• J. D. Hamkins, “A multiverse perpsective on the axiom of constructibility.” , pp. 1-23. (under review)  
  Citation arχiv

  @INBOOK{Hamkins:MultiverseOnVeqL,
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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:

• The Multiverse view in set theory
• A question for the mathematics oracle

• J. D. Hamkins and B. Löwe, “Moving up and down in the generic multiverse,” ICLA 2013 LNCS, vol. 7750, pp. 139-147, 2013.  
  Citation arχiv

  @ARTICLE{HamkinsLoewe:MovingUpAndDownInTheGenericMultiverse,
  AUTHOR = {Joel David Hamkins and Benedikt L\"owe},
  title = {Moving up and down in the generic multiverse},
  journal = {ICLA 2013 LNCS},
  year = {2013},
  volume = {7750},
  number = {},
  pages = {139--147},
  month = {},
  note = {},
  url = {http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse},
  url = {http://arxiv.org/abs/1208.5061},
  eprint = {1208.5061},
  abstract = {},
  keywords = {},
  source = {},
  }

In this extended abstract we investigate the modal logic of the generic multiverse, which is a bimodal logic with operators corresponding to the relations “is a forcing extension of”‘ and “is a ground model of”. The fragment of the first relation is the modal logic of forcing and was studied by us in earlier work. The fragment of the second relation is the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.

The main theorems are as follows:

Theorem.  If  ZFC is consistent, then there is a model of  ZFC  whose modal logic of forcing and modal logic of grounds are both S4.2.

Theorem.  If  the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S4.2 and whose modal logic of grounds is S5.

Theorem.  If  the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S5 and whose modal logic of grounds is S4.2.

Theorem. There is no model of set theory such that both its modal logic of forcing and its modal logic of grounds are S5.

The current article is a brief extended abstract (10 pages).  A fuller account with more detailed proofs and further information will be provided in a subsequent articl

eprints:  ar$\chi$iv | NI12059-SAS | Hamburg #450

• J. D. Hamkins, G. Leibman, and B. Löwe, “Structural connections between a forcing class and its modal logic,” , pp. 1-23. (under review)  
  Citation arχiv

  @ARTICLE{HamkinsLeibmanLoewe:StructuralConnectionsForcingClassAndItsModalLogic,
  AUTHOR = {Joel David Hamkins and George Leibman and Benedikt L\"owe},
  TITLE = {Structural connections between a forcing class and its modal logic},
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  url = {http://arxiv.org/abs/1207.5841},
  url = {http://jdh.hamkins.org/a-forcing-class-and-its-modal-logic},
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The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting $\Box$ as “in all forcing extensions” and $\Diamond$ as “in some forcing extension”. In this modal language one may easily express sweeping general forcing principles, such as $\Diamond\Box\varphi\to\Box\Diamond\varphi$, the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or $\Diamond\Box\varphi\to\varphi$, the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC (see A simple maximality principle).

Every definable forcing class similarly gives rise to the corresponding forcing modalities, for which one considers extensions only by forcing notions in that class. In previous work, we proved that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2 (see The modal logic of forcing). In this article, we prove that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of $\omega_1$-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise, including the connection between various control statements, such as buttons, switches and ratchets, and their corresponding forcing validities. These structural connections therefore support a forcing-only analysis of other diverse forcing classes.

Preprints available at:  ar$\chi$iv | NI12055-SAS | UvA ILLC PP-2012-19 | HBM 446

• J. D. Hamkins and D. Seabold, “Well-founded Boolean ultrapowers as large cardinal embeddings,” , pp. 1-40. (under review)  
  Citation arχiv

  @ARTICLE{HamkinsSeabold:BooleanUltrapowers,
  AUTHOR = "Joel David Hamkins and Daniel Seabold",
  TITLE = "Well-founded {Boolean} ultrapowers as large cardinal embeddings",
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Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory—forcing and large cardinals—by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.

The topic of this article was the focus of my tutorial lecture series at the Young Set Theorists Workshop at the Hausdorff Center for Mathematics in Königswinter near Bonn, Germany, March 21-25, 2011.

• J. D. Hamkins, “Is the dream solution of the continuum hypothesis attainable?,” , pp. 1-10. (under review)  
  Citation arχiv

  @ARTICLE{Hamkins:IsTheDreamSolutionToTheContinuumHypothesisAttainable,
  AUTHOR = {Joel David Hamkins},
  TITLE = {Is the dream solution of the continuum hypothesis attainable?},
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  eprint = {1203.4026},
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Many set theorists yearn for a definitive solution of the continuum problem, what I call a  dream solution, one by which we settle the continuum hypothesis (CH) on the basis of a new fundamental principle of set theory, a missing axiom, widely regarded as true, which determines the truth value of CH.  In an earlier article, I have described the dream solution template as proceeding in two steps: first, one introduces the new set-theoretic principle, considered obviously true for sets in the same way that many mathematicians find the axiom of choice or the axiom of replacement to be true; and second, one proves the CH or its negation from this new axiom and the other axioms of set theory. Such a situation would resemble Zermelo’s proof of the ponderous well-order principle on the basis of the comparatively natural axiom of choice and the other Zermelo axioms. If achieved, a dream solution to the continuum problem would be remarkable, a cause for celebration.

