A CARDINAL PRESERVING EXTENSION MAKING THE SET OF POINTS OF COUNTABLE V COFINALITY NONSTATIONARY

Abstract. Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ + nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ +. Finally we show that our large cardinal assumption is optimal. The results in this paper were inspired by the following question, posed in a preprint (http://arxiv.org/abs/math/0509633v1, 27 September 2005) to the paper Viale [9]: Suppose V ⊂ W and V and W have the same cardinals and the same reals. Can it be shown, in ZFC alone, that for every cardinal κ, there is in V a partition {As | s ∈ κ <ω} of the points of κ + of countable V cofinality, into disjoint sets which are stationary in W? In this paper we show that under some assumptions on κ there is a reals and cardinal preserving generic extension W which satisfies that the set of points of κ + of countable V cofinality is nonstationary. In particular, a partition as abov