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 } 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 above cannot be found for each κ. Continuing to force further, we produce a reals and cardinal preserving extension in which the set of points of λ of countable V cofinality is nonstationary for every regular λ ≥ κ. All this is done under the large cardinal assumption that for each α < κ there exists θ < κ with Mitchell order at least α. We prove that this assumption is optimal. It should be noted that our counterexample (Theorem 1) leaves open the possibility that a partition as above, but of the points of κ of countable W (rather than V ) cofinality, can be found provably in ZFC. This is enough for Viale’s argument, and this weaker question is posed in the published paper. There has been work in the past leading to forcing extensions making the set of points of κ of countable V cofinality nonstationary in the extension, specifically in the context of making the nonstationary ideal on κ precipitous, see Gitik [1]. But preservation of cardinals was not an issue in that context, and the extensions involved did not in fact preserve cardinals. There has also been work on forcing to add clubs consisting of regulars in V , see Gitik [2]. Theorem 1. Suppose that cf(κ) = ω, (∀α < κ)(∃θ < κ)(o(θ) ≥ α), and 2 = κ. Then there is a generic extension W of V such that V and W have the same cardinals and same reals and W |= A is nonstationary, where A = {α < κ | cf (α) = ω}. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0094174.

DOI: 10.1007/s00153-007-0048-0

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Cite this paper

@article{Gitik2007ACP, title={A cardinal preserving extension making the set of points of countable V cofinality nonstationary}, author={Moti Gitik and Itay Neeman and Dima Sinapova}, journal={Arch. Math. Log.}, year={2007}, volume={46}, pages={451-456} }