On skinny stationary subsets of Pkλ

Yo Matsubara, Toshimichi Usuba

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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 languageEnglish
Pages (from-to)667-680
Number of pages14
JournalJournal of Symbolic Logic
Volume78
Issue number2
DOIs
Publication statusPublished - 2013 Jun 1

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ASJC Scopus subject areas

  • Philosophy
  • Logic

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