On skinny stationary subsets of

E-ISSN： 1943-5886|78|2|667-680

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.78, Iss.2, 2013-06, pp. : 667-680

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Abstract

We introduce the notion of skinniness for subsets of and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2^λ-saturation of NS_κλ ∣ X, where NS_κλ denotes the non-stationary ideal over , implies the existence of a skinny stationary subset of X. We also show that if λ is a singular cardinal, then there is no skinnier stationary subset of . Furthermore, if λ is a strong limit singular cardinal, there is no skinny stationary subset of . Combining these results, we show that if λ is a strong limit singular cardinal, then NS_κλ ∣ X can satisfy neither precipitousness nor 2^λ-saturation for every stationary X ⊆ . We also indicate that , where , is equivalent to the existence of a skinnier (or skinniest) stationary subset of under some cardinal arithmetical hypotheses.