linearly pre-ordered by embeddability: given any two models, one of them embeds into the other; or equivalently, one of them is isomorphic to a submodel of the other. Indeed, one model $\langle M,\in^M\rangle$ embeds into another $\langle N,\in^N\rangle$ just in case the ordinals of the first $\text{Ord}^M$ order-embed into the ordinals of the second $\text{Ord}^N$. (And this implies the theorem above.)

In the proof of that theorem, the embeddings $j:M\to L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. And I realized that I could not generally refute the possibility that such a $j$ might even be a class in $M$.

Currently, the question remains open, but we have some partial progress, and have settled it in a number of cases, including the following, on which I’ll speak:

• If there is an embedding $j:V\to L$, then for a proper class club of cardinals $\lambda$, we have $(2^\lambda)^V=(\lambda^+)^L$.
• If $0^\sharp$ exists, then there is no embedding $j:V\to L$.
• If $0^\sharp$ exists, then there is no embedding $j:V\to L$ and indeed no embedding $j:P(\omega)\to L$.
• If there is an embedding $j:V\to L$, then the GCH holds above $\aleph_0$.
• In the forcing extension $V[G]$ obtained by adding $\omega_1$ many Cohen reals (or more), there is no embedding $j:V[G]\to L$, and indeed, no $j:P(\omega)^{V[G]}\to V$. More generally, after adding $\kappa^+$ many Cohen subsets to $\kappa$, for any regular cardinal $\kappa$, then in $V[G]$ there is no $j:P(\kappa)\to V$.
• If $V$ is a nontrivial set-forcing extension of an inner model $M$, then there is no embedding $j:V\to M$. Indeed, there is no embedding $j:P(\kappa^+)\to M$, if the forcing has size $\kappa$. In particular, if $V$ is a nontrivial forcing extension, then there is no embedding $j:V\to L$.
• Every countable set $A$ has an embedding $j:A\to L$.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut.

See my related MathOverflow question: Can there be an embedding $j:V\to L$ from the set-theoretic universe $V$ to the constructible universe $L$, when $V\neq L$?

Talk Abstract

# The foundation axiom and elementary self-embeddings of the universe

• A. S. Daghighi, M. Golshani, J. Hamkins, and E. Jeřábek, “The foundation axiom and elementary self-embeddings of the universe,” in Infinity, computability, and metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch, S. Geschke, B. Löwe, and P. Schlicht, Eds., Coll. Publ., London, 2014, vol. 23, pp. 89-112.
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In this article, we examine the role played by the axiom of foundation in the well-known Kunen inconsistency, the theorem asserting that there is no nontrivial elementary embedding of the set-theoretic universe to itself. All the standard proofs of the Kunen inconsistency make use of the axiom of foundation (see Kanamori’s books and also Generalizations of the Kunen inconsistency), and this use is essential, assuming that $\ZFC$ is consistent, because as we shall show there are models of $\ZFCf$ that admit nontrivial elementary self-embeddings and even nontrivial definable automorphisms. Meanwhile, a fragment of the Kunen inconsistency survives without foundation as the claim in $\ZFCf$ that there is no nontrivial elementary self-embedding of the class of well-founded sets. Nevertheless, some of the commonly considered anti-foundational theories, such as the Boffa theory $\BAFA$, prove outright the existence of nontrivial automorphisms of the set-theoretic universe, thereby refuting the Kunen assertion in these theories.  On the other hand, several other common anti-foundational theories, such as Aczel’s anti-foundational theory $\ZFCf+\AFA$ and Scott’s theory $\ZFCf+\text{SAFA}$, reach the opposite conclusion by proving that there are no nontrivial elementary embeddings from the set-theoretic universe to itself. Our summary conclusion, therefore, is that the resolution of the Kunen inconsistency in set theory without foundation depends on the specific nature of one’s anti-foundational stance.

This is joint work with Ali Sadegh Daghighi, Mohammad Golshani, myself and Emil Jeřábek, which grew out of our interaction on Ali’s question on MathOverflow, Is there any large cardinal beyond the Kunen inconsistency?

# The role of the axiom of foundation in the Kunen inconsistency, CUNY September 2013

This will be a talk for the CUNY Set Theory Seminar on September 20, 2013 (date tentative).

