Embedding Cohen algebras using pcf theory

Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν adds a generic for the Cohen forcing notion on ν. The following question (problem 5.1 in Miller’s list [Mi91]) is attributed to Rene David and Sy Friedman: Does the product of the forcing notions n2 add a generic for the forcing ω+12? We show here that the answer is yes in ZFC. Previously Zapletal [Za] has shown this result under the assumption אω+1. In fact, a similar theorem can be shown about other singular cardinals as well. The reader who is interested only in the original problem should read אω+1 for λ, אω for μ and {אn : n ∈ (1, ω)} for a. Definition 1 1. Let a be a set of regular cardinals. ∏ a is the set of all functions f with domain a satisfying f(κ) ∈ κ for all κ ∈ a. 2. A set b ⊆ a is bounded if sup b < sup a. b is cobounded if a \ b is bounded. ∗The research partially supported by “The Israel Science Foundation” administered by of The Israel Academy of Sciences and Humanities. Publication 595.