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<ωκn, 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κ, implying failure of the (ω,κ)-d.l. These hitherto unpublished results are included, setting the stage for the following recent results.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 language | English (US) |
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Article number | 102827 |
Journal | Annals of Pure and Applied Logic |
Volume | 171 |
Issue number | 9 |
DOIs | |
State | Published - 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