Abstract
Schimmerling asked whether \(\square ^{\ast }_{\lambda }+\mathsf {GCH}\) entails the existence of a λ+-Souslin tree, for a singular cardinal λ. We provide an affirmative answer under the additional assumption that there exists a non-reflecting stationary subset of \({E}^{{\lambda }^{+}}_{{\neq } \text {cf}(\lambda )}\). As a bonus, the outcome λ+-Souslin tree is moreover free.
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Abraham, U., Shelah, S.: A \({\Delta }^{2}_{2}\) well-order of the reals and incompactness of L(Q MM). Ann. Pure Appl. Logic 59(1), 1–32 (1993)
Brodsky, A.M., Rinot, A.: A microscopic approach to Souslin-tree constructions. Part I. Ann. Pure Appl. Logic 168(11), 1949–2007 (2017)
Brodsky, A.M., Rinot, A.: Reduced powers of Souslin trees. Forum Math. Sigma 5(e2), 1–82 (2017)
Brodsky, A.M., Rinot, A.: A microscopic approach to Souslin-tree constructions. Part II. in preparation (2019)
Brodsky, A.M., Rinot, A.: Distributive Aronszajn trees. Fund. Math., to appear, https://doi.org/10.4064/fm542-4-2018. http://assafrinot.com/paper/29 (2019)
Brodsky, A.M., Rinot, A.: More notions of forcing add a Souslin tree. Notre Dame J. Form. Log., to appear. http://assafrinot.com/paper/26 (2019)
David, R.: Some results on higher Suslin trees. J. Symbolic Logic 55(2), 526–536 (1990)
Devlin, K.J.: A note on the combinatorial principles \(\lozenge \)(E). Proc. Amer. Math. Soc. 72(1), 163–165 (1978)
Fuchs, G., Rinot, A.: Weak square and stationary reflection. Acta Math. Hungar. 155(2), 393–405 (2018)
Gregory, J.: Higher Souslin trees and the generalized continuum hypothesis. J. Symbolic Logic 41(3), 663–671 (1976)
Jensen, R.B.: Souslin’s hypothesis is incompatible with V=L. Notices Amer. Math. Soc., 15(6), 935 (1968)
Kunen, K.: Set theory, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, New York (1980). An introduction to independence proofs
Kojman, M., Shelah, S.: μ-Complete Souslin trees on μ +. Arch. Math. Logic 32(3), 195–201 (1993)
Lambie-Hanson, C., Rinot, A.: Reflection on the coloring and chromatic numbers. Combinatorica, to appear, https://doi.org/10.1007/s00493-017-3741-6. http://assafrinot.com/paper/28 (2019)
Lücke, P.: Ascending paths and forcings that specialize higher Aronszajn trees. Fund. Math. 239(1), 51–84 (2017)
Magidor, M., Lambie-Hanson, C.: On the strengths and weaknesses of weak squares. Appalachian Set Theory: 2006–2012 406, 301–330 (2012)
Rinot, A.: A relative of the approachability ideal, diamond and non-saturation. J. Symbolic Logic 75(3), 1035–1065 (2010)
Rinot, A.: Putting a diamond inside the square. Bull. Lond. Math. Soc. 47(3), 436–442 (2015)
Rinot, A.: Higher Souslin trees and the GCH, revisited. Adv. Math. 311(C), 510–531 (2017)
Rinot, A.: Souslin trees at successors of regular cardinals. Mathematical Logic Quarterly, to appear, 2019. https://doi.org/10.1002/malq.201800065. http://assafrinot.com/paper/37
Schimmerling, E.: A question about Suslin trees and the weak square hierarchy. Notre Dame J. Formal Logic 46(3), 373–374 (2005). (electronic)
Shelah, S.: Diamonds. Proc. Am. Math. Soc. 138, 2151–2161 (2010)
Zeman, M.: Diamond, GCH and weak square. Proc. Amer. Math. Soc. 138(5), 1853–1859 (2010)
Acknowledgments
The first author was supported by the Center for Absorption in Science, Ministry of Aliyah and Integration, State of Israel. The second author was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18). We thank the anonymous referee for a careful reading of this paper and for providing valuable feedback.
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Brodsky, A.M., Rinot, A. A Remark on Schimmerling’s Question. Order 36, 525–561 (2019). https://doi.org/10.1007/s11083-019-09482-7
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DOI: https://doi.org/10.1007/s11083-019-09482-7