Tag Archives: forcing

Strongly uplifting cardinals and the boldface resurrection axioms

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[bibtex key=HamkinsJohnstone2017:StronglyUpliftingCardinalsAndBoldfaceResurrection]

Abstract. We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

The strongly uplifting cardinals, which we introduce in this article, are a boldface analogue of the uplifting cardinals introduced in our previous paper, Resurrection axioms and uplifting cardinals, and are equivalently characterized as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals. In consistency strength, these new large cardinals lie strictly above the weakly compact, totally indescribable and strongly unfoldable cardinals and strictly below the subtle cardinals, which in turn are weaker in consistency than the existence of $0^\sharp$. The robust diversity of equivalent characterizations of this new large cardinal concept enables constructions and techniques from much larger large cardinal contexts, such as Laver functions and forcing iterations with applications to forcing axioms. Using such methods, we prove that the existence of a strongly uplifting cardinal (or equivalently, a superstrongly unfoldable or almost hugely unfoldable cardinal) is equiconsistent over ZFC with natural instances of the boldface resurrection axioms, including the boldface resurrection axiom for proper forcing, for semi-proper forcing, for c.c.c. forcing and others. Thus, whereas in our prior article we proved that the existence of a mere uplifting cardinal is equiconsistent with natural instances of the (lightface) resurrection axioms, here we adapt both of these notions to the boldface context.

Definitions.

  • An inaccessible cardinal $\kappa$ is strongly uplifting if for every ordinal $\theta$ it is strongly $\theta$-uplifting, which is to say that for every $A\subset V_\kappa$ there is an inaccessible cardinal $\gamma\geq\theta$ and a set $A^*\subset V_\gamma$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle V_\gamma,{\in},A^*\rangle$ is a proper elementary extension.
  • A cardinal $\kappa$ is superstrongly unfoldable, if for every ordinal $\theta$ it is superstrongly $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subset N$.
  • A cardinal $\kappa$ is almost-hugely unfoldable, if for every ordinal $\theta$ it is almost-hugely $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $N^{<j(\kappa)}\subset N$.

Remarkably, these different-seeming large cardinal concepts turn out to be exactly equivalent to one another. A cardinal $\kappa$ is strongly uplifting if and only if it is superstrongly unfoldable, if and only if it is almost hugely unfoldable. Furthermore, we prove that the existence of such a cardinal is equiconsistent with several natural instances of the boldface resurrection axiom.

Theorem. The following theories are equiconsistent over ZFC.

  • There is a strongly uplifting cardinal.
  • There is a superstrongly unfoldable cardinal.
  • There is an almost hugely unfoldable cardinal.
  • The boldface resurrection axiom for all forcing.
  • The boldface resurrection axiom for proper forcing.
  • The boldface resurrection axiom for semi-proper forcing.
  • The boldface resurrection axiom for c.c.c. forcing.
  • The weak boldface resurrection axiom for countably-closed forcing, axiom-A forcing, proper forcing and semi-proper forcing, plus $\neg\text{CH}$.

 

 

Large cardinals need not be large in HOD, Rutgers logic seminar, April 2014

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I shall speak at the Rutgers Logic Seminar on April 21, 2014, 5:00-6:20 pm, Room 705, Hill Center, Busch Campus, Rutgers University.

Abstract. I will show that large cardinals, such as measurable, strong and supercompact cardinals, need not exhibit their large cardinal nature in HOD.  Specifically, it is relatively consistent that a supercompact cardinal is not weakly compact in HOD, and one may construct models with a proper class of supercompact cardinals, none of them weakly compact in HOD.  This is current joint work with Cheng Yong.

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Large cardinals need not be large in HOD, CUNY Set Theory Seminar, January 2014

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This will be a talk for the CUNY Set Theory Seminar, January 31, 2014, 10:00 am.

Abstract. I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I will begin with the example of a measurable cardinal that is not measurable in HOD. After this, I will describe how to force a more extreme divergence.  For example, among other possibilities, it is relatively consistent that there is a supercompact cardinal that is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

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Superstrong and other large cardinals are never Laver indestructible, ASL 2014, Boulder, May 2014

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The Flatirons, Boulder, ColoradoThis will be an invited talk at the ASL 2014 North American Annual Meeting (May 19-22, 2014) in the special session Set Theory in Honor of Rich Laver, organized by Bill Mitchell and Jean Larson.

