Mutual stationarity and singular Jonsson cardinals


We prove that if the sequence \(\langle k_n:1 \le n < \omega\rangle\) contains a so-called gap then the sequence \(\langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle\) of stationary sets is not mutually stationary, provided that \(k_n<n\) for every \(n \in \omega\). We also prove a sufficient condition for being singular Jonsson cardinals.

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  1. 1.

    James Cummings, Matthew Foreman, and Menachem Magidor, Canonical structure in the universe of set theory. II, Ann. Pure Appl. Logic, 142 (2006), 55–75

  2. 2.

    Matthew Foreman and Menachem Magidor, Mutually stationary sequences of sets and the nonsaturation of the nonstationary ideal on \(P_{k}(\lambda)\), Acta Math., 186 (2001), 271–300

    MathSciNet  Article  Google Scholar 

  3. 3.

    Matthew Foreman, Some problems in singular cardinals combinatorics, Notre Dame J. Formal Logic, 46 (2005), 309–322

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kecheng Liu and Saharon Shelah, Cofinalities of elementary substructures of structures on \(\aleph_{\omega}\), Israel J. Math., 99 (1997), 189–205

    MathSciNet  Article  Google Scholar 

  5. 5.

    Saharon Shelah, Dependent dreams: recounting types, arXiv: 1202.5795

  6. 6.

    Saharon Shelah, Independence of strong partition relation for small cardinals, and the freesubset problem, J. Symbolic Logic, 45 (1980), 505–509

    MathSciNet  Article  Google Scholar 

  7. 7.

    Saharon Shelah, Proper Forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag (Berlin–New York, 1982)

  8. 8.

    Saharon Shelah, Cardinal Arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press (New York, 1994)

    Google Scholar 

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Correspondence to S. Shelah.

Additional information

The author thanks Alice Leonhardt for the beautiful typing. References like [5, Th0.2=Ly5] means the label of Th.0.2 is y5. The reader should note that the version in my website is usually more updated than the one in the mathematical archive. This is work 1158 in the author’s list.

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Shelah, S. Mutual stationarity and singular Jonsson cardinals. Acta Math. Hungar. 163, 140–148 (2021).

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Key words and phrases

  • combinatorial set theory
  • Jonsson cardinal
  • mutual stationarity

Mathematics Subject Classification

  • primary 03E55
  • secondary 03E40