Ascending Paths and Forcings That Specialize Higher Aronszajn Trees

In this paper, we study trees of uncountable regular heights containing ascending paths of small width. This combinatorial property of trees generalizes the concept of a cofinal branch and it causes trees to be non-special not only in V, but also in every cofinality-preserving outer model of V. Moreover, under certain cardinal arithmetic assumptions, the non-existence of such paths through a tree turns out to be equivalent to the statement that the given tree is special in a cofinality preserving forcing extension of the ground model. We will present a number of consistency results on the non-existence of trees without cofinal branches containing ascending paths of small width. In contrast, we will construct such trees using certain combinatorial principles. As an application of our results, we show that the consistency strength of a potential forcing axiom for σ-closed, well-met partial orders satisfying the א2-chain condition and collections of א2-many dense subsets is at least a weakly compact cardinal. In addition, we will use our results to show that the infinite productivity of the Knaster property characterizes weak compactness in canonical inner models. Finally, we study the influence of the Proper Forcing Axiom on trees containing ascending paths.