More fine structural global square sequences

Zeman, M.

doi:10.1007/s00153-009-0156-0

Cited by 6 publications

(10 citation statements)

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“…In contrast, we will use Theorem 1.11 to show that the infinite productivity of the ϑ-Knaster property characterizes weak compactness in canonical inner models of set theory (so-called Jensen-style extender models, see [29]). The proof of the following theorem relies on Todorčević's method of walks on ordinals and results of Schimmerling and Zeman on the existence of square sequences in canonical inner models (see [17] and [30]) that extend seminal results of Jensen from [8].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

Ascending paths and forcings that specialize higher Aronszajn trees

Lücke

2017

Fund. Math.

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“…Seminal results of Jensen (see [6,Section 6]) show that the above reflection property characterizes weak compactness in Gödel's constructible universe L. These result were extended by Zeman in [23] to a much larger class of canonical inner models. Together with [23, Corollary 0.2.…”

Section: Introductionmentioning

confidence: 83%

“…κ-c.c. is productive [22], the failure of (κ) from the productivity of the κ-chain condition is due to Rinot [16], the implication from a failure of (κ) to the weak compactness of κ in L is a consequence of results of Jensen [6] and the equivalence of stationary reflection and weak compactness in canonical inner models L[E] is due to Jensen [6] and Zeman [23]. All other implications and non-implications in the diagram are either trivial, or are proved in the current paper.…”

Section: Questions and Concluding Remarksmentioning

confidence: 99%

Characterizing large cardinals in terms of layered posets

Cox

Lücke

2017

Annals of Pure and Applied Logic

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Given an uncountable regular cardinal κ, a partial order is κstationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in Pκ(P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial orders are κ-stationarily layered implies that κ is a Mahlo cardinal and every stationary subset of κ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.2010 Mathematics Subject Classification. 03E05, 03E55, 06A07.

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“…In [12] fine structural global square sequences are used to determine a lower bound for the consistency strength of the restricted proper forcing axiom PFA(c + -linked) in the following sense: If PFA(c + -linked) holds in a generic extension via proper forcing over a fine structural model M then M must contain the remarkable result, as it was proved earlier, in [11], that a Σ 2 1 -indescribable 1-gap suffices for obtaining a proper forcing extension of a model satisfying GCH where PFA(c + -linked) holds. Paper [21] is a sequel to this paper where the current methods are further extended to give a characterization of stationary reflection at inaccessible cardinals similar to that in L, that is, in terms of coherent club sequences. Paper [9] is a sequel to the current paper, [21] and [14], where the methods developed in these three papers are further extended to constructions of nonthreadable square sequences at successor cardinals in extender models.…”

Section: Introductionmentioning

confidence: 98%

“…Paper [21] is a sequel to this paper where the current methods are further extended to give a characterization of stationary reflection at inaccessible cardinals similar to that in L, that is, in terms of coherent club sequences. Paper [9] is a sequel to the current paper, [21] and [14], where the methods developed in these three papers are further extended to constructions of nonthreadable square sequences at successor cardinals in extender models.…”

Section: Introductionmentioning

confidence: 98%