Boban Velickovic at University of Illinois at Chicago on October 15, 2011

Appalachian set theory

Saturday, October 15, 2011

University of Illinois at Chicago

Boban Velickovic : "Proper forcing remastered"

List of participants in this workshop

Lecture notes by Boban Velickovic and Giorgio Venturi (PDF)

Workshop description

The theory of proper forcing, which has been developed by Shelah over the last 30 years, is a powerful tool for proving independence results. However, quite often when we wish to introduce some object by forcing, the obvious partial order is not proper. In the early 1980s Baumgartner introduced the idea of using side conditions which constrain the way in which the forcing conditions are extended. This approach was later reformulated in terms of finite chains of countable elementary submodels of some H(θ) by Todorcevic and has found numerous applications. However, in some situations when one wishes to preserve not only ω1 but also some larger cardinal countable models as side conditions do not suffice and one is naturally lead to consider side conditions of two or more different types. The first such result is due to Mitchell and Friedman who generalized Baumgartner's poset for adding a club in ω1 to larger cardinals. Recently, the theory of generalized side conditions has been studied with great success by Neeman who used it to obtain versions of the Proper Forcing Axiom for larger cardinals. In this workshop we present the basic theory of generalized side conditions and use it to obtain several applications such as the Friedman-Mitchell forcing for adding a thin club to ω2, Kosmzider's poset for adding a strong ω2-chain of functions in ((ω1)ω1, <fin) and the Baumgarther-Shelah poset for adding a thin very tall superatomic Boolean algebra. Some open questions will be discussed at the end of the workshop.

Suggested reading

Advanced references

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 postdocs, and to 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.


Workshop lectures run from 9:30 a.m. to 6 p.m. with morning refreshments before the workshop, starting at around 8:30 or 8:45 a.m., several coffee breaks during the workshop, and a lunch break from 12:30 to 2:30 p.m. There is no formal dinner but groups of participants will head out to restaurants after the talks.