Forcing by non-scattered sets

Abstract

We show that for each non-scattered linear order 〈 L, < 〉 the set of non-scattered subsets of L ordered by the inclusion is forcing equivalent to the two-step iteration of the Sacks forcing and a σ-closed forcing. If the equality sh(S)=א1 or PFA holds in the ground model, then the second iterand is forcing equivalent to the algebra P(ω) / Fin of the Sacks extension.