### Abstract

We introduce the notion of skinniness for subsets ofP ë and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2ë-saturation of NSë | X, where NSë denotes the non-stationary ideal over Pë, implies the existence of a skinny stationary subset of X. We also show that if ë is a singular cardinal, then there is no skinnier stationary subset of Pë. Furthermore, if ë is a strong limit singular cardinal, there is no skinny stationary subset of Pë. Combining these results, we show that if ë is a strong limit singular cardinal, then NSë | X can satisfy neither precipitousness nor 2ë-saturation for every stationary X Pë. We also indicate that ë(Eë <), where Eë < def = { < ë cf() < }, is equivalent to the existence of a skinnier (or skinniest) stationary subset of Pë under some cardinal arithmetical hypotheses.

Original language | English |
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Pages (from-to) | 667-680 |

Number of pages | 14 |

Journal | Journal of Symbolic Logic |

Volume | 78 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 Jun 1 |

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### ASJC Scopus subject areas

- Philosophy
- Logic

### Cite this

*Journal of Symbolic Logic*,

*78*(2), 667-680. https://doi.org/10.2178/jsl.7802180