Axiomatic approaches to forcing techniques in set theory (13w5026)

Arriving in Banff, Alberta Sunday, November 3 and departing Friday November 8, 2013


Matthew Foreman (University of California, Irvine)

(Cornell University, Ithaca, NY, USA)

Stevo Todorcevic (University of Toronto and Institut de Mathématiques de Jussieu)


While the Baire Category Theorem is a relatively simple mathematical result, it has played a central role on several levels in modern set theory. While its facility for building complex mathematical structures was hinted at in the work of Banach in the 1920s, its importance within set theory became especially apparent after Cohen's invention of the method of forcing in the early 1960s.

One interpretation of the Baire Category Theorem is that no compact topological space can be covered by countably many nowhere dense subsets.

Shortly after the invention of forcing, the importance of the following question emerged: Can this assertion can be strengthened by replacing ``countably many'' by ``at most $theta$'' for some uncountable cardinal $theta$? The answer is negative, but it is possible if some restrictions are placed on the class of compact spaces.

For instance if one restricts to the class of compact spaces with no uncountable family of pairwise disjoint open sets, then one obtains the statement of $mathrm{MA}_theta$, where MA abbreviates Martin's Axiom. In every reasonable class of compact spaces, however, the corresponding assertion about Baire category is not a theorem of ZFC.

For broader classes of compact spaces, it becomes much more natural to formulate the corresponding Baire category statement in terms of partial orders, also known as forcings. The existence of a point not contained in a collection of nowhere dense sets in a compact space translates into the existence of a filter which intersects all members of a collection of dense sets in a given forcing.

Such a filter is said to be generic with respect to this collection of dense sets and the collection of dense sets is said to admit a generic filter. We say that the forcing axiom for a class of partial orders holds if whenever $Q$ is in the class, any family of $aleph_1$ many dense sets in $Q$ admits a generic filter.

This is just the Baire category assertion cast into the language of forcings. Forcings which do not preserve the stationarity of a subset of $omega_1$ necessarily contain a family of $aleph_1$ many dense sets which does admit a generic filter. Remarkably, this is the only obstruction to having a consistent forcing axiom.

(Foreman, Magidor, Shelah) The forcing axiom for partial orders which preserve stationary subsets of $omega_1$ is consistent relative to the existence of a supercompact cardinal.

The forcing axiom specified above is known Martin's Maximum (MM). It is also known that MM implies $|mathbb{R}| = aleph_2$ and in particular that $aleph_1$ in the formulation of MM can not be increased to $aleph_2$.

Our understanding of forcing and forcing axioms has evolved dramatically over the last decade with both the development of new techniques and the realization of the potential for future applications.

This pursuit is timely in that it represents both the beginning of a deeper interaction with areas such as analysis and an area in which young researchers are increasingly active. The aim of the workshop is to train new researchers in the area and to deepen our understanding of the applications of forcing both to other areas of mathematics and to set theory itself.

While an emphasis will be placed on applications of forcing axioms, the program will also deal with other aspects of forcing, such as those relevant to proving results in ZFC.

The goal will be in part of explore new possibilities for applications in areas of mathematics such as analysis, ergodic theory, and group theory.

The study of forcing axioms takes many forms. On one hand, the initial interest in these axioms arose from their ability to settle a large number of statements arising in mathematics outside of set theory and foundations. This trend has continued up to the present.

(Moore) MM implies every uncountable linear order contains an isomorphic copy of one of the following: $X$, $omega_1$, $-omega_1$, $C$, $-C$. Here $X$ is any set of reals of cardinality $aleph_1$ and $C$ is any Countryman line.

(Todorcevic) MM implies that if $B$ is an infinite dimensional Banach space of density at most $aleph_1$ then $B$ has an infinite dimensional separable quotient.

While direct applications of MM generally require specialized knowledge of set theory, there are an increasing number of combinatorial principles that follow from MM which are at the same time powerful and approachable by the non specialist. Both applying them and isolating new principles of this type is an important theme in set theoretic research. Two such principles are the P-Ideal Dichotomy and Todorcevic's formulation of the Open Coloring Axiom:

PID If $X$ is a set and $mathcal{I}$ is an ideal of countable subsets of $X$ which is countably directed under mod finite containment, then either (a) there is an uncountable $Z subseteq X$, all of whose countable subsets are in $mathcal{I}$ or (b) $X$ can be covered by countably many sets none of which contain an infinite subset in $mathcal{I}$.

OCA If $G$ is a graph on a separable metric space whose edge set is topologically open, then either (a) $G$ contains an uncountable complete subgraph or (b) $G$ admits a vertex coloring with countably many colors.

We will now describe some ways in which these principles have been applied. Recall that the Calkin Algebra is the quotient of the bounded operators on a separable Hilbert space modulo the compact operators.

(Farah) OCA implies that all automorphisms of the Calkin algebra are inner.

Von Neumann asked whether the countable chain condition and weak distributivity were sufficient conditions to ensure the existence of a strictly positive measure on a complete Boolean algebra. Maharam broke this problem into two halves: whether these conditions implied the existence of a strictly positive continuous submeasure and whether the existence of such a submeasure implies the existence of a strictly positive measure. The PID played an important role in the solution to the first of these problems.

(Balcar, Jech, Pazak) Assume the PID. If a complete Boolean algebra satisfies the countable chain condition and is weakly distributive, then it supports a strictly positive continuous submeasure.

The latter problem was solved negatively by Talagrand.

In some cases, applications of forcing axioms yield ZFC theorems as bi-products. For instance, the Todorcevic modified the proof of the above result to prove the following.

(Todorcevic) If a complete Boolean algebra satisfies the $sigma$-bounded chain condition and is weakly distributive, then it supports a strictly positive continuous submeasure.

Todorcevic's structural analysis of Rosenthal compacta is another example.

In addition to exploring further applications of forcing axioms, an emphasis will also be placed on exploring uses of the method of forcing itself outside of set theory. The workshop will include a list of participants both from within set theory as well as other areas of mathematics which the organizers felt there was potential for future applications of the method of forcing.