Singular Cardinal Combinatorics

May 1 - 6, 2004 (1/2 workshop)

Organizers: Claude Laflamme (University of Calgary), Matthew Foreman (University of California, Irvine), Stevo Todorcevic (University of Toronto and CNRS Paris).

Objectives

Set Theory is not only one of the areas of mathematics where the Axiom of Choice is very widely utilized, but also an area of mathematics which systematically searches for structure that is independent of the axiom. Since the mid 1980's there has been increased recognition in Set Theory of the existence of \emph{natural structure}: structure that may require the Axiom of Choice to prove its existence, but is independent of the choices made. This structure includes natural stationary sets (that are not the cofinalities), natural ideals and the existence of previously unstudied objects such as \emph{scales}. Much of this natural structure is centered around a connected body of work that can be labelled ``singular cardinal combinatorics".

An example of this phenomenon is the arithmetic of cardinal numbers. At one time it was generaly believed that the Axiom of Choice simplifies the arithmetic of cardinal numbers to the point of making it almost trivial. In fact this is quite false. Even with the assumption of the Axiom of Choice, there is a tremendous amount to be said about the behaviour of arithmetic operations on the cardinal numbers. Shelah's 1995 book titled `Cardinal Arithmetic' contained much of his work on the subject and won for its author the prestigious Bolyai prize and eventualy the much esteemed Wolf prize.

The proposed workshop is designed to bring together researchers from around the world who work on singular cardinal combinatorics. The various communities in Europe, Israel, Japan, Canada and the United States have ofter worked independently; in some cases with remarkably little communication. The workshop will give the participants the opportunity to share their results and allow cross-fertilization between the various groups.

At the forefront of singular cardinal combinatorics is Shelah's discovery of PCF-theory (a theory of `possible cofinalities'). Shelah used this theory to give remarkable bounds on the powers of Singular Cardinals, and it has yielded important advances on classical questions such as stationary set reflection, the partition calculus of Erd\"os and Rado, and so on. Progress on these problems previously depended on simplifying assumptions like GCH or V=L.

There are various strands of the theory of singular cardinal combinatorics. One is the PCF theory and the development of the theory of scales, the relation to square properties and the singular cardinals problem. A major conjecture in this part of the area is:

\begin{quotation} Suppose that $\kappa$ is a singular cardinal of countable cofinality and the singular cardinals hypothesis fails at $\kappa$, then there is an Aronszajn tree on $\kappa^+$. \end{quotation}

This problem has been attacked by several authors who have produced a series of long papers with partial results on the subject. While not solved, the problem seems intimately tied up with questions about the existence of scales with strong properties, the stationary set of ``good" ordinals and various other PCF objects.

Another important strand of singular cardinal combinatorics is the investigation of bounds on $2^\kappa$ for $\kappa$ singular. This has been successfully done by Shelah, but it is not know if his bounds are sharp. A very important problem in this area is the \emph{PCF Conjecture}:

\begin{quotation} Suppose that $A$ is a progressive set of cardinals with supremum $\kappa$ and $\kappa$ is not a cardinal fixed point. Then the cardinality of the PCF(A) is less than or equal to the cardinality of $A$. \end{quotation}

The most important special case of this problem is: \begin{quotation} Suppose that $\aleph_\omega$ is a strong limit cardinal. Then \[2^{\aleph_\omega}<\aleph_{\omega_1}.\] \end{quotation}

Both of these problems are investigated in the positive sense by combinatorial methods. However the methods that give counterexamples to conjectures seem to require intricate forcing techniques. This technology, originally discovered by Magidor, has been developed into a fearsome machine by Gitik, among others. This machine is the main source of examples that provide limitations on what can be proved using purely combinatorial methods.

Another important issue is PCF axiomatics. This can be summarized by saying that the study has two parts: \begin{enumerate} \item PCF completeness \item Topological properties of PCF structures. \end{enumerate} The first part attempts to build models with cardinals whose PCF structure is isomorphic to a given abstract PCF structure. The second attempts to build abstract PCF structures with particular cardinal and topological properties. Success on these two projects would give a picture of the exact relationship between the PCF theory and cardinal arithmetic.

The final strand is given by ``inner model theory". Many, if not most, examples of interesting singular cardinals phenomena produced by forcing require assumptions beyond ZFC. The \emph{inner model theory} is a large body of work showing that these assumptions are necessary. Inner model theory combines purely combinatorial techniques with techniques from the study of the fine structure of Godel's universe $L$.

It is hoped that by bringing this group of emminent researchers together, significant progress can be made on the problems listed above, as well as the large number of associated problems and sub-problems.

The proposed participants come from 11 countries. They include 8 speakers from various Internation Congresses of Mathematicians, one of whom was a plenary speaker. Listed by nationality of workplace: Canada (4), USA (9), Germany (5), Israel (5), Hungary (4), Japan (4), Spain (3), England (2), Czech Republic(2), Finland (2), France (1).

Roughly speaking: the Americans work on the combinatorial side and on inner model theory. The Canadians work largely on the combinatorial side. The German's have worked on the inner model theory and three of them have co-authored a book on Cardinal Arithmetic. Israel has been a center for Singular Cardinals Combinatorics, being home to both Shelah and Gitik. The Hungarians have a long tradition of applying singular cardinal combinatorics to partition problems. Questions and results of Erd\"os played a crucial role in the history of the subject, and Erd\"os's collaborators continue to be very active in the area and are suggested participants.

Japan has an emerging school in set theory and members of that school are quite enthusiastic and active. The suggested participants from Spain have results related to the PCF axiomatics. England has workers in inner model theory and PCF. The Czech Republic has important members of the community, as do Finland and France.

Six of the proposed applicants are ``young" as determined by holding a postdoctoral position or being within 5 years of their Ph.D.