Abstract.  The large cardinal indestructibility phenomenon, discovered by Richard Laver with his seminal result on supercompact cardinals, is by now often seen as pervasive in the large cardinal hierarchy. Nevertheless, a new never-indestrucible phenomenon has emerged.  Superstrong cardinals, for example, are never Laver indestructible.  Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible.  The proof involves a detailed technical analysis of the complexity of the definition in Laver’s theorem on the definability of the ground model, thereby involving and extending results in set-theoretic geology.  This is joint work between myself and Joan Bagaria, Kostas Tasprounis and Toshimichi Usuba.

Article | Slides

Exploring the Frontiers of Incompleteness, Harvard, August 2013

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I will be participating in the culminating workshop of the Exploring the Frontiers of Incompleteness conference series at Harvard University, to take place August 31-September 1, 2013.  Rather than conference talks, the program will consist of extended discussion sessions by the participants of the year-long series, with the discussion framed by very brief summary presentations.  Peter Koellner asked me to prepare such a presentation on the multiverse conception, and you can see the slides in The multiverse perspective in set theory (Slides).

My previous EFI talk was The multiverse perspective on determinateness in set theory, based in part on my paper The set-theoretical multiverse.

Resurrection axioms and uplifting cardinals

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[bibtex key=HamkinsJohnstone2014:ResurrectionAxiomsAndUpliftingCardinals]

Abstract. We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of uplifting cardinal.

Many classical forcing axioms can be viewed, at least informally, as the claim that the universe is existentially closed in its forcing extensions, for the axioms generally assert that certain kinds of filters, which could exist in a forcing extension $V[G]$, exist already in $V$. In several instances this informal perspective is realized more formally: Martin’s axiom is equivalent to the assertion that $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $H_{\frak{c}}\prec_{\Sigma_1}V[G]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $H_{\omega_2}$ is existentially closed in all proper forcing extensions, or $H_{\omega_2}\prec_{\Sigma_1}V[G]$; and there are other similar instances.

In model theory, a submodel $M\subset N$ is existentially closed in $N$ if existential assertions true in $N$ about parameters in $M$ are true already in $M$, that is, if $M$ is a $\Sigma_1$-elementary substructure of $N$, which we write as $M\prec_{\Sigma_1} N$. Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.

Elementary Fact. If $\mathcal{M}$ is a submodel of $\mathcal{N}$, then the following are equivalent.

  1. The model $\mathcal{M}$ is existentially closed in $\mathcal{N}$.
  2. $\mathcal{M}\subset \mathcal{N}$ has resurrection. That is, there is a further extension $\mathcal{M}\subset\mathcal{N}\subset\mathcal{M}^+$ for which $\mathcal{M}\prec\mathcal{M}^+$.

We call this resurrection because although certain truths in $\mathcal{M}$ may no longer hold in the extension $\mathcal{N}$, these truths are nevertheless revived in light of $\mathcal{M}\prec\mathcal{M}^+$ in the further extension to $\mathcal{M}^+$.

In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $\mathcal{M}^+$ arising in the fact, and in this context the equivalence of (1) and (2) breaks own, although the converse implication $(2)\to(1)$ always holds, and every instance of resurrection implies the corresponding instance of existential closure. This key observation leads us to the main unifying theme of this article, the idea that

resurrection may allow us to formulate more robust forcing axioms 

than existential closure or than combinatorial assertions about filters and dense sets. We therefore introduce in this paper a spectrum of new forcing axioms utilizing the resurrection concept.

Main Definition. Let $\Gamma$ be a fixed definable class of forcing notions.

  1. The resurrection axiom $\text{RA}(\Gamma)$ is the assertion that for every forcing notion $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, with $\vdash_{\mathbb{Q}}\mathbb{R}\in\Gamma$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
  2. The weak resurrection axiom $\text{wRA}(\Gamma)$ is the assertion that for every $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.

