Menachem Magidor at Carnegie Mellon University on March 19, 2011
Appalachian set theory
Saturday, March 19, 2011
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
"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
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
which is a weaker principle than □κ.
there is Schimmerling's hierarchy of principles
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:
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
- The definitions of weak squares and a few examples of applications
in infinite combinatorics, topology and algebra.
- The conneciton between weak squares and the existence of good and very
- 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
(This generalizes a result of Cummings and Schimmerling
about Prikry forcing and weak squares.)
- Weak squares and large cardinals.
- Weak squares and forcing axioms.
A good source for the definition of
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
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:
- Mirna Džamonja and Saharon Shelah,
On squares, outside guessing of clubs and If[λ],
Fundamenta Mathematicae 148 (1995), no. 2, 165-198
- Matthew Foreman and Menachem Magidor,
A very weak square principle,
The Journal of Symbolic Logic 62 (1997), no. 1, 175-196
- James Cummings, Matthew Foreman and Menachem Magidor,
Squares, scales and stationary reflection,
Journal of Mathematical Logic 1 (2001), no. 1, 35-98
- Ernest Schimmerling and Martin Zeman,
□κ in core models,
Journal of Mathematical Logic 4 (2004), no. 1, 1-72
- James Cummings and Menachem Magidor,
Martin's Maximum and weak square,
Proceedings of the American Mathematical Society,
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.
- Your name, university affiliation,
mailing address, phone number and email address
- Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of
your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant
- A brief statement about your interest in the workshop
- An itemized estimate of your expected transportation expenses
The most popular choice is
Other options are listed
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
at $27 per person in a shared van.
Parking at CMU
East Campus Garage is free on weekends.