Increasing u2 by a stationary set preserving forcing

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 , 〈Hθ;∈, I〉 is a generic iterate of a countable structure 〈M ;∈, I〉. 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 δ ̃ 1 2. Moreover, if Bounded Martin’s Maximum holds and the nonstationary ideal on ω1 is precipitous, then δ ̃ 1 2 = u2 = ω2. In this paper we modify Jensen’s L-forcing (cf. [Jen90a] and [Jen90b]) and apply this to the theory of precipitous ideals and the question about the size of u2. Forcings which increase the size of u2 were already presented in the past. After Steel and van Wesep had shown that u2 = ω2 is consistent in the presence of large cardinal hypotheses (cf. [SVW82]), Woodin proved that if the nonstationary ideal on ω1 is ω2-saturated and P(ω1) # exists, then u2 = ω2 (cf. [Woo99, Theorem 3.17]; in particular, u2 = ω2 follows from Martin’s Maximum by work of Foreman, Magidor and Shelah, cf. [FS88].) More recently, Ketchersid, Larson, and Zapletal also constructed forcings which increase u2 (cf. [KLZ07]). Recall that δ ̃ 1 2 is the supremum of the lengths of all ̃ 1 2 well-orderings of the reals, and that if the reals are closed under sharps, then u2, the second uniform indiscernible, is defined to be the least ordinal above ω1 ∗Both authors gratefully acknowledge support by DFG grant no. SCHI 484/3-1. The second author also gratefully acknowledges support by NSF grant no. DMS-0401312. The authors wish to thank Philipp Doebler and the referee for valuable comments.