On skinny stationary subsets of

Journal of Symbolic Logic 78 (2):667-680 (2013)
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Abstract

We introduce the notion of skinniness for subsets of $\mathcal{P}_\kappa \lambda$ and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or $2^\lambda$-saturation of $\mathrm{NS}_{\kappa\lambda}\mid X$, where $\mathrm{NS}_{\kappa\lambda}$ denotes the non-stationary ideal over $\mathcal{P}_\kappa \lambda$, implies the existence of a skinny stationary subset of $X$. We also show that if $\lambda$ is a singular cardinal, then there is no skinnier stationary subset of $\mathcal{P}_\kappa \lambda$. Furthermore, if $\lambda$ is a strong limit singular cardinal, there is no skinny stationary subset of $\mathcal{P}_\kappa \lambda$. Combining these results, we show that if $\lambda$ is a strong limit singular cardinal, then $\mathrm{NS}_{\kappa\lambda}\mid X$ can satisfy neither precipitousness nor $2^\lambda$-saturation for every stationary $X \subseteq \mathcal{P}_\kappa \lambda$. We also indicate that $\diamondsuit_\lambda(E^{\lambda}_{<\kappa})$, where $E^{\lambda}_{<\kappa} \stackrel{\mathrm{def}}{=} \{\alpha < \lambda \mid \mathrm{cf}(\alpha) < \kappa\}$, is equivalent to the existence of a skinnier (or skinniest) stationary subset of $\mathcal{P}_\kappa \lambda$ under some cardinal arithmetical hypotheses

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Citations of this work

On the existence of skinny stationary subsets.Yo Matsubara, Hiroshi Sakai & Toshimichi Usuba - 2019 - Annals of Pure and Applied Logic 170 (5):539-557.

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References found in this work

Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
Nowhere precipitousness of the non-stationary ideal over.Yo Matsubara & Saharon Shelah - 2002 - Journal of Mathematical Logic 2 (01):81-89.

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