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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580 -Cardinal arithmetic (9)

March 7, 2009

2. The ultrapower construction

 

The study of ultrapowers originates in model theory, although it has found applications both in algebra and in analysis. However, it is accurate to say that it is mainly exploited in set theory. Here I present the basic idea, showing its close connection to the study of measurable cardinals, defined last lecture.

Suppose first that {{\mathcal U}} is an ultrafilter over a set {X.} We want to define the ultrapower of the universe {V} of sets by {{\mathcal U}.} The basic idea is to consider the product of {X} many copies of the structure {(V,\in).} We want to “amalgamate” them somehow into a new structure {(\tilde V,\tilde\in).} For this, we look for the “typical” properties of the elements {\{f(x): x\in X\}} of each “thread” {f:X\rightarrow V,} and add an element {\tilde f} to {\tilde V} whose properties in {(\tilde V,\tilde\in)} are precisely these typical properties. We use {{\mathcal U}} to make this precise, by saying that a property {\varphi} is typical of the range of {f} iff {\{x\in X:\varphi(f(x))\}\in{\mathcal U}.} This leads us to the following definition, due to Dana Scott, that adapts the ultrapower construction to the context of proper classes:

Definition 1 Let {{\mathcal U}} be an ultrafilter over a nonempty set {X.} We define the ultrapower {(V^X/{\mathcal U},\hat\in)} of {V} by {{\mathcal U}} as follows:

For {f,g:X\rightarrow V,} say that

\displaystyle  f=_{\mathcal U} g\mbox{ iff }\{x\in X:f(x)=g(x)\}\in{\mathcal U}.

This is easily seen to be an equivalence relation. We would like to make the elements of {V^X/{\mathcal U}} to be the equivalence classes of this relation. Unfortunately, these are all proper classes except for the trivial case when {X} is a singleton, so we cannot within the context of our formal theory form the collection of all equivalence classes.

Scott’s trick solves this problem by replacing the class of {f} with

\displaystyle  [f]:=\{g:X\rightarrow V: g=_{\mathcal U}f\mbox{ and }{\rm rk}(g)\mbox{ is least possible}\}.

Here, as usual, {{\rm rk}(g)={\rm min}\{\alpha:g\in V_{\alpha+1}\}=\sup\{{\rm rk}(x)+1:x\in g\}.} All the “classes” {[f]} are now sets, and we set {V^X/{\mathcal U}=\{[f]: f:X\rightarrow V\}.}

We define {\hat\in} by saying that for {f,g:X\rightarrow V} we have

\displaystyle  [f]\hat\in[g]\mbox{ iff }\{x\in X:f(x)\in g(x)\}\in{\mathcal U}.

(It is easy to see that {\hat\in} is indeed well defined, i.e., if {f=_{\mathcal U}f'} and {g=_{\mathcal U}g'} then {\{x\in X:f(x)\in g(x)\}\in{\mathcal U}} iff {\{x\in X:f'(x)\in g'(x)\}\in{\mathcal U}.})

 

(The ultrapower construction is more general than as just defined; what I have presented is the particular case of interest to us.) The remarkable observation, due to \mbox{\L o\'s,} is that this definition indeed captures the typical properties of each thread in the sense described above:

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