Perfect tree forcings for singular cardinals

Natasha Dobrinen, Dan Hathaway, Karel Prikry

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals 〈κn:n<ω〉, Prikry defined the forcing P of all perfect subtrees of ∏n<ωκn, and proved that for κ=supn<ω⁡κn, assuming the necessary cardinal arithmetic, the Boolean completion B of P is (ω,μ)-distributive for all μ<κ but (ω,κ,δ)-distributivity fails for all δ<κ, implying failure of the (ω,κ)-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. P satisfies a Sacks-type property, implying that B is (ω,∞,<κ)-distributive. The (h,2)-d.l. and the (d,∞,<κ)-d.l. fail in B. P(ω)/fin completely embeds into B. Also, B collapses κω to h. We further prove that if κ is a limit of countably many measurable cardinals, then B adds a minimal degree of constructibility for new ω-sequences. Some of these results generalize to cardinals κ with uncountable cofinality.

Original languageEnglish (US)
Article number102827
JournalAnnals of Pure and Applied Logic
Volume171
Issue number9
DOIs
StatePublished - Oct 1 2020

Bibliographical note

Funding Information:
Dobrinen's research was partially supported by National Science Foundation Grant DMS-1600781 .

Publisher Copyright:
© 2020 Elsevier B.V.

Keywords

  • Cardinal characteristics of the continuum
  • Complete Boolean algebras
  • Distributive laws
  • Forcing
  • Large cardinals

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