Goldstern–Judah–Shelah preservation theorem for countable support iterations

In [4] a preservation theorem for countable support iterated forcing is proved with restriction to forcing notions which are not ω-distributive. We give the proof of the theorem without this restriction. 1. The preservation theorem. In [4] a preservation theorem for countable support iteration of proper forcing notions was proved with the additional assumption that all forcing notions which are iterated add a new sequence of ordinals. In this section we will prove the same theorem (Theorem 1.7) without this additional assumption. We use the terminology introduced in [4]; definitions and lemmata 1.1–1.8 correspond to 5.4, 5.5, 5.6, 5.8, 5.11, 5.12, 5.14, 5.13 of [4]. Lemma 1.9 is a version of 5.15 without the additional assumption and essentially it marks the difference between these two proofs of the preservation theorem. Let 〈vn: n ∈ ω〉 be an increasing sequence of two-place relations on ω. We let v = n vn. We assume the following: (i) {f ∈ ω : f vn g} is a closed set for any n ∈ ω and g ∈ ω; (ii) the set C = dom(v) is a closed subset of ω; (iii) for every countable set A ⊆ C there is g ∈ ω such that ∀f ∈ A, f v g; (iv) the closed sets mentioned in conditions (i) and (ii) have an absolute definition (i.e. as Borel sets they have the same Borel codes in all transitive models we will consider). 1991 Mathematics Subject Classification: Primary 03E40.