The main result is to prove that various formulations of the resurrection axioms are equiconsistent with the existence of an uplifting cardinal, where an inaccessible cardinal $\kappa$ is uplifting, if there are arbitrarily large inaccessible cardinals $\gamma$ for which $H_\kappa\prec H_\gamma$.  This is a rather weak large cardinal notion, having consistency strength strictly less than the existence of a Mahlo cardinal, which is traditionally considered to be very low in the large cardinal hierarchy.  One highlight of the article is our development of “the world’s smallest Laver function,” the Laver function concept for uplifting cardinals, and we perform an analogue of the Laver preparation in order to achieve the resurrection axiom for c.c.c. forcing.

Main Theorem. The following theories are equiconsistent over ZFC:

  1. There is an uplifting cardinal.
  2. $\text{RA}(\text{all})$.
  3. $\text{RA}(\text{ccc})$.
  4. $\text{RA}(\text{semiproper})+\neg\text{CH}$.
  5. $\text{RA}(\text{proper})+\neg\text{CH}$.
  6. For some countable ordinal $\alpha$, the axiom $\text{RA}(\alpha\text{-proper})+\neg\text{CH}$.
  7. $\text{RA}(\text{axiom-A})+\neg\text{CH}$.
  8. $\text{wRA}(\text{semiproper})+\neg\text{CH}$.
  9. $\text{wRA}(\text{proper})+\neg\text{CH}$.
  10. For some countable ordinal $\alpha$, the axiom $\text{wRA}(\alpha\text{-proper})+\neg\text{CH}$.
  11. $\text{wRA}(\text{axiom-A})+\neg\text{CH}$.
  12. $\text{wRA}(\text{countably closed})+\neg\text{CH}$.

The proof outline proceeds in two directions: on the one hand, the resurrection axioms generally imply that the continuum $\frak{c}$ is uplifting in $L$; and conversely, given any uplifting cardinal $\kappa$, we may perform a suitable lottery iteration of $\Gamma$ forcing to obtain the resurrection axiom for $\Gamma$ in a forcing extension with $\kappa=\frak{c}$.

In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $A\subset\frak{c}$ and ask for extensions of the form $\langle H_{\frak{c}},{\in},A\rangle\prec\langle H_{\frak{c}}^{V[g\ast h]},{\in},A^\ast\rangle$, for some $A^\ast\subset\frak{c}^{V[g\ast h]}$ in the extension.  In that article, we prove the equiconsistency of various formulations of boldface resurrection with the existence of a strongly uplifting cardinal, which we prove is the same as a superstrongly unfoldable cardinal.

Superstrong and other large cardinals are never Laver indestructible

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[bibtex key=BagariaHamkinsTsaprounisUsuba2016:SuperstrongAndOtherLargeCardinalsAreNeverLaverIndestructible]

Abstract.  Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, $1$-extendible cardinals, $0$-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals (all for $n\geq 3$) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ or larger.

The large cardinal indestructibility phenomenon, occurring when certain preparatory forcing makes a given large cardinal become necessarily preserved by any subsequent forcing from a large class of forcing notions, is pervasive in the large cardinal hierarchy. The phenomenon arose in Laver’s seminal result that any supercompact cardinal $\kappa$ can be made indestructible by ${\lt}\kappa$-directed closed forcing. It continued with the Gitik-Shelah treatment of strong cardinals; the universal indestructibility of Apter and myself, which produced simultaneous indestructibility for all weakly compact, measurable, strongly compact, supercompact cardinals and others; the lottery preparation, which applies generally to diverse large cardinals; work of Apter, Gitik and Sargsyan on indestructibility and the large-cardinal identity crises; the indestructibility of strongly unfoldable cardinals; the indestructibility of Vopenka’s principle; and diverse other treatments of large cardinal indestructibility. Based on these results, one might be tempted to the general conclusion that all the usual large cardinals can be made indestructible.

In this article, my co-authors and I temper that temptation by proving that certain kinds of large cardinals cannot be made nontrivially indestructible. Superstrong cardinals, we prove, are never Laver indestructible. Consequently, neither are almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals and $1$-extendible cardinals, to name a few. Even the $0$-extendible cardinals are never indestructible, and neither are weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, strongly uplifting cardinals, superstrongly unfoldable cardinals, $\Sigma_n$-reflecting cardinals, $\Sigma_n$-correct cardinals and $\Sigma_n$-extendible cardinals, when $n\geq 3$. In fact, all these large cardinal properties are superdestructible, in the sense that if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the large cardinal properties with target $\theta$ or larger. Many quite ordinary forcing notions, which one might otherwise have expected to fall under the scope of an indestructibility result, will definitely ruin all these large cardinal properties. For example, adding a Cohen subset to any cardinal $\kappa$ will definitely prevent it from being superstrong—as well as preventing it from being uplifting, $\Sigma_3$-correct, $\Sigma_3$-extendible and so on with all the large cardinal properties mentioned above—in the forcing extension.

