Foundations of Mathematics

February 1, 2019

Foundations of Mathematics, Andrés E. Caicedo, James Cummings, Peter Koellner, and Paul B. Larson, eds., Contemporary Mathematics, vol. 690, Amer. Math. Soc., Providence, RI, 2017. DOI: 10.1090/conm/690. MR3656304. Zbl 06733965.

This book contains the proceedings of the conference in honor of Hugh Woodin’s 60th birthday, that I previously discussed on this blog (here, here, and here).

The AMS page for the volume can be found here, including the table of contents and links to the front- and endmatter (which I think are available to everybody) and links to the individual papers (which I imagine may not be).


Help!

July 5, 2016

IMG_3359

Help us identify all mathematicians in this picture (click on it for a larger version). Please post comments here, on G+, or email me or Paul Larson.

The picture will appear in the book of proceedings of the Woodin conference, http://logic.harvard.edu/woodin_meeting.html. (Thanks to David Schrittesser for allowing us to use it.)


Woodin meeting

April 1, 2015

IMG_3359

IMAG1270

(Don’t know who to credit for the group picture, but as pointed out by Paul in the comments, it was with David Schrittesser‘s camera. Toast picture by Paul Larson. Toast by Ted Slaman.)

I’m sad I had to miss the meeting, although it was for obvious reasons.


Woodin conference

March 11, 2015

Woodin_Poster

The conference in honor of Hugh Woodin’s 60th birthday will take place at Harvard University, on March 27-29, 2015. The meeting is partially supported by the Mid-Atlantic Mathematical Logic Seminar and the National Science Foundation. Funding is available to support participant travel. Please write to woodinbirthdayconference@gmail.com to apply for support, and to notify the organizers if you are planning to attend.

The list of speakers is as follows:

  • H. Garth Dales
  • Qi Feng
  • Matthew D. Foreman
  • Ronald Jensen
  • Alexander S. Kechris
  • Menachem Magidor
  • Donald A. Martin
  • Grigor Sargsyan
  • Theodore A. Slaman
  • John R. Steel.

We expect to publish proceedings of the conference, together with select additional research and survey papers, through the series Contemporary Mathematics, of the AMS. The editors of the proceedings are myself, James Cummings, Peter Koellner, and Paul Larson. Please contact me for information regarding the proceedings.

Additional information can be found at the conference website.


Cryptic marks

February 5, 2015

New scientist recently ran a series on articles on “How to think about…” One of them, by Richard Webb and published December 13, 2014, was about infinity. It contains this quote:

Woodin’s notepads consist mainly of cryptic marks he uses to focus his attention, to the occasional consternation of fellow plane passengers. “If they don’t try to change seats they ask me if I’m an artist,” he says.

David Roberts wondered on Google+ what these cryptic marks look like. This reminded me of some pictures I took of them at the Conference on inner model theory at UC Berkeley last year.

2014-06-10 17.59.20

2014-06-10 17.37.00

2014-06-10 17.36.08


Woodin’s proof of the second incompleteness theorem for set theory

November 4, 2010

[Edit, Oct. 1, 2013: Robert Solovay has pointed out an inaccuracy in my presentation of Woodin’s argument: Rather than simply requiring that P is a hereditary property of models, we must require that \mathsf{ZFC} proves this. A corrected presentation of the argument will be posted shortly.]

As part of the University of Florida Special Year in Logic, I attended a conference at Gainesville on March 5–9, 2007, on Singular Cardinal Combinatorics and Inner Model Theory. Over lunch, Hugh Woodin mentioned a nice argument that quickly gives a proof of the second incompleteness theorem for set theory, and somewhat more. I present this argument here.

The proof is similar to that in Thomas Jech, On Gödel’s second incompleteness theorem, Proceedings of the American Mathematical Society 121 (1) (1994), 311-313. However, it is semantic in nature: Consistency is expressed in terms of the existence of models. In particular, we do not need to present a proof system to make sense of the result. Of course, thanks to the completeness theorem, if consistency is first introduced syntactically, we can still make use of the semantic approach.

Woodin’s proof follows.

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Luminy – Hugh Woodin: Ultimate L (III)

October 27, 2010

For the first lecture, see here.

For the second lecture, see here.

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Luminy – Hugh Woodin: Ultimate L (II)

October 21, 2010

For the first lecture, see here.

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Luminy – Hugh Woodin: Ultimate L (I)

October 19, 2010

The XI International Workshop on Set Theory took place October 4-8, 2010. It was hosted by the CIRM, in Luminy, France. I am very glad I was invited, since it was a great experience: The Workshop has a tradition of excellence, and this time was no exception, with several very nice talks. I had the chance to give a talk (available here) and to interact with the other participants. There were two mini-courses, one by Ben Miller and one by Hugh Woodin. Ben has made the slides of his series available at his website.

What follows are my notes on Hugh’s talks. Needless to say, any mistakes are mine. Hugh’s talks took place on October 6, 7, and 8. Though the title of his mini-course was “Long extenders, iteration hypotheses, and ultimate L”, I think that “Ultimate L” reflects most closely the content. The talks were based on a tiny portion of a manuscript Hugh has been writing during the last few years, originally titled “Suitable extender sequences” and more recently, “Suitable extender models” which, unfortunately, is not currently publicly available.

The general theme is that appropriate extender models for supercompactness should provably be an ultimate version of the constructible universe L. The results discussed during the talks aim at supporting this idea.

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580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that {\omega\rightarrow(\omega)^n_m} (Ramsey) and {\omega\rightarrow[\omega]^n_\omega} ({\mbox{Erd\H os}}-Rado) for any {n,m<\omega.} On the other hand, we also have that {2^\kappa\not\rightarrow(3)^2_\kappa} ({\mbox{Sierpi\'nski}}) and {2^\kappa\not\rightarrow(\kappa^+)^2} ({\mbox{Erd\H os}}-Kakutani) for any infinite {\kappa.}

Positive results can be obtained for larger cardinals than {\omega} if we relax the requirements in some of the colors. A different extension, the {\mbox{Erd\H os}}-Rado theorem, will be discussed later.

Theorem 1 ({\mbox{Erd\H os}}-Dushnik-Miller) For all infinite cardinals {\lambda,} {\lambda\rightarrow(\lambda,\omega)^2.}

This was originally shown by Dushnik and Miller in 1941 for {\lambda} regular, with {\mbox{Erd\H os}} providing the singular case. For {\lambda} regular one can in fact show something stronger:

Theorem 2 ({\mbox{Erd\H os}}-Rado) Suppose {\kappa} is regular and uncountable. Then
\displaystyle  \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2, which means: If {f:[\kappa]^2\rightarrow2} then either there is a stationary {H\subseteq\kappa} that is {0}-homogeneous for {f}, or else there is a closed subset of {\kappa} of order type {\omega+1} that is {1}-homogeneous for {f}.

(Above, top stands for “topological.”)

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