Category Archives: Talks

Universality, saturation and the surreal number line, Shanghai, June 2013

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Fudan bridgeThis will be a short lecture series given at the conclusion of the graduate logic class in the Mathematical Logic group at Fudan University in Shanghai, June 13, 18 (or 20), 2013.

I will present an elementary introduction to the theory of universal orders and relations and saturated structures.  We’ll start with the classical fact, proved by Cantor, that the rational line is the universal countable linear order.  But what about universal partial orders, universal graphs and other mathematical structures?  Is there a computable universal partial order?  What is the countable random graph? Which orders embed into  the power set of the natural numbers under the subset relation ⟨P(N),⊂⟩? Proceeding to larger and larger universal orders, we’ll eventually arrive at the surreal numbers and the hypnagogic digraph.

Fudan University seal

 

Playing go with Jiachen

Playful paradox with large numbers, infinity and logic, Shanghai, June 2013

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Playful paradox

This will be a talk at Fudan University in Shanghai, China, June 12, 2013, sponsored by the group in Mathematical Logic at Fudan, for a large audience of students.

Abstract: For success in mathematics and science, I recommend an attitude of playful curiosity about one’s subject. We shall accordingly explore a number of puzzling conundrums at the foundations of mathematics concerning issues with large numbers, infinity and logic. These are serious issues—and we’ll have serious things to say—while still having fun. Can one complete a task involving infinitely many steps? Are there some real numbers that in principle cannot be described? Is every true statement provable? Does every mathematical problem ultimately reduce to computational procedure? What is the largest natural number that can be written or described in ordinary type Fudan University seal on an index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every sentence either true or false or neither true nor false? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics and philosophy.   The Largest-Number Contest.  In preparation for the talk, and with a nod to Douglas Hofstadter, we shall be holding a contest:  Who can describe the largest number on an ordinary index card?   See the contest announcement poster.

  1. A submission entry consists of the description of a positive integer written on an ordinary index card, using common mathematical operations and notation or English words and phrases.
  2. Write legibly, and use at most 100 symbols from the ordinary ASCII character set.  Bring your submission to the talk.
  3. Descriptions that fail to describe a number are disqualified.
  4. The submission with the largest number wins.
  5. The prize will be $1 million USD divided by the winning number itself, rounded to the nearest cent, plus another small token prize.

Example submissions: 

99999.

10*(10*99)+5

The population of Shanghai at this moment.

Read a more detailed account of the contest and its results.

Algebraicity and implicit definability in set theory, CUNY, May 2013

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This is a talk May 10, 2013 for the CUNY Set Theory Seminar.

Abstract.  An element a is definable in a model M if it is the unique object in M satisfying some first-order property. It is algebraic, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing a. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interesting new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.

This is joint work with Cole Leahy (MIT).

NYlogic abstract | MathOverflow post

The theory of infinite games, with examples, including infinite chess

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This will be a talk on April 30, 2013 for a joint meeting of the Yeshiva University Mathematics Club and the  Yeshiva University Philosophy Club.  The event will take place in 5:45 pm in Furst Hall, on the corner of Amsterdam Ave. and 185th St.

Abstract. I will give a general introduction to the theory of infinite games, suitable for mathematicians and philosophers.  What does it mean to play an infinitely long game? What does it mean to have a winning strategy for such a game?  Is there any reason to think that every game should have a winning strategy for one player or another?  Could there be a game, such that neither player has a way to force a win?  Must every computable game have a computable winning strategy?  I will present several game paradoxes and example infinitary games, including an infinitary version of the game of Nim, and several examples from infinite chess.

NYlogic entry | Yeshiva University | Infinite chess | Video

 

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:

The omega one of chess, CUNY, March, 2013

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This is a talk for the New York Set Theory Seminar on March 1, 2013.

This talk will be based on my recent paper with C. D. A. Evans, Transfinite game values in infinite chess.

Infinite chess is chess played on an infinite chessboard.  Since checkmate, when it occurs, does so after finitely many moves, this is technically what is known as an open game, and is therefore subject to the theory of open games, including the theory of ordinal game values.  In this talk, I will give a general introduction to the theory of ordinal game values for ordinal games, before diving into several examples illustrating high transfinite game values in infinite chess.  The supremum of these values is the omega one of chess, denoted by ω1Ch in the context of finite positions and by ω1Ch ∼ in the context of all positions, including those with infinitely many pieces. For lower bounds, we have specific positions with transfinite game values of ω, ω2, ω2⋅k and ω3. By embedding trees into chess, we show that there is a computable infinite chess position that is a win for white if the players are required to play according to a deterministic computable strategy, but which is a draw without that restriction. Finally, we prove that every countable ordinal arises as the game value of a position in infinite three-dimensional chess, and consequently the omega one of infinite three-dimensional chess is as large as it can be, namely, true ω1.

On the axiom of constructibility and Maddy’s conception of restrictive theories, Logic Workshop, February 2013

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This is a talk for the CUNY Logic Workshop on February 15, 2013.

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

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 specific proposal.  For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive.  Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.

Ultimately, I argue that the V≠L via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of V=L.  In particular, every countable model of set theory is a transitive set inside a model of V=L.  I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

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 κ can be made indestructible by <κ-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 κ are destroyed by Add(κ,1), by Add(κ,κ+), by Add(κ+,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

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

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

Abstract.  I will speak on my recent theorem that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding j:M→LM 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 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 LM contains a submodel that is a universal acyclic digraph of rank OrdM. 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 ω1+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 OrdM; 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 HFM of M. Indeed, HFM is universal for all countable acyclic binary relations.

Article | Rutgers Logic Seminar

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

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

Vestíbul Universitat de Barcelona

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→LM 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 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 LM contains a submodel that is a universal acyclic digraph of rank OrdM. 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 ω1+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 OrdM; 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 HFM of M. Indeed, HFM is universal for all countable acyclic binary relations.

Article | Barcelona research group in set theory

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 ⟨N,S,0⟩ 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 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

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly θ-supercompact, New York, September 14, 2012

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This will be a talk for the CUNY Set Theory seminar on September 14, 2012.

Abstract.  Starting from suitable large cardinal hypothesis, I will explain how to force the least weakly compact cardinal to be unfoldable, weakly measurable and, indeed, nearly θ-supercompact.  These results, proved in joint work with Jason Schanker, Moti Gitik and Brent Cody, exhibit an identity-crises phenomenon for weak compactness, similar to the phenomenon identified by Magidor for the case of strongly compact cardinals.

 

Recent progress on the modal logic of forcing and grounds, CUNY Logic Workshop, September 2012

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This will be a talk for the CUNY Logic Workshop on September 7, 2012.

Abstract. The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such asthe assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, orthe assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC.  Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”.  In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.

 

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

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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→LM 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 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 Image: Igougo.comwith the surreal numbers. The proof shows that LM contains a submodel that is a universal acyclic digraph of rank OrdM. 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 ω1+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 OrdM; 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 HFM of M. Indeed, HFM is universal for all countable acyclic binary relations.

Article | my profile at set theory talks | Toronto Set Theory Seminar post