Ernest Schimmerling

### Mathematical logic seminar - April 27, 2004

 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 non-reflecting 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 anti-chains 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.