February 1
Gunter Fuchs,
CUNY
Bounded forcing axioms and the preservation of wide Aronszajn trees
I will talk about connections between three properties of a forcing class $\Gamma$: the most well-known one is the bounded forcing axiom at $(\omega_1,\lambda)$ for this class, introduced by Goldstern and Shelah for proper forcing. The second property is a two cardinal version of $\Sigma^1_1$ absoluteness, at $(\omega_1,\lambda)$, and the third is the preservation of Aronszajn trees of height $\omega_1$ and width $\lambda$. Under the assumption that $\lambda=\lambda^\omega$ and forcings in $\Gamma$ don't add countable subsets of $\lambda$, I will show that these properties are equivalent. The class of forcing notions satisfying Jensen's property of subcompleteness is a canonical class that does not add reals and it is a corollary of the above result that the bounded forcing axiom for subcomplete forcing at $(\omega_1,2^\omega)$ is equivalent to the preservation of Aronszajn trees of height $\omega_1$ and width $2^\omega$. This generalizes a prior result, obtained jointly with Kaethe Minden. I will discuss some further results, as well as some open questions, on the preservation of Aronszajn trees of height $\omega_1$ and other widths.