BOISE EXTRAVAGANZA IN SET THEORY (BEST) -Announcement 3 and Call for papers

March 7, 2009

Announcement 3: Call for papers, Lodging Deadlines.

The 18-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Friday) – March 29 (Sunday), 2009. It is organized by Liljana Babinkostova, Andres Caicedo, Stefan Geschke, Richard Ketchersid, and Marion Scheepers. Contributed and invited talks will be held on Friday, Saturday and Sunday at the Department of Mathematics, Boise State University. The four invited speakers are:

Steve Jackson (University of North Texas)

Ljubisa Kocinac (University of Nis, Republic of Serbia)

Assaf Rinot (Tel Aviv University, Israel)

Grigor Sargsyan (University of California, Berkeley)

Please consult the conference webpage at URL

There are three remaining important deadlines regarding the conference. CRITICAL DEADLINE: 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. Please follow this link to the Hampton Inn’s online reservation site for BEST. Anyone interested in participating should contact the organizers as soon as possible by sending an email to

DEADLINE 2: Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 18. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Links are available at the BEST 18 web site. DEADLINE 3: Call for papers: The organizers will be editors for a volume in the Contemporary Mathematics series. Research papers on topics related to Set Theory and its Applications will be considered for publication in this volume. All papers will go through a thorough referee process. Former and current participants of the BEST conferences or their collaborators are especially encouraged to consider submitting a research paper. Anyone interested in submitting a paper should contact Marion Scheepers as soon as possible at

with this information. Subsequently information regarding preparation of papers will be sent to contributing authors by Contemporary Mathematics. The deadline for submitting a paper is JULY 21. On Saturday, March 28, Billy and Kris Hudson will host a social gathering at their home. All participants are cordially invited to ths social event. Kindly inform Billy at e-mail address

of plans to attend. More information is available at the conference web site. The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Contemporary Mathematics is published by the American Mathematical Society. Reduced lodging rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

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:

Read the rest of this entry »