Pseudo-Prikry sequences

Lambie-Hanson, Chris

doi:10.1090/proc/13996

Cited by 3 publications

(1 citation statement)

References 11 publications

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“…Now move to W and note that, since , we have . By assumptions (1)–(3) in the statement of the theorem, [18, Corollary 4.2] implies that we can find an increasing sequence of ordinals that is cofinal in and such that, for all , for all sufficiently large , we have . (This fact is also implicit in the earlier [11, Theorem 2.0].…”

Section: Indexed Square Sequencesmentioning

confidence: 82%

Knaster and Friends Iii: Subadditive Colorings

Lambie-Hanson¹,

Rinot²

2022

J. symb. log.

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We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa $ , the existence of a strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is a theorem of $\textsf{ZFC}$ . Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is independent of $\textsf{ZFC}$ . We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of $\kappa $ -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring $c:[\kappa ]^2 \rightarrow \theta $ is equivalent to a certain weak indexed square principle $\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa , \theta )$ . We conclude the paper with an application to the failure of the infinite productivity of $\kappa $ -stationarily layered posets, answering a question of Cox.

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Section: Indexed Square Sequencesmentioning

confidence: 82%

Knaster and Friends Iii: Subadditive Colorings

Lambie-Hanson¹,

Rinot²

2022

J. symb. log.

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More Notions of Forcing Add a Souslin Tree

Brodsky¹,

Rinot²

2019

Notre Dame J. Formal Logic

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Squares and uncountably singularized cardinals

Levine

Sinapova

2021

Fund. Math.

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It is known that if κ is inaccessible in V , and W is an outer model of V such that (κ +) V = (κ +) W , and cf W (κ) = ω, then κ,ω holds in W. Many strengthenings of this theorem have been investigated as well. We show that this theorem does not generalize to uncountable cofinalities: There is a model V in which κ is inaccessible and there is a forcing extension W of V in which (κ +) V = (κ +) W , ω < cf W (κ) < κ, and κ,τ fails in W for all τ < κ. We make use of Magidor's forcing for singularizing an inaccessible κ to have uncountable cofinality. Along the way, we analyze stationary reflection in this model, and we show that it is possible for κ,cf(κ) to hold in a forcing extension by Magidor's poset if the ground model is prepared with a partial square sequence.

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Pseudo-Prikry sequences

Cited by 3 publications

References 11 publications

Knaster and Friends Iii: Subadditive Colorings

Knaster and Friends Iii: Subadditive Colorings

More Notions of Forcing Add a Souslin Tree

Squares and uncountably singularized cardinals

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