Iterations with Mixed Support

In this talk we will consider three properties of iterations with mixed (finite/countable) supports: iterations of arbitrary length preserve ω1, iterations of length ≤ ω2 over a model of CH have the א2-chain condition and iterations of length < ω2 over a model of CH do not increase the size of the continuum. Definition 1. Let Pκ be an iterated forcing construction of length κ, with iterands 〈Q̇α : α < κ〉 such that for every α < κ α ”Q̇α is σ-centered” or α ”Q̇α is countably closed”. Then Pκ is finite/countable iteration if and only if for every p ∈ Pκ, support(p) = {α < κ : p(α) 6= 1̇α} is countable and Fsupport(p) = {α : ”Qα is σ-centered”, p(α) 6= 1̇α} is finite. Remark 1. In the context of the above definition, whenever α ”Q̇α is σ-centered” we will say that α is a σ-centered stage and correspondingly, whenever ”Q̇α is countably closed” we will say that α is a countably closed stage. From now on Pκ is a finite/countable iteration of length κ. Definition 2. Let p, q ∈ Pκ. We say than p ≤D q if and only if p ≤ q and for every σ-centered stage α < κ, p α p(α) = q(α). Similarly p ≤C q if and only if for every countably closed stage α, p α p(α) = q(α). Claim. Both ≤D and ≤C are transitive relations. Lemma 1. Let 〈pn〉n∈ω be a sequence in Pκ such that for every n ∈ ω, pn+1 ≤D pn. Then there is a condition p ∈ Pκ such that for every n ∈ ω, p ≤D pn. Date: August 2, 2007. 1