Abstract
Shelah and Stanley (Proc Am Math Soc 104(3):887–897, 1988) constructed a \(\kappa ^+\)-Aronszjan tree with an ascent path using \(\square _{\kappa }\). We show that \(\square _{\kappa ,2}\) does not imply the existence of Aronszajn trees with ascent paths. The proof goes through an intermediate combinatorial principle, which we investigate further.
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Chen, W., Neeman, I.: Square principles with tail-end agreement. Arch. Math. Logic 54, 439–452 (2015)
Cummings, J.: Iterated forcing and elementary embeddings. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 2, pp. 775–884. Springer, Berlin (2010)
Cummings, J., Foreman, M., Magidor, M.: Squares, scales and stationary reflection. J. Math. Logic 01(01), 35–98 (2001)
Devlin, K.J.: Reduced products of \(\aleph _{2}\)-trees. Fundam. Math. 118, 129–134 (1983)
Hamkins, J.D.: Small forcing makes any cardinal superdestructible. J. Symb. Logic 63(1), 51–58 (1998)
Magidor, M., Lambie-Hanson, C.: On the strengths and weaknesses of weak squares. In: Cummings, J., Schimmerling, E. (eds.) Appalachian Set Theory, 2006–2012. Cambridge University Press, Cambridge (2012)
Neeman, I.: Two applications of finite side conditions at \(\omega _{2}\). http://www.math.ucla.edu/~ineeman/
Shelah, S., Stanley, L.: Weakly compact cardinals and nonspecial Aronszajn trees. Proc. Am. Math. Soc. 104(3), 887–897 (1988)
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Shani, A. Fresh subsets of ultrapowers. Arch. Math. Logic 55, 835–845 (2016). https://doi.org/10.1007/s00153-016-0497-4
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DOI: https://doi.org/10.1007/s00153-016-0497-4