Time:  12  1:30 p.m. 
Room: 
7220 Wean Hall

Speaker:  Richard Dore Undergraduate Honors Student Department of Mathematical Sciences Carnegie Mellon University 
Title: 
$\omega_2$Suslin Trees

Abstract: 
The focus of this talk will be $\omega_2$Suslin trees. It is know that
GCH is not strong enough to guarantee the existence of $\omega_1$Suslin
trees, but for $\omega_2$, this problem remains open. It is fairly easy to
show (under CH) that $\Diamond_{\omega_2}(S)$ implies the existence of
$\omega_2$Suslin trees if $S$ is a stationary subset of $\omega_2$ of
cofinality $\omega_1$ points. $S$ being a nonreflecting stationary set of
cofinality $\omega$ points also works. GCH also implies
$\Diamond_{\omega_2}(S)$ when $S$ is a set of cofinality $\omega$
points, but it does not guarantee that a nonreflecting stationary set
exists.
Contrastingly, Laver and Shelah proved Con(CH + no $\omega_2$Suslin
trees). The proof involves Levy collapsing $\kappa$ measurable cardinal
(can be improved to a weak compact), and then iteratively forcing with
countable antichains using countable support. The difficult part of this
proof involves inductively showing that the forcing has the $\kappa$ chain
condition. I will focus on the illustrative case of killing the first
Suslin tree. Then I will discuss how this can be iterated
appropriately.
