# Moving up and down in the generic multiverse

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

# 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.

# Is the dream solution of the continuum hypothesis attainable?

• J. D. Hamkins, “Is the dream solution of the continuum hypothesis attainable?,” Notre Dame J. Form. Log., vol. 56, iss. 1, pp. 135-145, 2015.
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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.

# Philosophy of set theory, Fall 2011, NYU PH GA 1180

I taught a course in Fall 2011 at NYU entitled Topics in Logic: set theory and the philosophy of set theory, aimed at graduate students in philosophy and others who want to gain greater understanding of some of the set-theoretic topics central to work in the philosophy of set theory.  The course began with a review of the mathematical ideas, including a presentation of large cardinals, strong axioms of infinity and their associated elementary embeddings of the universe, and forcing, emphasizing the connection with the Boolean ultrapower and Boolean-valued models, but discussing the alternative formalizations. The second part of the course covers some of the philosophical literature, including what it means to accept or believe mathematical axioms, whether mathematics needs new axioms, the criteria one might use when adopting new axioms, and the question of pluralism and categoricity in set theory.

Here is a partial list of our readings:

1. Mathematical background.

2.  Penelope Maddy, “Believing the axioms”, in two parts.  JSL vols. 52 and 53. Part 1Part 2

3. Chris Freiling, “Axioms of Symmetry: throwing darts at the real number line,”
JSL, vol. 51.   http://www.jstor.org/stable/2273955

4. W. N. Reinhardt, “Remarks on reflection principles, large cardinals, and elementary embeddings,” Proceedings of Symposia in Pure Mathematics, Vol 13, Part II, 1974, pp. 189-205.

5. Donald Martin, “Multiple universes of sets and indeterminate truth values,” Topoi 20, 5–16, 2001.

6. Hartry Field, “Which undecidable mathematical sentences have determinate truth values,” as reprinted in his book Truth and the Absence of Fact, Oxford University Press, 2001.

7. A brief selection from Marc Balaguer, Platonism and Anti-Platonism in Mathematics, Oxford University Press, 1998, describing the plenitudinous Platonism position.

8. Daniel Isaacson, “The reality of mathematics and the case of set theory,” 2007.

9. J. D. Hamkins, “The set-theoretic multiverse,” to appear in the Review of Symbolic Logic.

10.  Solomon Feferman, Does mathematics need new axioms? Text of an invited AMS-MAA joint meeting, San Diego, January, 1997.

11. Solomon Feferman, Is the continuum hypothesis a definite mathematical problem? Draft article for the Exploring the Frontiers of Independence lecture series at Harvard University, October, 2011.

12. Peter Koellner, Feferman On the Indefiniteness of CH, a commentary on Feferman’s EFI article.

13. Interpretability of theories, the interpretability degrees and Orey sentences in set theory and arithmetic.  Some of the basic material is found in Per Lindström’s book Aspects of Incompleteness, available at  http://projecteuclid.org/euclid.lnl/1235416274, particularly chapter 6, and some later chapters.

14. Haim Gaifman, “On ontology and realism in mathematics,” to appear in the Review of Symbolic Logic (special issue connected with the NYU philosophy of mathematics conference 2009).

15. Saharon Shelah, “Logical dreams,”  Bulletin of the AMS, 40(20):203–228, 2003. (Pre-publication version available at:http://arxiv.org/abs/math.LO/0211398)

16.  For mathematical/philosophical amusement, Philip Welch and Leon Horsten, “The aftermath.”

It’s been a great semester!

# Set theory: universe or multiverse? Vienna, 2011

This talk was held as part of the Logic Cafe series at the Institute of Philosophy at the University of Vienna, October 31, 2011.

A 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. On the multiverse view, in constrast, there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. In this talk, after framing the debate, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that 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.

# The set-theoretical multiverse

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

# The set-theoretical multiverse: a natural context for set theory, Japan 2009

• J. D. Hamkins, “The Set-theoretic Multiverse : A Natural Context for Set Theory,” Annals of the Japan Association for Philosophy of Science, vol. 19, pp. 37-55, 2011.
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This article is based on a talk I gave at the conference in honor of the retirement of Yuzuru Kakuda in Kobe, Japan, March 7, 2009. I would like to express my gratitude to Kakuda-sensei and the rest of the logic group in Kobe for the opportunities provided to me to participate in logic in Japan. In particular, my time as a JSPS Fellow in the logic group at Kobe University in 1998 was a formative experience. I was part of a vibrant research group in Kobe; I enjoyed Japanese life, learned to speak a little Japanese and made many friends. Mathematically, it was a productive time, and after years away how pleasant it is for me to see that ideas planted at that time, small seedlings then, have grown into tall slender trees.

