Category Archives: Talks

The theory of infinite games: how to play infinite chess and win, VCU Math Colloquium, November 2014

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Releasing the hordesI shall speak at the Virginia Commonwealth University Math Colloquium on November 21, 2014.

Abstract. I shall give a general introduction to the theory of infinite games, using infinite chess—chess played on an infinite chessboard stretching without bound in every direction—as a central example. Since chess, when won, is always won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values, which provide a measure in a position of the distance remaining to victory. I shall exhibit several interesting positions in infinite chess with very high transfinite ordinal game values. Some of these positions involve large numbers of pieces, and the talk will include animations of infinite chess in play, with hundreds of pieces (or infinitely many) making coordinated attacks on the board. Meanwhile, the precise ordinal value of the omega one of chess is an open mathematical question.

Slides | Transfinite game values in infinite chess | The mate-in-n problem of infinite chess is decidable

The span of infinity, roundtable discussion at The Helix Center, October 2014

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I was a panelist at The Span of Infinity, a roundtable discussion held at The Helix Center, at the New York Psychoanalytic Society & Institute, 247 E 82nd Street, on October 25, 2014, 2:30 – 4:30 pm.

The Helix Center describes the discussion topic as:

Perhaps no thing conceived in the mind has enjoyed a greater confluence of cosmological, mathematical, philosophical, psychological, and theological inquiry than the notion of the infinite. The epistemological tension between the concrete and the ideal, between the phenomenological and the ontological, is nowhere clearer in outline yet more obscure in content. These inherent paradoxes limn the vital, eternal questions we will explore about humankind’s place in the universe and the comprehensibility of existence.

The Helix Center Roundtable Series is described by:

Our roundtable format is designated the Theaetetus Table, an extempore discussion among five participants, all leaders in their respective fields, and named for the classical Greek mathematician and eponym for the Platonic dialogue investigating the nature of knowledge, who proved that there are five regular convex polyhedra, or Platonic solids. Each Theaetetus Table aspires to emulate the dialogue’s unhurried search for wisdom; and, like the five Platonic solids held to be the fundamental building blocks of the classical elements, the contributions of our five participants become the fundamental constituents of interdisciplinary insights emerging in the alchemy of the roundtable, insights that, in turn, transform the elemental thinking of those participants. The gathering of five discussants also symbolizes the five interrelated qualities of mind our interdisciplinary forums are intended to facilitate in our participants, and inculcate in our audience: curiosity, playfulness, inspiration, reflection, and wonder.

The video of the actual event is now available:

The pluralist perspective on the axiom of constructibility, MidWest PhilMath Workshop, Notre Dame, October 2014

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University of Notre DameThis will be a featured talk at the Midwest PhilMath Workshop 15, held at Notre Dame University October 18-19, 2014.  W. Hugh Woodin and I will each give one-hour talks in a session on Perspectives on the foundations of set theory, followed by a one-hour discussion of our talks.

Abstract. I shall 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.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible. Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory. In this talk, I shall criticize Maddy’s proposal, pointing out that neither the fairly-interpreted-in relation nor the (strongly) maximizes-over relation is transitive, and furthermore, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s account, contrary to what had been desired. Ultimately, I shall argue that the V≠L via maximize position loses its force on a multiverse conception of set theory with an upwardly extensible concept of set, in light of the classical facts that models of set theory can generally be extended to models of V=L. I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Slides

Large cardinals need not be large in HOD, International Workshop on Set Theory, CIRM, Luminy, September 2014

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I shall speak at the 13th International Workshop on Set Theory, held at the CIRM Centre International de Rencontres Mathématiques in Luminy near Marseille, France, September 29 to October 3, 2014. 

Abstract.  I shall prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals. There are many open questions.

This talk will include joint work with Cheng Yong and Sy-David Friedman.

Article | Participants | Slides

A meeting at the crossroads – science, performance and the art of possibility, panel discussion, Underground Zero Festival, Intrinsic Value Project, July 2014

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I shall be a panelist at A meeting at the crossroads – science, performance and the art of possibility, a panel discussion considering the intrinsic value of Art and Science, a part of the Intrinsic Value series at the Undergroundzero Festival 2014.

