Club isomorphisms on higher Aronszajn trees

Krueger, John

doi:10.1016/j.apal.2018.05.003

Cited by 7 publications

(8 citation statements)

References 5 publications

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“…Generalizing this further, Golshani and Hayut have recently shown ( [19]), using posets which specialize with anticipation, that it is consistent that, simultaneously, for every regular cardinal κ, SATP(κ + ) holds. Krueger has generalized the result of Laver-Shelah (and also Abraham-Shelah, [2]) in a different direction ( [28]), showing that it is consistent that any two countably closed Aronszjan trees on ω 2 are club isomorphic. And finally, Asperó and Golshani ([3]) have announced a positive solution to the question of whether SATP(ω 2 ) is consistent with the GCH.…”

Section: Introductionmentioning

confidence: 85%

“…Much of this material was originally developed by Mitchell [33]. Our exposition here is a summary of what can be found in [28], Section 1, to which we refer the reader for proofs.…”

Section: Review Of Strongly Generic Conditionsmentioning

confidence: 99%

“…The following definition, which is inspired by the presentation in [28], is a variation on the standard notion of a suitable sequence, modified to have a domain in F WC . Definition 2.13.…”

Section: Review Of Strongly Generic Conditionsmentioning

confidence: 99%

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Club Stationary Reflection and the Special Aronszajn Tree Property

Ben-Neria¹,

Gilton²

2021

Preprint

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We prove that it is consistent that Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on ω 2 , thereby contributing to the study of the tension between compactness and incompactness in set theory. The poset which produces the final model follows the collapse of a weakly compact cardinal first with an iteration of club adding (with anticipation) and second with an iteration specializing Aronszajn trees.In the first part of the paper, we prove a general theorem about specializing Aronszajn trees after forcing with what we call F WC -Strongly Proper posets. This type of poset, of which the Levy collapse is a degenerate example, uses systems of exact residue functions to create many strongly generic conditions. We prove a new result about stationary set preservation by quotients of this kind of poset; as a corollary, we show that the original Laver-Shelah model satisfies a strong stationary reflection principle, though it fails to satisfy the full Club Stationary Reflection. In the second part, we show that the composition of collapsing and club adding (with anticipation) is an F WC -Strongly Proper poset. After proving a new result about Aronszajn tree preservation, we show how to obtain the final model.

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Section: Introductionmentioning

confidence: 85%

“…Much of this material was originally developed by Mitchell [33]. Our exposition here is a summary of what can be found in [28], Section 1, to which we refer the reader for proofs.…”

Section: Review Of Strongly Generic Conditionsmentioning

confidence: 99%

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Club Stationary Reflection and the Special Aronszajn Tree Property

Ben-Neria¹,

Gilton²

2021

Preprint

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“…It would certainly be interesting to see to what extent our results generalize to higher cardinals above ω 1 . In particular, we mention a recent result of Krueger [13] on the club-isomorphism of higher Aronszajn trees that could substitute the Abraham-Shelah model. The construction schemes developed by Brodsky, Lambie-Hanson and Rinot [4,15,18] for higher Suslin and Aronszajn trees also seem rather relevant.…”

Section: Proof (1) Suppose Thatmentioning

confidence: 99%

On the complexity of classes of uncountable structures: trees on $\aleph _1$

Friedman

Soukup

2021

Fund. Math.

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We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire space ω ω 1 1. First, we show that none of these classes have the Baire property (unless they are empty). Moreover, under V = L, (a) the class of Aronszajn and Suslin trees is Π 1 1-complete, (b) the class of special Aronszajn trees is Σ 1 1-complete, and (c) the class of Kurepa trees is Π 1 2-complete. We achieve these results by finding nicely definable reductions that map subsets X of ω1 to trees TX so that TX is in a given tree class T if and only if X is stationary/non-stationary (depending on the class T). Finally, we present models of CH where these classes have lower projective complexity. 1 and 2 ω 1. Basic open sets correspond to countable partial functions, which, in turn, give rise to an ω 1-Borel structure on ω ω 1 1 , 2 ω 1 and so P(ω 1) as well. This allows us to measure the complexity of subsets of ω ω 1 1 or, equivalently, of families of natural combinatorial structures on ω 1. In this paper, we will focus on models of CH, i.e., 2 ℵ 0 = ℵ 1. This is a fairly natural assumption in

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“…It would certainly be interesting to see to what extent our results generalise to higher cardinals above ω 1 . In particular, we mention a recent result of Krueger [13] on the club-isomorphism of higher Aronszajn trees that could substitute the Abraham-Shelah model. The construction schemes developed by Brodsky, Lambie-Hanson and Rinot [4,18,15] for higher Suslin and Aronszajn trees also seems rather relevant.…”

Section: Kurepa Treesmentioning

confidence: 99%

On the complexity of classes of uncountable structures: trees on $\aleph_1$

Friedman¹,

Soukup²

2019

Preprint

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We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space ω 1 ω 1 . First, we will show that none of these classes have the Baire property (unless they are empty). Moreover, under (V = L), (a) the class of Aronszajn and Suslin trees is Π 1 1 -complete, (b) the class of special Aronszajn trees is Σ 1 1 -complete, and (c) the class of Kurepa trees is Π 1 2 -complete. We achieve these results by finding nicely definable reductions that map subsets X of ω 1 to trees T X so that T X is in a given tree-class T if and only if X is stationary/non-stationary (depending on the class T ). Finally, we present models of CH where these classes have lower projective complexity.

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Club isomorphisms on higher Aronszajn trees

Cited by 7 publications

References 5 publications

Club Stationary Reflection and the Special Aronszajn Tree Property

Club Stationary Reflection and the Special Aronszajn Tree Property

On the complexity of classes of uncountable structures: trees on $\aleph _1$

On the complexity of classes of uncountable structures: trees on $\aleph_1$

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