Main Theorem. 

  1. Superstrong cardinals are never Laver indestructible.
  2. Consequently, almost huge, huge, superhuge and rank-into-rank cardinals are never Laver indestructible.
  3. Similarly, extendible cardinals, $1$-extendible and even $0$-extendible cardinals are never Laver indestructible.
  4. Uplifting cardinals, pseudo-uplifting cardinals, weakly superstrong cardinals, superstrongly unfoldable cardinals and strongly uplifting cardinals are never Laver indestructible.
  5. $\Sigma_n$-reflecting and indeed $\Sigma_n$-correct cardinals, for each finite $n\geq 3$, are never Laver indestructible.
  6. Indeed—the strongest result here, because it is the weakest notion—$\Sigma_3$-extendible cardinals are never Laver indestructible.

In fact, each of these large cardinal properties is superdestructible. Namely, if $\kappa$ exhibits any of them, with corresponding target $\theta$, then in any forcing extension arising from nontrivial strategically ${\lt}\kappa$-closed forcing $\mathbb{Q}\in V_\theta$, the cardinal $\kappa$ will exhibit none of the mentioned large cardinal properties with target $\theta$ or larger.

The proof makes use of a detailed analysis of the complexity of the definition of the ground model in the forcing extension.  These results are, to my knowledge, the first applications of the ideas of set-theoretic geology not making direct references to set-theoretically geological concerns.

Theorem 10 in the article answers (the main case of) a question I had posed on MathOverflow, namely, Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals?  I had been specifically interested there to know whether a cardinal $\kappa$ necessarily becomes definable after adding a Cohen subset to it, and theorem 10 shows indeed that it does:  after adding a Cohen subset to a cardinal, it becomes $\Sigma_3$-definable in the extension, and this fact can be seen as explaining the main theorem above.

Related MO question | CUNY talk

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact

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[bibtex key=CodyGitikHamkinsSchanker2015:LeastWeaklyCompact]

Abstract.   We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, for nearly any desired function $\kappa\mapsto\theta_\kappa$. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.

In this article, we prove that the least weakly compact cardinal can exhibit any of several much stronger large cardinal properties. Namely, the least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact for any desired $\theta$.

Main Theorem.  Assuming a suitable large cardinal hypothesis, the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$.

Meanwhile, the least weakly compact cardinal can never exhibit these extra large cardinal properties in $L$, and indeed, the existence of a weakly measurable cardinal in the constructible universe is impossible. Furthermore, in each case the extra properties are strictly stronger than weak compactness in consistency strength.

We show in addition a more global result, that the entire class of weakly compact cardinals can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals, and with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, with enormous flexibility in the map $\kappa\mapsto\theta_\kappa$.

Our results therefore extend the `identity-crises’ phenomenon—first identified (and named) by Magidor—which occurs when a given large cardinal property can be made in various models to coincide either with much stronger or with much weaker large cardinal notions. Magidor had proved that the least strongly compact cardinal can be the least supercompact cardinal in one model of set theory and the least measurable cardinal in another. Here, we extend the phenomenon to weak measurability, partial near supercompactness and unfoldability. Specifically, the least weakly measurable cardinal coincides with the least measurable cardinal under the GCH, but it is the least weakly compact cardinal in our main theorem. Similarly, the least cardinal $\kappa$ that is nearly $\kappa^{+}$-supercompact is measurable with nontrivial Mitchell order under the GCH, but it is the least weakly compact cardinal here (and similar remarks apply to near $\kappa^{++}$-supercompactness and so on). The least unfoldable cardinal is strongly unfoldable in $L$, and therefore a $\Sigma_2$-reflecting limit of weakly compact cardinals there, but it is the least weakly compact cardinal in our main theorem. The global results of section 6 show just how malleable these notions are.