Abstract. The axiom of foundation plays an interesting role in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself, for the truth or falsity of the Kunen assertion depends on one’s specific anti-foundational stance.  The fact of the matter is that different anti-foundational theories come to different conclusions about this assertion.  On the one hand, it is relatively consistent with ZFC without foundation that the Kunen assertion fails, for there are models of  ZFC-F  in which there are definable nontrivial elementary embeddings $j:V\to V$. Indeed, in Boffa’s anti-foundational theory BAFA, the Kunen assertion is outright refutable, and in this theory there are numerous nontrivial elementary embeddings of the universe to itself. Meanwhile, on the other hand, Aczel’s anti-foundational theory GBC-F+AFA, as well as Scott’s theory GBC-F+SAFA and other anti-foundational theories, continue to prove the Kunen assertion, ruling out the existence of a nontrivial elementary embedding $j:V\to V$.

This talk covers very recent joint work with Emil Jeřábek, Ali Sadegh Daghighi and Mohammad Golshani, based on an interaction growing out of Ali’s question on MathOverflow, which lead to our recent article, The role of the axiom of foundation in the Kunen inconsistency.

# 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.

# The countable models of set theory are linearly pre-ordered by embeddability, Rutgers, November 2012

This will be a talk for the Rutgers Logic Seminar on November 19, 2012.

Abstract.  I will speak on my recent theorem that every countable model of set theory $M$, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  Indeed, the bi-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC—is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.

# Every countable model of set theory is isomorphic to a submodel of its own constructible universe, Barcelona, December, 2012

This will be a talk for a set theory workshop at the University of Barcelona on December 15, 2012, organized by Joan Bagaria.

Abstract. Every countable model of set theory $M$, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  Indeed, the bi-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC—is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.

# Every countable model of set theory embeds into its own constructible universe, Fields Institute, Toronto, August 2012

This will be a talk for the  Toronto set theory seminar at the Fields Institute, University of Toronto, on August 24, 2012.

Abstract.  Every countable model of set theory $M$, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random $\mathbb{Q}$-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $\text{Ord}^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other.  Indeed, the bi-embeddability classes form a well-ordered chain of length $\omega_1+1$.  Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $\text{Ord}^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC—is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.

# Singular cardinals and strong extenders

• A. W. Apter, J. Cummings, and J. D. Hamkins, “Singular cardinals and strong extenders,” Cent. Eur. J. Math., vol. 11, iss. 9, pp. 1628-1634, 2013.
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Brent Cody asked the question whether the situation can arise that one has an elementary embedding $j:V\to M$ witnessing the $\theta$-strongness of a cardinal $\kappa$, but where $\theta$ is regular in $M$ and singular in $V$.

In this article, we investigate the various circumstances in which this does and does not happen, the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing “$\kappa$ is $\mu$-strong”, such that $\mu$ is singular in $Ult(V, E)$.

# Generalizations of the Kunen inconsistency, KGRC, Vienna 2011

This is a talk at the research seminar of the Kurt Gödel Research Center, November 3, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

# Generalizations of the Kunen inconsistency

• J. D. Hamkins, G. Kirmayer, and N. L. Perlmutter, “Generalizations of the Kunen inconsistency,” Annals of Pure and Applied Logic, vol. 163, iss. 12, pp. 1872-1890, 2012.
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We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.

# The wholeness axioms and $V=\rm HOD$

• J. D. Hamkins, “The wholeness axioms and $V=\rm HOD$,” Arch.~Math.~Logic, vol. 40, iss. 1, pp. 1-8, 2001.
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The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding $j:V\to V$. Formalized by augmenting the usual language of set theory with an additional unary function symbol j to represent the embedding, they avoid the Kunen inconsistency by restricting the base theory ZFC to the usual language of set theory. Thus, under the Wholeness Axioms one cannot appeal to the Replacement Axiom in the language with j as Kunen does in his famous inconsistency proof. Indeed, it is easy to see that the Wholeness Axioms have a consistency strength strictly below the existence of an $I_3$ cardinal. In this paper, I prove that if the Wholeness Axiom $WA_0$ is itself consistent, then it is consistent with $V=HOD$. A consequence of the proof is that the various Wholeness Axioms $WA_n$ are not all equivalent. Furthermore, the theory $ZFC+WA_0$ is finitely axiomatizable.

# Gap forcing: generalizing the Lévy-Solovay theorem

• J. D. Hamkins, “Gap forcing: generalizing the Lévy-Solovay theorem,” Bull.~Symbolic Logic, vol. 5, iss. 2, pp. 264-272, 1999.
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The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

# Canonical seeds and Prikry trees

• J. D. Hamkins, “Canonical seeds and Prikry trees,” J.~Symbolic Logic, vol. 62, iss. 2, pp. 373-396, 1997.
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Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.