Set theorists often take their subject as constituting a foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets. In this way, they regard the set-theoretic universe as the universe of all mathematics. And although many set-theorists affirm the Platonic view that there is just one universe of all sets, nevertheless the most powerful set-theoretic tools developed over the past half century are actually methods of constructing alternative universes. With forcing and other methods, we can now produce diverse models of ZFC set theory having precise, exacting features. The fundamental object of study in set theory has thus become the model of set theory, and the subject consequently begins to exhibit a category-theoretic second-order nature. We have a multiverse of set-theoretic worlds, connected by forcing and large cardinal embeddings like constellations in a dark sky. In this article, I will discuss a few emerging developments illustrating this second-order nature. The work engages pleasantly with various philosophical views on the nature of mathematical existence.

Slides

# A natural model of the multiverse axioms

• 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.
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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.”

# Some second order set theory

• J. D. Hamkins, “Some second order set theory,” in Logic and its applications, R.~Ramanujam and S.~Sarukkai, Eds., Berlin: Springer, 2009, vol. 5378, pp. 36-50.
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This article surveys two recent developments in set theory sharing an essential second-order nature, namely, the modal logic of forcing, oriented upward from the universe of set theory to its forcing extensions; and set-theoretic geology, oriented downward from the universe to the inner models over which it arises by forcing. The research is a mixture of ideas from several parts of logic, including, of course, set theory and forcing, but also modal logic, finite combinatorics and the philosophy of mathematics, for it invites a mathematical engagement with various philosophical views on the nature of mathematical existence.

# The Ground Axiom

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

# The multiverse perspective on determinateness in set theory, Harvard, 2011

This talk, taking place October 19, 2011, is part of the year-long Exploring the Frontiers of Incompleteness (EFI) series at Harvard University, a workshop focused on the question of determinateness in set theory, a central question in the philosophy of set theory.  Streaming video will be available on-line, and each talk will be associated with an on-line discussion forum, to which links will be made here later.

In this talk, I will discuss the multiverse perspective on determinateness in set theory.  The multiverse view in set theory 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 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.

# The multiverse view in set theory, Singapore 2011

A talk at the Asian Initive for Infinity: Workshop on Infinity and Truth, July 25-29, 2011, National University of Singapore.

I shall outline and defend the Multiverse view in set theory, the view that there are many set-theoretic universes, each instantiating its own concept of set, and contrast it with the Universe view, the view that there is an absolute background set-theoretic universe.  In addition, I will discuss some recent set-theoretic developments that have been motivated and informed by a multiverse perspective, including the modal logic of forcing and the emergence of set-theoretic geology.

# The set-theoretic multiverse: a model-theoretic philosophy of set theory, Paris, 2010

A talk at the Philosophy and Model Theory conference held June 2-5, 2010 at the Université Paris Ouest Nanterre.

Set theorists commonly regard set theory as an ontological foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, enjoying a real mathematical existence as sets accumulate to form the universe of all sets. The Universe view—perhaps it is the orthodox view among set theorists—takes this universe of sets to be unique, and holds that a principal task of set theory is to discover its fundamental truths. For example, on this view, interesting set-theoretical questions, such as the Continuum Hypothesis, will have definitive final answers in this universe. Proponents of this view point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy as indicating in broad strokes that we are on the right track towards these final answers.

A paradox for the orthodox view, however, is the fact that the most powerful tools in set theory are most naturally understood as methods for constructing alternative set-theoretic universes. With forcing and other methods, we seem to glimpse into alternative mathematical worlds, and are led to consider a model-theoretic, multiverse philosophical position. In this talk, I shall describe and defend the Multiverse view, which takes these other worlds at face value, holding that there are many set-theoretical universes. This is a realist position, granting these universes a full mathematical existence and exploring their interactions. The multiverse view remains Platonist, but it is second-order Platonism, that is, Platonism about universes. I shall argue that set theory is now mature enough to fruitfully adopt and analyze this view. I shall propose a number of multiverse axioms, provide a multiverse consistency proof, and describe some recent results in set theory that illustrate the multiverse perspective, while engaging pleasantly with various philosophical views on the nature of mathematical existence.

Slides  | Article | see related Singapore talk