Are theatre and the arts vital to life here and now? Does science creatively address the larger questions of our time? This panel will bring together distinguished scientists and theatre professionals to answer these questions. They will consider how both areas are intrinsically valuable to society and investigate the performative possibilities when the two fields overlap.

At the the Abrazo Interno Gallery, Clemente Soto Vélez, 107 Suffolk Street New York NY 10002. (Venue and Tickets) July 9 & 10, 2014, 7-8 pm.

Higher infinity and the foundations of mathematics, plenary General Public Lecture, AAAS, June, 2014

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I have been invited to give a plenary General Public Lecture at the 95th annual meeting of the American Association for the Advancement of Science (Pacific Division), which will be held in Riverside, California, June 17-20, 2014.  The talk is sponsored by the BEST conference, which is meeting as a symposium at the larger AAAS conference.

This is truly a rare opportunity to communicate with a much wider community of scholars, to explain some of the central ideas and methods of set theory and the foundations of mathematics to a wider group of nonspecialist but mathematics-interested researchers. I hope to explain a little about the exciting goings-on in the foundations of mathematics.  Frankly, I feel deeply honored for the opportunity to represent my field in this way.

The talk will be aimed at a very general audience, the general public of the AAAS meeting, which is to say, mainly, scientists.  I also expect, however, that there will be a set-theory contingent present of participants from the BEST conference, which is a symposium at the conference — but I shall not take a stand here on whether mathematics is a science; you’ll have to come to my talk for that!

MissionInnPanoramaBestAbstract. Let me tell you the story of infinity and what is going on in the foundations of mathematics. For over a century, mathematicians have explored the soaring transfinite tower of different infinity concepts. Yet, fundamental questions at the foundation of this tower remain unsettled. Indeed, researchers in set theory and the foundations of mathematics have uncovered a pervasive independence phenomenon, whereby foundational mathematical questions are often in principle neither provable nor refutable. Presented with what may be these inherent limitations on our mathematical reasoning, we now face difficult philosophical questions on the nature of mathematical truth and the meaning of mathematical existence. Does mathematics need new axioms? Some mathematicians point the way the way towards what they describe as an ultimate theory of mathematical truth. Some adopt a scientific attitude, judging new mathematical axioms and theories by their predictions and explanatory power. Others propose a multiverse mathematical foundation with pluralist truth. In this talk, I shall take you from the basic concept of infinity and some simple paradoxes up to the continuum hypothesis and on to the higher infinity of large cardinals and the raging philosophical debates.

Slides | AAAS PD 2014 | Schedule | BEST | My other BEST talk

Boldface resurrection and the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, BEST 2014

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I will speak at the BEST conference, which is held as a symposium in the much larger 95th Annual Meeting of the American Association for the Advancement of Science, at the University of California at Riverside, June 18-20, 2014.

This talk will be for specialists in the BEST symposium.

Abstract.  I shall introduce several new large cardinal concepts, namely, the strongly uplifting cardinals, the superstrongly unfoldable cardinals and the almost-hugely unfoldable cardinals, and prove their tight connection with one another — actually, they are equivalent! — as well as their equiconsistency with several natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.  This is joint work with Thomas A. Johnstone.

I am also scheduled to give a plenary General Pubic Lecture, entitled Higher infinity and the foundations of mathematics, as a part of the larger AAAS program, to which the general public is invited.

Slides | Article | Program | AAAS General Public Lecture

Tutorial on Boolean ultrapowers, BLAST 2015, Las Cruces, NM

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I shall give a tutorial lecture series on Boolean ultrapowers, two or three lectures, at the BLAST conference in Las Cruces, New Mexico, January 5-9, 2015.  (The big AMS meeting in San Antonio, reportedly a quick flight, begins on the 10th.)

BLAST is a conference series focusing on

B = Boolean Algebras
L = Lattices, Algebraic and Quantum Logic
A = Universal Algebra
S = Set Theory
T = Set-theoretic and Point-free Topology

In this tutorial, I shall give a general introduction to the Boolean ultrapower construction.