Norman Lewis Perlmutter

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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

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

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.

Pluralism in mathematics: the multiverse view in set theory and the question of whether every mathematical statement has a definite truth value, Rutgers, March 2013

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This is a talk for the Rutgers Logic Seminar on March 25th, 2013.  Simon Thomas specifically requested that I give a talk aimed at philosophers.

Abstract.  I shall describe the debate on pluralism in the philosophy of set theory, specifically on the question of whether every mathematical and set-theoretic assertion has a definite truth value. A traditional Platonist view in set theory, which I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. I shall try to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Some of this material arises in my recent articles:

Superstrong cardinals are never Laver indestructible, and neither are extendible, almost huge and rank-into-rank cardinals, CUNY, January 2013

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This is a talk for the CUNY Set Theory Seminar on February 1, 2013, 10:00 am.

Abstract.  Although the large cardinal indestructibility phenomenon, initiated with Laver’s seminal 1978 result that any supercompact cardinal $\kappa$ can be made indestructible by $\lt\kappa$-directed closed forcing and continued with the Gitik-Shelah treatment of strong cardinals, is by now nearly pervasive in set theory, nevertheless I shall show that no superstrong strong cardinal—and hence also no $1$-extendible cardinal, no almost huge cardinal and no rank-into-rank cardinal—can be made indestructible, even by comparatively mild forcing: all such cardinals $\kappa$ are destroyed by $\text{Add}(\kappa,1)$, by $\text{Add}(\kappa,\kappa^+)$, by $\text{Add}(\kappa^+,1)$ and by many other commonly considered forcing notions.

This is very recent joint work with Konstantinos Tsaprounis and Joan Bagaria.

nylogic.org | Set Theory Seminar |

Pluralism in set theory: does every mathematical statement have a definite truth value? GC Philosophy Colloquium, 2012

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This will be my talk for the CUNY Graduate Center Philosophy Colloquium on November 28, 2012.

I will be speaking on topics from some of my recent articles:

I shall give a summary account of some current issues in the philosophy of set theory, specifically, the debate on pluralism and the question of the determinateness of set-theoretical and mathematical truth.  The traditional Platonist view in set theory, what I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value.  What I would like to do is to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.  A competing view, which I call the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths.  After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Slides

A multiverse perspective on the axiom of constructiblity

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[bibtex key=Hamkins2014:MultiverseOnVeqL]

This article expands on an argument that I made during my talk at the Asian Initiative for Infinity: Workshop on Infinity and Truth, held July 25–29, 2011 at the Institute for Mathematical Sciences, National University of Singapore, and will be included in a proceedings volume that is being prepared for that conference.

Abstract. I argue that the commonly held $V\neq L$ via maximize position, which rejects the axiom of constructibility $V=L$ on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of $V=L$ inside larger set-theoretic universes.

In section two, I provide a few new criticisms of Maddy’s proposed concept of `restrictive’ theories, pointing out that her concept of fairly interpreted in is not a transitive relation: there is a first theory that is fairly interpreted in a second, which is fairly interpreted in a third, but the first is not fairly interpreted in the third.  The same example (and one can easily construct many similar natural examples) shows that neither the maximizes over relation, nor the properly maximizes over relation, nor the strongly maximizes over relation is transitive.  In addition, the theory ZFC + “there are unboundedly many inaccessible cardinals” comes out as formally restrictive, since it is strongly maximized by the theory ZF + “there is a measurable cardinal, with no worldly cardinals above it.”

To support the main philosophical thesis of the article, I survey a series of mathemtical results,  which reveal various senses in which the axiom of constructibility $V=L$ is compatible with strength in set theory, particularly if one has in mind the possibility of moving from one universe of set theory to a much larger one.  Among them are the following, which I prove or sketch in the article:

Observation. The constructible universe $L$ and $V$ agree on the consistency of any constructible theory. They have models of the same constructible theories.

Theorem. The constructible universe $L$ and $V$ have transitive models of exactly the same constructible theories in the language of set theory.

Corollary. (Levy-Shoenfield absoluteness theorem)  In particular, $L$ and $V$ satisfy the same $\Sigma_1$ sentences, with parameters hereditarily countable in $L$. Indeed, $L_{\omega_1^L}$ and $V$ satisfy the same such sentences.

