Ascending paths and forcings that specialize higher Aronszajn trees

Philipp Lücke¹

¹ Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn Endenicher Allee 60 53115 Bonn, Germany

Fundamenta Mathematicae, Tome 239 (2017) no. 1, pp. 51-84.

Résumé

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.

Détail

DOI : 10.4064/fm224-11-2016

Keywords: study trees uncountable regular heights containing ascending paths small width combinatorial property trees generalizes concept cofinal branch causes trees non special only every cofinality preserving outer model moreover under certain cardinal arithmetic assumptions non existence paths through tree turns out equivalent statement given tree special cofinality preserving forcing extension ground model certain combinatorial principles construct trees without cofinal branches containing ascending paths small width contrast present number consistency results non existence trees application results consistency strength potential forcing axiom sigma closed well met partial orders satisfying aleph chain condition collections aleph many dense subsets least weakly compact cardinal addition results infinite productivity knaster property characterizes weak compactness canonical inner models finally study influence proper forcing axiom trees containing ascending paths