Increasing u2 by a stationary set preserving forcing

Abstract

We show that if I is a precipitous ideal on ω₁ and if θ > ω₁ is a regular cardinal, then there is a forcing ℛ = ℛ(I,θ) which preserves the stationarity of all I-positive sets such that in V^ℛ, 〈 H_θ;∈,I 〉 is a generic iterate of a countable structure 〈 M;∈,Ī 〉. This shows that if the nonstationary ideal on ω₁ is precipitous and H_θ^# exists, then there is a stationary set preserving forcing which increases \utilde{δ}¹₂. Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω₁ is precipitous, then \utilde{δ}¹₂ = u₂ = ω₂.