Theorem. Every countable transitive set is a countable transitive set in the well-founded part of an $\omega$-model of V=L.

Theorem. If there are arbitrarily large $\lambda<\omega_1^L$ with $L_\lambda\models\text{ZFC}$, then every countable transitive set $M$ is a countable transitive set inside a structure $M^+$  that is a pointwise-definable model of ZFC + V=L, and $M^+$ is well founded as high in the countable ordinals as desired.

Theorem. (Barwise)  Every countable model of  ZF has an end-extension to a model of ZFC + V=L.

Theorem. (Hamkins, see here)  Every countable model of set theory $\langle M,{\in^M}\rangle$, including every transitive model, is isomorphic to a submodel of its own constructible universe $\langle L^M,{\in^M}\rangle$. In other words,  there is an embedding $j:M\to L^M$, which is elementary for quantifier-free assertions.

Another way to say this is that every countable model of set theory is a submodel of a model isomorphic to $L^M$. If we lived inside $M$, then by adding new sets and elements, our universe could be transformed into a copy of the constructible universe $L^M$.

(Plus, the article contains some nice diagrams.)

Related Singapore links:

A question for the mathematics oracle

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At the Workshop on Infinity and Truth in Singapore last year, we had a special session in which the speakers were asked to imagine that they had been granted an audience with an all-knowing mathematical oracle, given the opportunity to ask a single yes-or-no question, to be truthfully answered.  These questions will be gathered together and published in the conference volume.  Here is my account.

 

A question for the mathematics oracle

Joel David Hamkins, The City University of New York

 

Granted an audience with an all-knowing mathematics oracle, we may ask a single yes-or-no question, to be truthfully answered……

I might mischievously ask the question my six-year-old daughter Hypatia often puts to our visitors:  “Answer yes or no.  Will you answer `no’?” They stammer, caught in the liar paradox, as she giggles. But my actual question is:

Are we correct in thinking we have an absolute concept of the finite?

An absolute concept of the finite underlies many mathematician’s understanding of the nature of mathematical truth. Most mathematicians, for example, believe that we have an absolute concept of the finite, which determines the natural numbers as a unique mathematical structure—$0,1,2,$ and so on—in which arithmetic assertions have definitive truth values. We can prove after all that the second-order Peano axioms characterize $\langle\mathbb{N},S,0\rangle$ as the unique inductive structure, determined up to isomorphism by the fact that $0$ is not a successor, the successor function $S$ is one-to-one and every set containing $0$ and closed under $S$ is the whole of $\mathbb{N}$. And to be finite means simply to be equinumerous with a proper initial segment of this structure. Doesn’t this categoricity proof therefore settle the matter?

I don’t think so. The categoricity proof, which takes place in set theory, seems to my way of thinking merely to push the absoluteness question for finiteness off to the absoluteness question for sets instead. And surely this is a murkier realm, where already mathematicians do not universally agree that we have a single absolute background concept of set. We know by forcing and other means how to construct alternative set concepts, which seem fully as legitimate and set-theoretic as the set concepts from which they are derived. Thus, we have a plurality of set concepts, and our confidence in a unique absolute set-theoretic background is weakened. How then can we sensibly base our confidence in an absolute concept of the finite on set theory? Perhaps this absoluteness is altogether illusory.

My worries are put to rest if the oracle should answer positively. A negative answer, meanwhile, would raise alarms. A negative answer could indicate, on the one hand, that our understanding of the finite is simply incoherent, a catastrophe, where our cherished mathematical theories are all inconsistent. But, more likely in my view, a negative answer could also mean that there is an undiscovered plurality of concepts of the finite. I imagine technical developments arising that would provide us with tools to modify the arithmetic of a model of set theory, for example, with the same power and flexibility that forcing currently allows us to modify higher-order features, while not providing us with any reason to prefer one arithmetic to another (unlike our current methods with non-standard models). The discovery of such tools would be an amazing development in mathematics and lead to radical changes in our conception of mathematical truth.

Let’s have some fun—please post your question for the oracle in the comment fields below.

A question for the math oracle (pdf) | My talk at the Workshop