Chang’s conjecture and semiproperness of nonreasonable posets



Let \(\mathbb {Q}\) denote the poset which adds a Cohen real then shoots a club through the complement of \(\big ( [\omega _2]^\omega \big )^V\) with countable conditions. We prove that the version of Strong Chang’s conjecture from Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) implies semiproperness of \(\mathbb {Q}\), and that semiproperness of \(\mathbb {Q}\)—in fact semiproperness of any poset which is sufficiently nonreasonable in the sense of Foreman and Magidor (Ann Pure Appl Log 76(1):47–97, 1995)—implies the version of strong Chang’s conjecture from Woodin (The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, 2nd ed., vol 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010) and Todorčević (Conjectures of Rado and Chang and cardinal arithmetic, Kluwer Acadamic, Dordrecht, 1993). In particular, semiproperness of \(\mathbb {Q}\) has large cardinal strength, which answers a question of Friedman and Krueger (Trans. Am. Math. Soc. 359(5):2407–2420, 2007). One corollary of our work is that the version of Strong Chang’s Conjecture from Todorčević and Torres-Pérez (MLQ Math Log Q 58(4–5):342–347, 2012) does not imply the existence of a precipitous ideal on \(\omega _1\).


Semiproper forcing Chang’s conjecture Reasonable forcing Forcing axioms Martin’s maximum Stationary reflection 

Mathematics Subject Classification

03E05 03E35 03E55 03E57 03E65 


  1. 1.
    Abraham, U., Shelah, S.: Forcing closed unbounded sets. J. Symb. Log. 48(3), 643–657 (1983). MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Doebler, P.: Rado’s conjecture implies that all stationary set preserving forcings are semiproper. J. Math. Log. 13(1), 1350001 (2013). MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Doebler, P., Schindler, R.: \(\Pi _2\) consequences of BMM plus \(\text{ NS }_{\omega _1}\) is precipitous and the semiproperness of stationary set preserving forcings. Math. Res. Lett. 16(5), 797–815 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Foreman, M.: Stationary sets, Chang’s conjecture and partition theory. Set theory, (Piscataway, NJ, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 58, Amer. Math. Soc., Providence, RI, pp. 73–94 (2002)Google Scholar
  5. 5.
    Foreman, M., Magidor, M.: Large cardinals and definable counterexamples to the continuum hypothesis. Ann. Pure Appl. Log. 76(1), 47–97 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Foreman, M., Magidor, M., Shelah, S.: Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. Math. 127(1), 1–47 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Foreman, M., Todorčević, S.: A new Löwenheim–Skolem theorem. Trans. Am. Math. Soc. 357(5), 1693–1715 (2005). (electronic)CrossRefMATHGoogle Scholar
  8. 8.
    Friedman, S.-D., Krueger, J.: Thin stationary sets and disjoint club sequences. Trans. Am. Math. Soc. 359(5), 2407–2420 (2007). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gitik, M.: Nonsplitting subset of \({\cal{P}}_{\kappa } (\kappa ^{+})\). J. Symb. Log. 50(4), 881–894 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jech, T., Magidor, M., Mitchell, W., Prikry, K.: Precipitous ideals. J. Symb. Log. 45(1), 1–8 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jensen, R., Steel, J.: \(K\) without the measurable. J. Symb. Log. 78(3), 708–734 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Krueger, J.: Internal approachability and reflection. J. Math. Log. 8(1), 23–39 (2008). MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Larson, P., Shelah, S.: Bounding by canonical functions, with CH. J. Math. Log. 3(2), 193–215 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sakai, H.: Semiproper ideals. Fund. Math. 186(3), 251–267 (2005). MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sharpe, I., Welch, P.D.: Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann. Pure Appl. Log. 162(11), 863–902 (2011). CrossRefMATHGoogle Scholar
  16. 16.
    Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  17. 17.
    Steel, J.R.: The Core Model Iterability Problem, vol. 8. Springer, Berlin (1996)MATHGoogle Scholar
  18. 18.
    Todorčević, S.: Conjectures of Rado and Chang and cardinal arithmetic. In: Finite and Infinite Combinatorics in Sets and Logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 411, Kluwer Acadamic Publisher, Dordrecht, 1993, pp. 385–398Google Scholar
  19. 19.
    Todorčević, S., Torres-Pérez, V.: Conjectures of Rado and Chang and special Aronszajn trees. MLQ Math. Log. Q. 58(4–5), 342–347 (2012).
  20. 20.
    Torres-Pérez, V., Wu, L.: Strong Chang’s conjecture and the tree property at \(\omega _2\). Topol. Appl. 196, 999–1004 (2015)CrossRefMATHGoogle Scholar
  21. 21.
    Usuba, T.: Bounded dagger principles. MLQ Math. Log. Q. 60(4–5), 266–272 (2014). MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Usuba, T.: Reflection principles for \(\omega _2\) and the semi-stationary reflection principle. J. Math. Soc. Jpn. 68(3), 1081–1098 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Veličković, B.: Forcing axioms and stationary sets. Adv. Math. 94(2), 256–284 (1992). MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Woodin, W.: Hugh: The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, vol. 1, 2nd edn. Walter de Gruyter GmbH & Co. KG, Berlin (2010). CrossRefGoogle Scholar
  25. 25.
    Zeman, M.: Inner Models and Large Cardinals, de Gruyter Series in Logic and its Applications, vol. 5. Walter de Gruyter & Co., Berlin (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsVirginia Commonwealth UniversityRichmondUSA

Personalised recommendations