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On the relationship between mutual and tight stationarity. (English) Zbl 1489.03015

Let \(\kappa\) be a singular cardinal and \(\langle \kappa_{\xi}: \xi < \operatorname{cf}(\kappa)\rangle\) a sequence of regular cardinals cofinal in \(\kappa\). A stationary sequence is a sequence \(\vec{\xi} = \langle S_{\xi} : \xi < \operatorname{cf}(\kappa) \rangle\) where \(S_{\xi} \subseteq \kappa_{\xi}\) is stationary in \(\kappa_{\xi}\) for all but boundedly many \(\xi < \operatorname{cf}(\kappa)\).
M. Foreman and M. Magidor [Acta Math. 186, No. 2, 271–300 (2001; Zbl 1017.03022)] introduced two new concepts of stationarity, mutual stationarity and tight stationarity. Here, the authors are interested in the case where \(\operatorname{cf}(\kappa) = \omega\) and where each \(S_{\xi}\) consists of ordinals of some fixed uncountable cofinality \(\eta\). J. Cummings et al. [Ann. Pure Appl. Logic 142, No. 1–3, 55–75 (2006; Zbl 1096.03060)] constructed models where \(\langle \omega_n : n<\omega \rangle\) have mutually stationary sequences which are not tightly stationary.
The main result of this paper is a forcing construction so that there are mutually stationary but not tightly stationary sequences on every increasing \(\omega\)-sequence of regular cardinals. They show that this property is preserved under a class of Prikry type forcing. They give examples in the Cohen and Prikry models of \(\omega\)-sequences of regular cardinals for which there is a non-tightly stationary sequence of stationary subsets consisting of cofinality \(\omega_1\) ordinals. They show that such stationary sequences are mutually stationary in case of interleaved supercompact cardinals.

MSC:

03E35 Consistency and independence results
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E55 Large cardinals
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[1] Ben-Neria, Omer, On singular stationarity I (mutual stationarity and ideal-based methods), Adv. Math., 356, Article 106790 pp. (2019) · Zbl 1441.03036
[2] Ben-Neria, Omer, On singular stationarity II (tight stationarity and extenders-based methods), J. Symb. Log., 84, 1, 320-342 (2019) · Zbl 1436.03243
[3] Chen, William, Tight stationarity and tree-like scales, Ann. Pure Appl. Log., 166, 10, 1019-1036 (2015) · Zbl 1373.03070
[4] Cummings, James, Notes on singular cardinal combinatorics, Notre Dame J. Form. Log., 46, 3, 251-282 (2005) · Zbl 1121.03053
[5] Cummings, James; Foreman, Matthew, Diagonal Prikry extensions, J. Symb. Log., 75, 4, 1383 (2010) · Zbl 1245.03078
[6] Cummings, James; Foreman, Matthew; Magidor, Menachem, Canonical structure in the universe of set theory: part one, Ann. Pure Appl. Log., 129, 1, 211-243 (2004) · Zbl 1058.03051
[7] Cummings, James; Foreman, Matthew; Magidor, Menachem, Canonical structure in the universe of set theory: part two, Ann. Pure Appl. Log., 142, 1, 55-75 (2006) · Zbl 1096.03060
[8] Cummings, James; Foreman, Matthew; Schimmerling, Ernest, Organic and tight, Ann. Pure Appl. Log., 160, 1, 22-32 (2009) · Zbl 1165.03029
[9] Dehornoy, Patrick, Iterated ultrapowers and Prikry forcing, Ann. Math. Log., 15, 2, 109-160 (1978) · Zbl 0417.03025
[10] Foreman, Matthew, More saturated ideals, (Cabal Seminar 79-81: Proceedings, Caltech-UCLA Logic Seminar 1979-81 (1983), Springer: Springer Berlin, Heidelberg), 1-27 · Zbl 0536.03032
[11] Foreman, Matthew, Stationary sets, Chang’s conjecture and partition theory, DIMACS Ser. Discret. Math. Theor. Comput. Sci., 58, 73-94 (2002) · Zbl 1013.03049
[12] Foreman, Matthew; Magidor, Menachem, Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on \(P_\kappa(\lambda)\), Acta Math., 186, 2, 271-300 (2001) · Zbl 1017.03022
[13] Gitik, Moti, On a question of Pereira, Arch. Math. Log., 47, 1, 53-64 (2008) · Zbl 1145.03024
[14] Jech, Thomas, On the cofinality of countable products of cardinal numbers, (A Tribute to Paul Erdős (1990)), 289-305 · Zbl 0711.03020
[15] Jech, Thomas, Set Theory (2013), Springer Science & Business Media · Zbl 0419.03028
[16] Kanamori, Akihiro, The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2008), Springer Science & Business Media · Zbl 0813.03034
[17] Koepke, Peter, Forcing a mutual stationarity property in cofinality \(\omega_1\), Proc. Am. Math. Soc., 135, 5, 1523-1533 (2007) · Zbl 1110.03040
[18] Koepke, Peter; Welch, Philip, On the strength of mutual stationarity, (Bagaria, Joan; Todorcevic, Stevo, Set Theory. Set Theory, Trends in Mathematics (2006), Birkhäuser: Birkhäuser Basel), 309-320 · Zbl 1111.03041
[19] Lambie-Hanson, Chris, Good and bad points in scales, Arch. Math. Log., 53, 7-8, 749-777 (2014) · Zbl 1339.03039
[20] Shelah, Saharon, Cardinal Arithmetic (1994), Clarendon Press: Clarendon Press Oxford · Zbl 0848.03025
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