Ascending paths and forcings that specialize higher Aronszajn trees

Abstract

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 ℵ 2-chain condition and collections of ℵ 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. 2010 Mathematics Subject Classification. 03E05, 03E35, 03E55. Key words and phrases. Special trees, Aronszajn trees, specialization forcings for trees, productivity of chain conditions, Knaster property, walks on ordinals, proper forcing axiom. During the preparation of this paper, the author was partially supported by DFG-grant LU2020/1-1. This research was partially done whilst the author was a visiting fellow at the Isaac Newton Institute for Mathematical Sciences in the programme 'Mathematical, Foundational and Computational Aspects of the Higher Infinite' (HIF). The author would like to thank the organizers for the opportunity to participate in this programme. Moreover, the author would like to thank Assaf Rinot for helpful discussions on the topic of the paper.

Reduced Powers of Souslin Trees

“…Such trees are even harder to specialize, since ℱ fin θ projects to ℱ bd µ for all infinite cardinals µ θ , and so a model in which such a λ + -tree becomes special would have to satisfy cf(λ) = cf(µ) for all infinite cardinals µ θ . On a dual front, Lücke [Luc17] proved that assuming λ <λ = λ, any λ + -tree with the property that for all infinite θ < λ, the tree admits no ℱ θ -ascent path, can be made special via a cofinality-preserving notion of forcing. That is, the tree is specializable.…”

Squares, Ascent Paths, and Chain Conditions

“…Note that this shows that trees containing ascent paths are non-special in a very absolute way, because it implies that they remain non-special in every outer model of V in which µ and µ + remain cardinals and cof(λ) = cof(µ) holds. This result was later strengthened by Todorčevic and Torres Pérez in [26] and by the second author in [17] (see Lemma 2.6).…”

“…With the help of a result form [17], it is easy to see that the conclusion of Theorem 1.3 implies that κ is a weakly compact cardinal in L. Proof. Assume, towards a contradiction, that there is a (κ)-sequence C and let T = T(ρ C 0 ) denote the tree of full codes of walks through C defined in [24, Section 1] (as in the proof of Theorem 3.5).…”

“…It is easy to see that, if κ is a weakly compact cardinal and T is tree of height κ containing a λ-ascent path with λ < κ, then T contains a cofinal branch. Moreover, basic arguments, presented in [17,Section 3], show that, if κ is a weakly compact cardinal, µ < κ is a regular, uncountable cardinal, and G is Col(µ, <κ)-generic over V, then every tree of height κ in V[G] that contains a λ-ascent path with λ < µ already has a cofinal branch. Since seminal results of Jensen and Todorčević show that, for uncountable regular cardinals κ, a failure of (κ) implies that κ is weakly compact in Gödel's constructible universe L (see [11,Section 6] and [24, (1.10)]), the above theorem directly yields the following corollary showing that the existence of regular cardinals λ < µ such that there are no µ + -Aronszajn trees with λ-ascent paths is equiconsistent with the existence of a weakly compact cardinal.…”

“…The results of [4] and [17] leave open the question whether it is consistent that the κ-Knaster property is countably productive for accessible uncountable regular cardinals, like ℵ 2 . It is easy to see that this productivity implies certain cardinal arithmetic statements.…”