Moti Gitik at CMU on April 3, 2010

Moti Gitik : "Short extender forcing"

Workshop description

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 En with critical point κn. But κn is not "too large" in that the strength of En is "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 preliminary forcing. 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 dropping cofinality.

References

• 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 (PDF)

Lodging

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 Super Shuttle at \$27 per person in a shared van.

Parking at CMU

The 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.