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.]

Downward transference of mice and universality of local core models

April 4, 2012

Martin Zeman and I have just submitted our paper Downward transference of mice and universality of local core models to the Journal of Symbolic Logic, downloadable from my papers page. We have also uploaded it on the ArXiv. (I should have been doing this for years; this is the first time I post there.)

It is a nice observation that goes back to Friedman that if 0^\sharp exists and {\mathbf M} is an inner model that correctly computes \omega_2, then 0^\sharp\in {\mathbf M}. Looking at a completely different problem, from the theory of forcing axioms, we were led to the question of how much this result can be generalized.

Our main result is that there is a significant transfer of structure going on, simply due to the agreement of cardinals. (The statement of the result and, of course, the argument, require familiarity with fine structure theory, as developed in Steel’s or Martin’s books.)

Theorem. Assume that {\mathbf M} is a proper class inner model, and that \delta is regular in {\mathbf V}.

  1. If there are no inner models of {\mathbf V} with Woodin cardinals, \delta>\omega_1, and

    \{x\in{\mathcal P}_\delta(\delta^+)\cap{\mathbf M}\mid{\rm cf}^{\mathbf M}(x\cap\delta)>\omega\}

    is stationary, then {\mathbf K}^{\mathbf M}\|\delta is universal for all iterable 1-small premice in {\mathbf V} of cardinality less than \delta.

  2. If, in {\mathbf M}, 0^\P does not exist, and {\mathcal P}_\delta(\delta^+)\cap{\mathbf M} is stationary, and \delta>\omega_1, then the same conclusion holds. If \delta=\omega_1, then {\mathbf K}^{\mathbf M}\|\omega_2 is universal for all countable iterable premice in {\mathbf V}.

Here, as usual, {\mathcal P}_\kappa(\lambda) denotes the collection of subsets of \lambda of size less than \kappa. It is easy to check that {\mathcal P}_{\omega_1} (\omega_2)\cap{\mathbf M} is stationary if {\mathbf M} computes \omega_2 correctly, so this result generalizes the statement about 0^\sharp mentioned above.

In fact, it follows that, if \omega_2 is computed correctly in {\mathbf M}, then any sound mouse in {\mathbf V} projecting to \omega and below 0^\P, is in {\mathbf M}. Beyond 0^\P, the argument becomes more complicated, and we need to assume a global anti-large cardinal assumption, namely, that there are no inner models in {\mathbf V} with Woodin cardinals.

We expect that this restriction can be weakened, perhaps even dispensed with.

(This paper is the second in a series, aiming to explore the structure of inner models for which some agreement of cardinals holds. I briefly mentioned the first paper here.)

On CH (After Hamkins)

April 2, 2012

(Wherein I quote from Twitter.)

This is my excuse to put this page to use. It all started, more or less, around here:

Moreno, Javier (bluelephant). “Es muy bueno este artículo de Hamkins sobre algo así como sociología de la teoría de conjuntos: http://arxiv.org/abs/1203.4026” 2 April 2012, 3:39 p.m. Tweet.

Villaveces, Andrés (gavbenyos). “@bluelephant lo siento bastante crudo – me parece que Joel simplemente pone en versión impresa cierto consenso, pero falta argumentar” 2 April 2012, 4:32 p.m. Tweet.

(Loosely translated: “This is a very good article by Hamkins on something akin to sociology of set theory” “I find it weak — I think that Joel is simply putting in writing a known consensus, but needs to argue for it more”) Long story short, I feel Andrés is right, but I thought I should elaborate my view, at least somewhat. Originally I considered writing a blog entry, but it quickly became apparent it would grow longer than I can afford time-wise. So, I used twitter instead.

(But there is a serious caveat, namely, it seems that the paper is intended for a general mathematical and philosophical audience, so the omission of technical issues is most likely intentional. Javier even remarked as much.)

What follows is the series of Tweets I posted. It is not a transcription; to ease reading, I have added a couple of links, reformatted the posts rather than continuing with the MLA suggested approach, very lightly edited the most obvious typos, and added a couple of phrases where I felt more clarity was needed. [Edit, April 22: I also removed a line on MM^{++} vs Woodin’s (*), as the proof of the underlying claim has been withdrawn.]

I started at 10:44 p.m., with a warning: “(Technical pseudo-philosophical thoughts for a few posts.)”

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