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Less Saturated Ideals

Authors:	Moti Gitik Saharon Shelah

Institution:	School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv 69978, Israel ; Hebrew University of Jerusalem, Department of Mathematics, Givat Ram, Jerusalem, Israel

Abstract:	We prove the following: (1) If $\kappa$ is weakly inaccessible then $NS_{\kappa }$ is not $\kappa ^{+}$ -saturated. (2) If $\kappa$ is weakly inaccessible and $\theta < \kappa$ is regular then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$ -saturated. (3) If $\kappa$ is singular then $NS^{cf \kappa }_{ \kappa ^{+}}$ is not $\kappa ^{++}$ -saturated. Combining this with previous results of Shelah, one obtains the following: (A) If $\kappa >\aleph _{1}$ then $NS_{\kappa }$ is not $\kappa ^{+}$ -saturated. (B) If $\theta ^{+}< \kappa$ then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$ -saturated.

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