Organ Mountains, NM, with snow

Boolean ultrapowers generalize the classical ultrapower construction on a power-set algebra to the context of an ultrafilter on an arbitrary complete Boolean algebra. Introduced by Vopěnka as a means of undertaking forcing constructions internally to ZFC, the method has many connections with forcing. Nevertheless, the Boolean ultrapower construction stands on its own as a general model-theoretic construction technique, and historically, researchers have come to the Boolean ultrapower concept from both set theory and model theory.  An emerging interest in Boolean ultrapowers arises from a focus on well-founded Boolean ultrapowers as large cardinal embeddings.

In this tutorial, we shall see that the Boolean ultrapower construction reveals that two central set-theoretic techniques–forcing and classical ultrapowers–are facets of a single underlying construction, namely, the Boolean ultrapower.  I shall provide a thorough introduction to the Boolean ultrapower construction, assuming only an elementary graduate student-level familiarity with set theory and the classical ultrapower and forcing techniques.

Article | Tutorial lecture notes  | Blast 2015 | BLAST 2013 | Boolean ultrapowers tutorial at YSTW Bonn, 2011

A natural strengthening of Kelley-Morse set theory, CUNY Logic Workshop, May 2014

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This will be a talk for the CUNY Logic Workshop on May 2, 2014.

Abstract. I shall introduce a natural strengthening of Kelley-Morse set theory KM to the theory we denote KM+, by including a certain class collection principle, which holds in all the natural models usually provided for KM, but which is not actually provable, we show, in KM alone.  The absence of the class collection principle in KM reveals what can be seen as a fundamental weakness of this classical theory, showing it to be less robust than might have been supposed.  For example, KM proves neither the Łoś theorem nor the Gaifman lemma for (internal) ultrapowers of the universe, and furthermore KM is not necessarily preserved, we show, by such ultrapowers. Nevertheless, these weaknesses are corrected by strengthening it to the theory KM+. The talk will include a general elementary introduction to the various second-order set theories, such as Gödel-Bernays set theory and Kelley-Morse set theory, including a proof of the folklore fact that KM implies Con(ZFC). This is joint work with Victoria Gitman and Thomas Johnstone.

 

Transfinite game values in infinite chess and other infinite games, Hausdorff Center, Bonn, May 2014

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Releasing the hordesI shall be very pleased to speak at the colloquium and workshop Infinity, computability, and metamathematics, celebrating the 60th birthdays of Peter Koepke and Philip Welch, held at the Hausdorff Center for Mathematics May 23-25, 2014 at the Universität Bonn.  My talk will be the Friday colloquium talk, for a general mathematical audience.

Abstract. I shall give a general introduction to the theory of infinite games, using infinite chess—chess played on an infinite edgeless chessboard—as a central example. Since chess, when won, is won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values. I shall exhibit several interesting positions in infinite chess with very high transfinite game values. The precise value of the omega one of chess is an open mathematical question.

 

Slides | Schedule | Transfinite game values in infinite chess | The mate-in-n problem of infinite chess is decidable

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.

Article

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.

Article

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

Universal structures, GC MathFest, February 2014

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Midtown in WinterThis will be a talk for the CUNY Graduate Center MathFest, held on the afternoon of Februrary 4, 2014, intended for graduate-school-bound undergraduate students, including prospective students for the CUNY Graduate Center, giving them a chance to meet graduate students and faculty at the CUNY Graduate Center and see the kind of mathematics that is done here.

In this 30 minute talk, I’ll introduce the concept of a universal structure, with various examples, including the countable random graph, the surreal number line and the hypnagogic digraph.

MathFest Program/schedule

Satisfaction is not absolute, Dartmouth Logic Seminar, January 2014

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dartmouth_campusThis will be a talk for the Dartmouth Logic Seminar on January 23rd, 2014.

Abstract. I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. The theorems are proved with elementary classical model-theoretic methods, and many of them can be considered folklore results in the subject of models of arithmetic.

Indefinite arithmetic truthOn the basis of these mathematical results, Ruizhi Yang (Fudan University, Shanghai) and I have argued that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

Main article: Satisfaction is not absolute