Squares and Narrow Systems

Lambie-Hanson, Chris

doi:10.1017/jsl.2017.38

Cited by 20 publications

(27 citation statements)

References 15 publications

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“…For full discussion about narrow systems and the Narrow System Property, see [8]. Systems and Narrow Systems appear naturally when dealing with the tree property at successor of singular cardinals.…”

Section: Definitionmentioning

confidence: 99%

Magidor–Malitz reflection

Hayut

2017

Arch. Math. Logic

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Section: Definitionmentioning

confidence: 99%

Magidor–Malitz reflection

Hayut

2017

Arch. Math. Logic

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“…Next, we present result concerning the existence and non-existence of trees without cofinal branches containing ascending paths of small width. The proofs of most these results make use of the notion of narrow system introduced by Magidor and Shelah in [15] and recent results of Lambie-Hanson about these systems contained in [12]. The statements (i), (ii) and (v) of the following theorem are direct consequences of results contained in [12].…”

Section: Introductionmentioning

confidence: 96%

Ascending paths and forcings that specialize higher Aronszajn trees

Lücke

2017

Fund. Math.

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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.

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“…The following notation, due to Magidor and Shelah [4], plays an important role in the investigation of the tree property at successors of singular cardinals. For more information about narrow systems and their connections to squares we refer to [3]. Definition 1.…”

Section: Preliminariesmentioning

confidence: 99%