In this article, however, I argue that a dream solution of CH has become impossible to achieve. Specifically, what I claim is that our extensive experience in the set-theoretic worlds in which CH is true and others in which CH is false prevents us from looking upon any statement settling CH as being obviously true. We simply have had too much experience by now with the contrary situation. Even if set theorists initially find a proposed new principle to be a natural, obvious truth, nevertheless once it is learned that the principle settles CH, then this preliminary judgement will evaporate in the face of deep experience with the contrary, and set-theorists will look upon the statement merely as an intriguing generalization or curious formulation of CH or $\neg$CH, rather than as a new fundamental truth. In short, success in the second step of the dream solution will inevitably undermine success in the first step.

This article is based upon an argument I gave during the course of a three-lecture tutorial on set-theoretic geology at the summer school Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Development, at the University of London, Birkbeck in August 2011.  Much of the article is adapted from and expands upon the corresponding section of material in my article The set-theoretic multiverse.

• G. Fuchs, J. D. Hamkins, and J. Reitz, “Set-theoretic geology.” (under review)  
  Citation arχiv

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  AUTHOR = "Gunter Fuchs and Joel David Hamkins and Jonas Reitz",
  TITLE = "Set-theoretic geology",
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  url = "http://arxiv.org/abs/1107.4776",
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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.

• J. D. Hamkins, “The set-theoretic multiverse,” Review of Symbolic Logic, vol. 5, pp. 416-449, 2012.  
  Journal Citation arχiv

  @ARTICLE{Hamkins2012:TheSet-TheoreticalMultiverse,
  AUTHOR = {Joel David Hamkins},
  TITLE = {The set-theoretic multiverse},
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  volume = {5},
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The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I 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.

Multiversive at n-Category Cafe | Multiverse on Mathoverflow

• V. Gitman and J. D. Hamkins, “A natural model of the multiverse axioms,” Notre Dame J. Form. Log., vol. 51, iss. 4, pp. 475-484, 2010.  
  Journal MR Citation arχiv

  @ARTICLE{GitmanHamkins2010:NaturalModelOfMultiverseAxioms,
  AUTHOR = {Gitman, Victoria and Hamkins, Joel David},
  TITLE = {A natural model of the multiverse axioms},
  JOURNAL = {Notre Dame J. Form. Log.},
  FJOURNAL = {Notre Dame Journal of Formal Logic},
  VOLUME = {51},
  YEAR = {2010},
  NUMBER = {4},
  PAGES = {475--484},
  ISSN = {0029-4527},
  MRCLASS = {03E40},
  MRNUMBER = {2741838},
  DOI = {10.1215/00294527-2010-030},
  URL = {http://dx.doi.org/10.1215/00294527-2010-030},
  eprint = {1104.4450},
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In this article, we prove that if ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms that I introduced in my paper, “The set-theoretic multiverse.”

• J. D. Hamkins, “Some second order set theory,” , R.~Ramanujam and S.~Sarukkai, Ed., Berlin: Springer, 2009, vol. 5378, pp. 36-50.  
  Journal MR Citation

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  TITLE = {Some second order set theory},
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  VOLUME = {5378},
  PAGES = {36--50},
  PUBLISHER = {Springer},
  EDITOR = {R.~Ramanujam and S.~Sarukkai},
  ADDRESS = {Berlin},
  YEAR = {2009},
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  MRNUMBER = {2540935 (2011a:03053)},
  DOI = {10.1007/978-3-540-92701-3_3},
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• J. D. Hamkins, J. Reitz, and H. W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc. Amer. Math. Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.  
  Journal MR Citation

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  AUTHOR = {Hamkins, Joel David and Reitz, Jonas and Woodin, W. Hugh},
  TITLE = {The ground axiom is consistent with {$V\ne{\rm HOD}$}},
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  VOLUME = {136},
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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.

• J. D. Hamkins and B. Löwe, “The modal logic of forcing,” Trans. Amer. Math. Soc., vol. 360, iss. 4, pp. 1793-1817, 2008.  
  Journal MR Citation arχiv

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  AUTHOR = {Hamkins, Joel David and L{\"o}we, Benedikt},
  TITLE = {The modal logic of forcing},
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  FJOURNAL = {Transactions of the American Mathematical Society},
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  MRREVIEWER = {Andreas Blass},
  DOI = {10.1090/S0002-9947-07-04297-3},
  URL = {http://dx.doi.org/10.1090/S0002-9947-07-04297-3},
  eprint = {math/0509616},
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What are the most general principles in set theory relating forceability and truth? As with Solovay’s celebrated analysis of provability, both this question and its answer are naturally formulated with modal logic. We aim to do for forceability what Solovay did for provability. A set theoretical assertion $\psi$ is forceable or possible, if $\psi$ holds in some forcing extension, and necessary, if $\psi$ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory known as S4.2.

Follow-up article:  Structural connections between a forcing class and its modal logic

• J. D. Hamkins, “A simple maximality principle,” J. Symbolic Logic, vol. 68, iss. 2, pp. 527-550, 2003.  
  Journal MR Citation arχiv

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  AUTHOR = {Hamkins, Joel David},
  TITLE = {A simple maximality principle},
  JOURNAL = {J. Symbolic Logic},
  FJOURNAL = {The Journal of Symbolic Logic},
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  month = {June},
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In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence$\varphi$ holding in some forcing extension $V^{\mathbb{P}}$ and all subsequent extensions $V^{\mathbb{P}*\mathbb{Q}}$ holds already in $V$. It follows, in fact, that such sentences must also hold in all forcing extensions of $V$. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\Diamond\square\varphi)\to\square\varphi$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $\varphi$, is equiconsistent with the scheme asserting that $V_\delta$ is an elementary substructure of $V$ for an inaccessible cardinal $\delta$, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0^\sharp$ exists, that $x^\sharp$ exists for every set $x$, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.