Résume | It is well-known that under CH one can attempt to specialize trees of height ω_2 without cofinal branches using a naive forcing with countable approximations. However, one has to require more (the nonexistence of ascending paths) than the lack of cofinal branches to make sure that the naive attempt does not fail.
I will discuss these possible obstacles to specialize trees of height ω_2 , and then
use models as side conditions to construct a forcing notion with finite conditions, which under PFA specializes a given tree of height ω_2 without cofinal branches.
If time permits, I will mention generalizations of this result to taller trees. |