Ascending paths and forcings that specialize higher Aronszajn trees
Volume 239 / 2017
Abstract
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 ${\rm {V}}$, but also in every cofinality-preserving outer model of ${\rm {V}}$. 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 use certain combinatorial principles to construct trees without cofinal branches containing ascending paths of small width. In contrast, we will also present a number of consistency results on the non-existence of such trees.
As an application of our results, we show that the consistency strength of a potential forcing axiom for $\sigma $-closed, well-met partial orders satisfying the $\aleph _2$-chain condition and collections of $\aleph _2$-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.
