Category Archives: Videos

The hierarchy of second-order set theories between GBC and KM and beyond

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This was a talk at the upcoming International Workshop in Set Theory at the Centre International de Rencontres Mathématiques at the Luminy campus in Marseille, France, October 9-13, 2017.

Hierarchy between GBC and KM

Abstract. Recent work has clarified how various natural second-order set-theoretic principles, such as those concerned with class forcing or with proper class games, fit into a new robust hierarchy of second-order set theories between Gödel-Bernays GBC set theory and Kelley-Morse KM set theory and beyond. For example, the principle of clopen determinacy for proper class games is exactly equivalent to the principle of elementary transfinite recursion ETR, strictly between GBC and GBC+$\Pi^1_1$-comprehension; open determinacy for class games, in contrast, is strictly stronger; meanwhile, the class forcing theorem, asserting that every class forcing notion admits corresponding forcing relations, is strictly weaker, and is exactly equivalent to the fragment $\text{ETR}_{\text{Ord}}$ and to numerous other natural principles. What is emerging is a higher set-theoretic analogue of the familiar reverse mathematics of second-order number theory.

Slides

https://plus.google.com/u/0/+JoelDavidHamkins1/posts/ekgdakenadv

How does a slinky fall?

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Have you ever observed carefully how a slinky falls? Suspend a slinky from one end, letting it hang freely in the air under its own weight, and then, let go! The slinky begins to fall. The top of the slinky, of course, begins to fall the moment you let go of it. But what happens at the bottom of the slinky? Does it also start to fall at the same moment you release the top? Or perhaps it moves upward, as the slinky contracts as it falls? Or does the bottom of the slinky simply hang motionless in the air for a time?

The surprising fact is that indeed the bottom of the slinky doesn’t move at all when you release the top of the slinky! It hangs momentarily motionless in the air in exactly the same coiled configuration that it had before the drop. This is the surprising slinky drop effect.

My son (age 13, eighth grade) took up the topic for his science project this year at school.  He wanted to establish the basic phenomenon of the slinky drop effect and to investigate some of the subtler aspects of it.  For a variety of different slinky types, he filmed the slinky drops against a graded background with high-speed camera, and then replayed them in slow motion to watch carefully and take down the data.  Here are a few sample videos. He made about a dozen drops altogether.  For the actual data collection, the close-up videos were more useful. Note the ring markers A, B, C, and so on, in some of the videos.

 

See more videos here.

For each slinky drop video, he went through the frames and recorded the vertical location of various marked rings (you can see the labels A, B, C and so on in some of the videos above) into a spreadsheet. From this data he then produced graphs such as the following for each slinky drop:

Small metal slinky graph

 

Large metal slinky graph

Plastic Slinky graph

In each case, you can see clearly in the graph the moment when the top of the slinky is released, since this is the point at which the top line begins to descend. The thing to notice next — the main slinky drop effect — is that the lower parts of the slinky do not move at the same time. Rather, the lower lines remain horizontal for some time after the drop point. Basically, they remain horizontal until the bulk of the slinky nearly descends upon them. So the experiments clearly establish the main slinky drop phenomenon: the bottom of the slinky remains motionless for a time hanging in the air unchanged after the top is released.

In addition to this effect, however, my son was focused on investigating a much more subtle aspect of the slinky drop phenomenon. Namely, when exactly does the bottom of the slinky start to move?  Some have said that the bottom moves only when the top catches up to it; but my son hypothesized, based on observations, as well as discussions with his father and uncles, that the bottom should start to move slightly before the bulk of the slinky meets it. Namely, he thought that when you release the top of the slinky, a wave of motion travels through the slinky, and this wave travels slightly fast than the top of the slinky falls. The bottom moves, he hypothesized, when the wave front first gets to the bottom.

His data contains some confirming evidence for this subtler hypothesis, but for some of the drops, the experiment was inconclusive on this smaller effect. Overall, he had a great time undertaking the science project.

June 2016 Update: On the basis of his science fair poster and presentation, my son was selected as nominee to the Broadcom Masters national science fair competition! He is now competing against other nominees (top 10% of participating science fairs) for a chance to present his research in Washington at the final national competition next October.

September 2016 Update: My son has now been selected as a Broadcom Masters semi-finalist, placing him in the top 300 amongst more than 6000 nominees. The finalists will be chosen in a few weeks, with the chance to present in Washington, D.C.

Slinky drop on YouTube | Modeling a falling slinky (Wired)
Explaining an astonishing slinky | Slinky drop on physics.stackexchange
Cross & Wheatland, “Modeling a falling slinky”

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.

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 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