| 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.
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