Set Theory Seminar, Fall 2008



 

                                  
September 12 - October  24
Andres Caicedo,     Forcing axioms and inner models
The lectures  are available on Caicedo's teaching page.

Abstract: We present some recent results (due to myself, Velickovic, and Viale, among others) that aim to illuminate the relation between models of strong forcing axioms (like Martin's Maximum) and their inner models. One of the ultimate goals of this line of research is to clarify what is needed of a forcing poset if it is to force a forcing axiom.

       
October 24 - November 14                                                                   
Richard Ketchersid

Abstract: We discuss a result due to Jackson and myself showing that under AD + V = L(R)  certain "HOD" like inner models have an initial segment of omega_1^{omega_1} mod NS. The technical difficulties have to do with defining canonical iteration strategies for mice and lifting to forcing extensions, essentially showing how to get universally Baire iteration strategies. We will start with a result of Woodin showing that assuming Delta^1_2 (lightface) determinacy plus sharps for all reals that there
is a countably iterable one Woodin model. This can be done with course models so that no fine structure is involved. I might then say something about what iterability gives, e.g., comparison and generic coding. The techniques are soft and related to what is required for the argument mentioned in the first paragraph. Lecture notes  

November 21
Stefan Geschke,   Efimov spaces of countable and uncountable coinitiality

Abstract:  The coinitiality of a compact space $X$ is the least ordinal $\delta$ such that $X$ is the limit of a nontrivial inverse system of length $\delta$.
There is an easy characterization of compact spaces of countable coinitiality.  We show that certain natural statements fail to characterize coinitiality $\omega_1$, at least assuming $\lozenge$. The main open question in this area is whether it is consistent that
there is a compact space of coinitiality $>\omega_1$.

December 5
Marion Scheepers, Infinite games and weak memory