Determinacy and Jónsson cardinals

April 9, 2012

It is a well known result of Kleinberg that the axiom of determinacy implies that the \aleph_n are Jónsson cardinals for all n\le\omega. This follows from a careful analysis of partition properties and the computation of ultrapowers of associated measures. See also here for extensions of this result, references, and some additional context.

Using his theory of descriptions of measures, Steve Jackson proved (in unpublished work) that, assuming determinacy, every cardinal below \aleph_{\omega_1} is in fact Rowbottom. See for example these slides: 12. Woodin mentioned after attending a talk on this result that the HOD analysis shows that every cardinal is Jónsson below \Theta.

During the Second Conference on the Core Model Induction and HOD Mice at Münster, Jackson, Ketchersid, and Schlutzenberg reconstructed what we believe is Woodin’s argument.

I have posted a short note with the proof on my papers page. The note is hastily written, and I would appreciate any

additions/comments/corrections/suggestions

you find appropriate. I posted a version of the note in August last year, and Grigor Sargsyan soon emailed me some comments, that I have incorporated in the current version.

[I mentioned this on Twitter last year, but apparently not here. These are the relevant links (while discussing the Münster meeting): 1, 2, 3, 4, 5.]

Advertisements

305 – Homework V

April 9, 2012

This is the last homework set of the term. It is due Friday, April 27, 2012, at the beginning of lecture, but I am fine collecting it during dead week, if that works better.

Read the rest of this entry »


50s

April 9, 2012

This is kind of cool, in its own way.

Reunión de la Junta Consultiva Internacional de la Universidad de los Andes, en 1952 en Princeton (New Jersey). En la foto Mario Laserna, fundador de Los Andes, y miembros de la Junta, entre otros, Albert Einstein. En la parte de atrás, el célebre profesor John Von Neumann (padre de los computadores, pionero de la teoría de juegos y uno de los cerebros más creativos de la historia de la humanidad), quien entró a hacer parte de la Junta en 1952.— at Princeton, New Jersey.

Roughly: “Meeting of the International Advisory Board of the University of the Andes in 1952 in Princeton (New Jersey). In the photo Mario Laserna, founder of Los Andes, and members of the Board, among others, Albert Einstein. In the back, the celebrated Professor John von Neumann (the father of computers, a pioneer of game theory and one of the most creative minds in the history of mankind), who joined the Board in 1952. – at Princeton, New Jersey.”

(Found at the Facebook thing. I was an undergrad at the Andes ages ago.)

[Now to make this more than perfect, I would only need a picture of the meeting of Leonel Parra, my first Calculus teacher, with Borges. They talked about infinity, of course.]