Distributivity and minimality in perfect tree forcings for singular cardinals

Abstract

AbstractDobrinen, Hathaway and Prikry studied a forcing ℙκ 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 ℙκ 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 ℙκ 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 ℙκ 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 ℙκ in particular is not (ω, ·, λ+)-distributive under these assumptions.While developing these ideas, we address natural questions regarding minimality and collapses of cardinals.