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6. Precipitouness of NS


    1. Add remark

      Problem 6.4.

      [Larson] Assume $\text{NS}_{\omega_1}$ is precipitous. Is it possible to force $\text{NS}_{\omega_1}$ to be non-precipitous without adding subsets of $\omega_1$?
        1. Remark. [Larson] It is consistent for this to be impossible, e.g. if $NS_{\omega_1}^+$ has a dense subset of size $\aleph_1$. Also, any such forcing must always add a subset of size $\omega_1$?

              Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.