Menachem Magidor at Carnegie Mellon University on March 19, 2011

Appalachian set theory

Saturday, March 19, 2011

Carnegie Mellon University in Pittsburgh PA

Lectures 9:30 a.m. - 12:30 p.m. and 2:30 - 6 p.m. in Doherty Hall 2302

Registration and morning refreshments 8:45 - 9:30 a.m. in Wean Hall 6220

Menachem Magidor : "On the strengths and weaknesses of weak squares"

List of participants in this workshop

Lecture notes from this workshop by Menachem Magidor and Chris Lambie-Hanson

Workshop description

The square principle κ for a cardinal κ, which was introduced by Jensen, is among the most useful combinatorial principles that hold in the Contructible Universe. It has many applications in set theory, algebra, topology and other fields. In these applications, square is typically used to produce an instance of "incompactness". Namely, a structure that lacks a certain property that all of its substructures of strictly smaller cardinality have. An example would be a non-free group all of whose subgroups of smaller cardinality are free.

Jensen also defined κ*, which is a weaker principle than κ. Between κ and κ* there is Schimmerling's hierarchy of principles κ,λ for 1 ≤ λ ≤ κ. Even weaker than κ* are Shelah's approximation properties.

There are several ways in which weak squares are useful. It is of great interest to identify the weakest form of square that suffices for a given application. As the strongest possible forms of square hold in the core models constructed for large cardinals, the strongest weak square that holds in a given model of ZFC is an intuitive measure of its distance from being a canonical inner model. The problem of determining the consistency strength of the failure of weak square (especially at singular cardinals) is a good litmus test for the success of the inner model program. Another fascinating subject is the impact of forcing axioms like PFA, MRP and MM on the existence or absence of weak squares.

In these lectures, we shall study some of the problems related to weak squares. A broad outline of the talks is:

  1. The definitions of weak squares and a few examples of applications in infinite combinatorics, topology and algebra.
  2. The conneciton between weak squares and the existence of good and very good scales.
  3. Weak squares at singular cardinals are difficult to avoid. The Džamonja-Shelah result that changing the cofinality of an inaccessible cardinal κ without collapsing cardinals automatically creates a model of κ,cf(κ). (This generalizes a result of Cummings and Schimmerling about Prikry forcing and weak squares.)
  4. Weak squares and large cardinals.
  5. Weak squares and forcing axioms.
For items 1-3 the prerequisites are minimal. We expect familiarity with basic notions of set theory like ordinals, cardinals, cofinalities, clubs, stationary sets, etc. Item 4 will require some familiarity with large cardinals like strongly compact, supercompact, etc., though the definitions will be given. Item 5 will require some familiarity with forcing axioms and forcing techniques.

Recommended reading

A good source for the definition of κ and some applications is Devlin's book Constructibility. The most relevant chapter is IV on κ+-Trees in L and the Fine Structure Theory.

A good place to study weak squares and some of their applications is Todd Eisworth's article in The Handbook of Set Theory. See chapter 15 in volume 2. The relevant section is on pages 1287-1317. Some of this material will be covered in the talks.

The large cardinal notions that will be used are all introduced in Kanamori's The Higher Infinite. The most relevant chapter is 5 on Strong Hypotheses.

A good reference for forcing axioms is Jech's Set Theory. The relevant chapters are 31 on Proper Forcing and 37 on Martin's Maximum.

Advanced references

Here we provide some more advanced references of the theorems that will be presented (time permitting as in some cases the theorem will be just quoted) in the talks. Some of the results are not published yet and in some cases we shall provide preprints:

Participant travel support

Funds provided by the NSF will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to Ernest Schimmerling by email.


The most popular choice is Shadyside Inn. Other options are listed here under the neighborhoods of Shadyside and Oakland.
Note: There is a shortcut from the Shadyside Inn to CMU that is very pleasant walk. Ask for directions at the registration desk.

Transportation to and from the airport

The least expense option is the 28X Airport Flyer with frequent service between the airport and CMU for $2.75.
The Shadyide Inn is less than 3/4 mile from CMU; you could walk, take a bus, or call the Shadyside Inn to pick you up.
(If you arrive early, you may want to meet others in the Mathematical Sciences Department lounge, 6220 Wean Hall.)

Taxis from the airport cost about $50.

Another door-to-door option is SuperShuttle at $27 per person in a shared van.

Parking at CMU

The East Campus Garage is free on weekends.