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A Forcing Axiom Deciding the Generalized Souslin Hypothesis

  • Chris Lambie-Hanson (a1) and Assaf Rinot (a1)

We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\unicode[STIX]{x1D706}$ , if $\unicode[STIX]{x1D706}^{++}$ is not a Mahlo cardinal in Gödel’s constructible universe, then $2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$ entails the existence of a $\unicode[STIX]{x1D706}^{+}$ -complete $\unicode[STIX]{x1D706}^{++}$ -Souslin tree.

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This research was partially supported by the Israel Science Foundation (grant #1630/14).

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[1] Brodsky, A. M. and Rinot, A., Distributive Aronszajn trees. To appear in Fundamenta Mathematicae, 2019.
[2] Brodsky, A. M. and Rinot, A., A microscopic approach to Souslin-tree constructions. Part I . Ann. Pure Appl. Logic 168(2017), no. 11, 19492007.
[3] Brodsky, A. M. and Rinot, A., Reduced powers of Souslin trees . Forum Math. Sigma 5(2017), e2.
[4] Devlin, K. J., Aspects of constructibility . Lecture Notes in Mathematics, 354, Springer-Verlag, Berlin-New York, 1973.
[5] Devlin, K. J., Constructibility Perspectives in mathematical logic. Springer-Verlag, Berlin, 1984.
[6] Foreman, M., An 1-dense ideal on 2 . Israel J. Math. 108(1998), 253290.
[7] Foreman, M., Magidor, M., and Shelah, S., Martin’s maximum, saturated ideals and nonregular ultrafilters. II . Ann. of Math. (2) 127(1988), no. 3, 521545.
[8] Gitik, M. and Rinot, A., The failure of diamond on a reflecting stationary set . Trans. Amer. Math. Soc. 364(2012), no. 4, 17711795.
[9] Jech, T., Non-provability of Souslin’s hypothesis . Comment. Math. Univ. Carolinae 8(1967), 291305.
[10] Jensen, R. B., The fine structure of the constructible hierarchy . Ann. Math. Logic 4(1972), 229308. erratum, ibid. 4(1972), 443.
[11] Jensen, R. B., Souslin’s hypothesis is incompatible with V= L . Notices Amer. Math. Soc 15(1968).
[12] Kurepa, G., Ensembles ordonnés et ramifiés. Publications de l’Institut Mathématique Beograd, 1935.
[13] Lambie-Hanson, C., Aronszajn trees, square principles, and stationary reflection . MLQ Math. Log. Q. 63(2017), no. 3–4, 265281.
[14] Laver, R. and Shelah, S., The 2-Souslin hypothesis . m Trans. Amer. Math. Soc. 264(1981), no. 2, 411417.
[15] Mitchell, W., Aronszajn trees and the independence of the transfer property . Ann. Math. Logic 5(1972/73), 2146.
[16] Raghavan, D. and Todorcevic, S., Suslin trees, the bounding number, and partition relations. Israel J. Math., to appear.
[17] Rinot, A., Higher Souslin trees and the GCH, revisited . Adv. Math. 311(2017), 510531.
[18] Shelah, S., Diamonds . Proc. Amer. Math. Soc. 138(2010), no. 6, 21512161.
[19] Shelah, S., Laflamme, C., and Hart, B., Models with second order properties. V. A general principle . Ann. Pure Appl. Logic 64(1993), no. 2, 169194.
[20] Shelah, S. and Stanley, L., S-forcing. I. A “black-box” theorem for morasses, with applications to super-Souslin trees . Israel J. Math. 43(1982), no. 3, 185224.
[21] Shelah, S. and Stanley, L., S-forcing. IIa. Adding diamonds and more applications: coding sets, Arhangelskii’s problem and . Israel J. Math. 56(1986), 1–65.
[22] Shelah, S. and Stanley, L., Weakly compact cardinals and nonspecial Aronszajn trees . Proc. Amer. Math. Soc. 104(1988), no. 3, 887897.
[23] Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin’s problem . Ann. of Math. (2) 94(1971), 201245.
[24] Souslin, M. Y., Problème 3 . Fundamenta Math. 1(1920), no. 1, 223.
[25] Specker, E., Sur un problème de Sikorski . Colloquium Math. 2(1949), 912.
[26] Tennenbaum, S., Souslin’s problem . Proc. Nat. Acad. Sci. U.S.A. 59(1968), 6063.
[27] Todorcevic, S., Walks on ordinals and their characteristics . Progress in Mathematics, 263, Birkhäuser Verlag, Basel, 2007.
[28] Todorcevic, S. and Torres Perez, V., Conjectures of Rado and Chang and special Aronszajn trees . MLQ Math. Log. Q. 58(2012), no. 4–5, 342347.
[29] Velleman, D., Souslin trees constructed from morasses. In: Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., 31, American Mathematical Society, Providence, RI, 1984, pp. 219–241.
[30] Velleman, D., Morasses, diamond, and forcing . Ann. Math. Logic 23(1982), no. 2–3, 199281.
[31] Zwicker, W. S.,  combinatorics. I. Stationary coding sets rationalize the club filter. In: Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., 31, American Mathematic Society, Providence, RI, 1984, pp. 243–259.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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