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 P = P(I, θ) which preserves the stationarity of all I-positive sets such that in $V^P $ , is a generic iterate of a countable structure . This shows that if the nonstationary ideal on ω₁ is precipitous and $H_\theta ^\# $ exists, then there is a stationary set preserving forcing which increases $\delta _2^1 $ · Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω₁ is precipitous, then $\delta _2^1 $ = u₂ = ω₂