# A cofinality-preserving small forcing may introduce a special Aronszajn tree

Extended Abstract:

Here, it is proved that adding a subset of $\omega_2$ may introduce a special Aronszajn tree of height $\aleph_{\omega_1+1}$ :

Starting with a model of two supercompact cardinals, we construct a model with no special $\aleph_{\omega_1+1}$-Aronszajn trees, in which there exists a notion of forcing $\mathbb P$ of cardinality $\omega_3$, which is $\sigma$-closed, $\omega_1$-distributive, $\omega_3$-Knaster, and such that in the generic extension by $\mathbb P$, there exists a special Aronszajn tree of height $\aleph_{\omega_1+1}$.

Abstract:

Abstract