Distributivity and minimality in perfect tree forcings for singular cardinals

Abstract

Dobrinen, Hathaway and Prikry studied a forcing ℙ_κ consisting of perfect trees of height λ and width κ where κ is a singular λ-strong limit of cofinality λ. They showed that if κ is singular of countable cofinality, then ℙ_κ is minimal for ω-sequences assuming that κ is a supremum of a sequence of measurable cardinals. We obtain this result without the measurability assumption.

Prikry proved that ℙ_κ is (ω, ν)-distributive for all ν < κ given a singular ω-strong limit cardinal κ of countable cofinality, and Dobrinen et al. asked whether this result generalizes if κ has uncountable cofinality. We answer their question in the negative by showing that ℙ_κ is not (λ, 2)-distributive if κ is a λ-strong limit of uncountable cofinality λ and we obtain the same result for a range of similar forcings, including one that Dobrinen et al. consider that consists of pre-perfect trees. We also show that ℙ_κ in particular is not (ω, ·, λ⁺)-distributive under these assumptions.

While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.