Moti Gitik at CMU on April 3, 2010
Appalachian set theory
Saturday, April 3, 2010
Registration and morning refreshments 9 - 9:30 a.m. in
Wean Hall 6220
Lectures 9:30 a.m. - 12:30 p.m. and 2:30 - 6 p.m. in
Doherty Hall 2302
"Short extender forcing"
List of participants in this workshop
Lecture notes by Moti Gitik and Spencer Unger (PDF)
The basic setting for such forcings is as follows.
We have a singular cardinal κ
of countable cofinality and an increasing sequence
κn for n < ω
whose supremum is κ.
Each κn is a large cardinal in that it carries an extender
with critical point κn.
But κn is not "too large" in that the strength of
"much less" than κ n+1.
In particular, we are not assuming that κ is overlapped
by any extender.
Using such a sequence of extenders, we define a forcing
that blows up the power of κ, that is, makes
2κ > κ+.
Such forcings allow us to reduce known consistency strength upper bounds of
certain instances of the Singular Cardinals Hypothesis to the optimal
ones. In addition, they allow us to construct models with PCF
configurations that other methods fail to produce. We hope that
further development of these forcing notions may lead to progress
on basic problems of cardinal arithmetic like the Shelah Weak
Hypothesis and the PCF Conjecture.
We plan to start the workshop with a forcing notion that uses an
overlapping sequence of extenders
En for n < ω to blow up the power of κ.
Then we modify the construction to one in which each En
has length (&kappan)+n+2
and calculate that 2κ
= &kappa++ in the corresponding forcing extension.
Next we make 2κ = &kappa+++
but for this we prepare the ground model with a certain
We discuss how these ideas can be used to make the gap
between κ and its power even larger.
Finally we turn to a certain cardinal arithmetic configuration called
- James Cummings, "Iterated forcing and elementary embeddings",
chapter in Handbook of Set Theory (PDF)
- Moti Gitik, "Prikry-type forcings",
chapter in Handbook of Set Theory
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.
Participant travel support
Funds provided by the National Science Foundation 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
James Cummings and
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