Tag Archives: CH

The continuum hypothesis and other set-theoretic ideas for non-set-theorists, CUNY Einstein Chair Seminar, April, 2015

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At Dennis Sullivan’s request, I shall speak on set-theoretic topics, particularly the continuum hypothesis, for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, April 27, 2015, in two parts:

  • An introductory background talk at 11 am, Room GC 6417
  • The main talk at 2 – 4 pm, Room GC 6417

I look forward to what I hope will be an interesting and fruitful interaction. There will be coffee/tea and lunch between the two parts.

Abstract. I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the introductory background talk, in the morning (11 am), I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $\omega_1$, the long line, and, time permitting, we’ll discuss Suslin’s hypothesis.

At the main talk, in the afternoon (2 pm), I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory.  If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.

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

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.

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:

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