BOISE EXTRAVAGANZA IN SET THEORY (BEST) – Announcement 1

December 21, 2009

The 19-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Saturday) – March 29 (Monday), 2010.

It is organized by Liljana Babinkostova, Andrés E. Caicedo, Masaru Kada, and Marion Scheepers (scientific committee), and Billy Hudson (social committee).

Contributed and invited talks will be held on Saturday, Sunday and Monday at the Department of Mathematics, Boise State University. The invited speakers are:

The conference webpage is available here. Anyone interested in participating should contact the organizers as soon as possible by sending an email to best@math.boisestate.edu

There are three important deadlines regarding the conference:

  • Lodging: The Hampton Inn & Suites is providing rooms at a reduced rate for BEST participants. To take advantage of the reduced rate, reservations must be made online by MARCH 12. After March 12 rooms will be available at prevailing rates.
  • Financial support: Limited financial support is available to partially offset lodging expenses of up to eight participants, and to partially offset lodging plus airfare expenses of up to two participants. Please see the conference website for details on applying for support. The deadline for applying for financial support is MARCH 3.
  • Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 19. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Abstracts can be submitted here, and viewed here.
The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Reduced lodging rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

BPFA and projective well-orderings of the reals

December 17, 2009

Sy Friedman and I recently submitted the paper {\sf BPFA} and projective well-orderings of the reals to The Journal of Symbolic Logic. The preprint is available at my papers page.

In a previous paper, Boban Velickovic and I showed that if {\sf BPFA}, the bounded version of the proper forcing axiom, holds, then one can define a well-ordering of the reals using what is in essence a subset C of \omega_1 as a parameter. The argument uses Justin Moore‘s technique of the Mapping Reflection Principle, and provides us with a \Delta_1(C) well-ordering. In this sense, the result is best possible. 

In earlier work, I had shown that {\sf BPFA} is consistent with a projective well-ordering of the reals. The result with Velickovic dramatically improves this; for example, if \omega_1=\omega_1^L and {\sf BPFA} holds, then there is a projective well-ordering of {\mathbb R}. Note that this is an implication rather than just a consistency result, and does not require that the universe is a forcing extension of L. The point here is that the parameter C can be chosen in L so that it is “projective in the codes,” and {\sf MA}_{\omega_1} then provides us with enough coding machinery to transform the \Delta_1(C) definition of the well-ordering into a projective one.

In the new paper, Friedman and I show that in fact under {\sf BPFA}+\omega_1=\omega_1^L, the projective definition is best possible, \Sigma^1_3. For this we need to combine the coding technique giving the \Delta_1(C) well-ordering with a powerful coding device in the absence of sharps, what Friedman calls David’s trick. The point now is that the forcing required to add the witnesses that make the coding work is proper, and {\sf BPFA} suffices to grant in the universe the existence of these objects.

As a technical problem, it would be interesting to see whether the appeal to the mapping reflection principle can be eliminated here. We only obtain a \Sigma^1_4 well-ordering in that case. Also, since {\sf BPFA} turns out, perhaps unexpectedly, to provides us with definable well-orderings, it would be interesting to see that {\sf MA}_{\omega_1} does not suffice for this.


Flashforward

December 17, 2009

Flashforward is a science-fiction novel by Canadian writer Robert Sawyer. I wanted to like the novel, I did. I first heard of it, of course, when ABC began to air commercials for their version of the story. The TV series has been kind of terrible so far. There is a lot of potential in the premise, I think, but neither the series nor the novel delivers it.

There are significant differences, too. In both cases, a global event occurs where everybody has a glimpse of their own future for a couple of minutes or so. Except for this, there is really not much in common in both cases. In the TV series, the `jump’ is six months or so. In the novel is several decades. One of the main characters finds out he is going to be killed, and when. Other than this, there is nothing much in common.

Sawyer has won several awards for his writing, but I do not understand it based on this novel. Perhaps his other works are better. In this one, what is clear is how little understanding he seems to have of the science he is using. His characters are actually well developed, and in a less plot driven story, I think his strengths would be evident. 

Here are three quotes:

Frau Drescher went as white as the snow cap on Mont Blanc. “Mein Gott,” she said. “Mein Gott.

I mean, seriously? Ok. Perhaps that is a bit unfair; his comparisons and metaphors usually work, and his characters tend to talk like people. Let’s forget that one.

“Of course,” said Michiko. “It’s not that three minutes passed during which planes and trains and cars and assembly lines operated without human intervention. Rather, three minutes passed during which nothing was resolved—all the possibilities existed, stacked into shimmering whiteness. But at the end of those three minutes, consciousness returned, and the world collapsed again into a single state. And, unfortunately but inevitably, it took the single state that made the most sense, given that there had been three minutes of no consciousness: it resolved itself into the world in which planes and cars had crashed. But the crashes didn’t occur during those three minutes; they never occurred at all. We simply went in one jump from the way things were before to the way they were after.”

“That’s… that’s crazy,” said Lloyd.

[...]

“No, it’s not. It’s quantum physics.”

No, it’s not. It’s crazy. It’s just absurd. It is what happens when rather than understanding the science, one takes the metaphors scientists use and erroneously assumes they are literal truths. The TV series quickly distanced itself from this nonsense. In the book, only humans see the future, since they are the only ones with “consciousness.” And no recording devices work during the `jump’, because nobody is observing them. Seriously. Sawyer seems to think that the tree actually makes no sound. Or, perhaps, there isn’t even a forest, if nobody is watching it. It gets worse, because of course it is not only quantum mechanics that is thoroughly misunderstood. 

Surely all consciousness everywhere had to agree on what constituted “now.”

This being what the main physicist of the story thinks. Everywhere being, of course, the entire universe. This in the same paragraph where the physicist explains some relativity theory to himself. 

Sigh.


502 – The Banach-Tarski paradox

December 17, 2009

1. Non-measurable sets

In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the concept of volume can lead to contradictions. A good reference for this topic is the very nice book The Banach-Tarski paradox by Stan Wagon.

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175 – Final exam

December 15, 2009

Here is the final exam, and here are the solutions.


502 – Exponentiation

December 9, 2009

This is the last homework assignment of the term: Assume {\sf CH}. Evaluate the cardinal number \aleph_3^{\aleph_0}, the size of the set of all functions f:\omega\to\omega_3.


502 – The constructible universe

December 9, 2009

In this set of notes I want to sketch Gödel’s proof that {{\sf CH}} is consistent with the other axioms of set theory. Gödel’s argument goes well beyond this result; his identification of the class {L} of constructible sets eventually led to the development of inner model theory, one of the main areas of active research within set theory nowadays.

A good additional reference for the material in these notes is Constructibility by Keith Devlin.

1. Definability

The idea behind the constructible universe is to only allow those sets that one must necessarily include. In effect, we are trying to find the smallest possible transitive class model of set theory.

{L} is defined as

\displaystyle L=\bigcup_{\alpha\in{\sf ORD}} L_\alpha,

where {L_0=\emptyset,} {L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha} for {\lambda} limit, and {L_{\alpha+1}={\rm D{}ef}(L_\alpha),} where

\displaystyle \begin{array}{rcl} {\rm D{}ef}(X)=\{a\subseteq X&\mid&\exists \varphi\,\exists\vec b\in X\\ && a=\{c\in X\mid(X,\in)\models\varphi(\vec b,c)\}\}. \end{array}

The first question that comes to mind is whether this definition even makes sense. In order to formalize this, we need to begin by coding a bit of logic inside set theory. The recursive constructions that we did at the beginning of the term now prove useful.

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