## 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 〈κ_{n}:n<ω〉, Prikry defined the forcing P of all perfect subtrees of ∏_{n<ω}κ_{n}, and proved that for κ=sup_{n<ω}κ_{n}, 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κ, implying failure of the (ω,κ)-d.l. These hitherto unpublished results are included, setting the stage for the following recent results.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 κ^{ω} 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