Less saturated ideals

Gitik, Moti; Shelah, Saharon

doi:10.1090/S0002-9939-97-03702-7

Less saturated ideals
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by Moti Gitik and Saharon Shelah PDF

Proc. Amer. Math. Soc. 125 (1997), 1523-1530 Request permission

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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