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Chang’s conjecture and semiproperness of nonreasonable posets

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Abstract

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 \(\omega _1\).

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Notes

  1. Shelah [16] Chapter XII proves that semiproperness of Namba forcing implies \(\text {SCC}^{\text {cof}}\); and a minor variation in the proof of Sect. 3 of Doebler [2] proves that \(\text {SCC}^{\text {cof}}\) implies semiproperness of Namba forcing.

  2. That is, for every stationary \(S \subseteq \omega _1\) there are stationarily many \(z \in [\omega _2]^\omega - V\) such that \(z \cap \omega _1 \in S\). The Gitik and Velickovic arguments actually prove something much more general: if W is an outer model of V and W has some real that is not in V, then for every W-regular \(\kappa \ge \omega _2^W\), W believes that \([\kappa ]^\omega - V\) is projective stationary.

  3. In fact one can arrange that the quotient is forcing equivalent to a \(\sigma \)-closed poset; and moreover the ideal can consistently be the nonstationary ideal on \(\omega _2\) restricted to ordinals of uncountable cofinality.

  4. CC is often expressed by \((\omega _2, \omega _1) \twoheadrightarrow (\omega _1, \omega )\).

  5. Doebler and Schindler [3] proved that their version implies \(\dagger \), which by Usuba [21] implies presaturation of \(\text {NS}_{\omega _1}\). Thus by Steel [17] and Jensen and Steel [11], the Doebler and Schindler version of \(\text {CC}^*\) has consistency strength at least a Woodin cardinal; whereas all the versions of Chang’s Conjecture considered in this paper can be forced from a measurable cardinal.

  6. Lemma 9.103 of [24] is the version for SCC; Lemma 13 above is the version for \(\text {SCC}^{\text {cof}}_{\text {gap}}\).

  7. Lemma 15 is the reason that our Definition 10 of SCC is equivalent to part 2 of Definition 9.101 of [24].

  8. In fact it is stationary and costationary, as shown in Usuba [22].

  9. The “cofinal” requirement of \(\text {SCC}^{\text {cof}}_{\text {gap}}\) isn’t used here, just the “gap” requirement. That is, the proof actually shows that if there is a nonreflecting ladder system for \(S^2_0\), then there are stationarily many models M for which there is no \(\beta \in \big ( \text {sup}(M \cap \omega _2),\omega _2 \big )\) such that \(\text {Sk}^{\mathfrak {A}}(M \cup \{ \beta \}) \cap \beta = M \cap \omega _2\).

  10. He shows that the model satisfies “Semistationary set reflection”, which is equivalent to \(\dagger \).

  11. Namely \(\mathcal {X}_n = \text {Active}(n)\), though the function \(\text {Active}\) is not assumed to be available to H.

  12. A set \(T \subseteq [\omega _2]^\omega \) is called local club iff \(T \cap [\beta ]^\omega \) contains a club for every \(\beta < \omega _2\).

  13. T is thin if for every \(\beta < \omega _2\): \(|\{ a \cap \beta \ | \ a \in T \}|\le \omega _1\).

  14. And SCC, though we won’t use that.

References

  1. Abraham, U., Shelah, S.: Forcing closed unbounded sets. J. Symb. Log. 48(3), 643–657 (1983). https://doi.org/10.2307/2273456

    Article  MathSciNet  MATH  Google Scholar 

  2. Doebler, P.: Rado’s conjecture implies that all stationary set preserving forcings are semiproper. J. Math. Log. 13(1), 1350001 (2013). https://doi.org/10.1142/S0219061313500013

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  5. Foreman, M., Magidor, M.: Large cardinals and definable counterexamples to the continuum hypothesis. Ann. Pure Appl. Log. 76(1), 47–97 (1995)

    Article  MathSciNet  Google Scholar 

  6. Foreman, M., Magidor, M., Shelah, S.: Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. Math. 127(1), 1–47 (1988)

    Article  MathSciNet  Google Scholar 

  7. Foreman, M., Todorčević, S.: A new Löwenheim–Skolem theorem. Trans. Am. Math. Soc. 357(5), 1693–1715 (2005). https://doi.org/10.1090/S0002-9947-04-03445-2. (electronic)

    Article  MATH  Google Scholar 

  8. Friedman, S.-D., Krueger, J.: Thin stationary sets and disjoint club sequences. Trans. Am. Math. Soc. 359(5), 2407–2420 (2007). https://doi.org/10.1090/S0002-9947-06-04163-8. (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gitik, M.: Nonsplitting subset of \({\cal{P}}_{\kappa } (\kappa ^{+})\). J. Symb. Log. 50(4), 881–894 (1985)

    Article  MathSciNet  Google Scholar 

  10. Jech, T., Magidor, M., Mitchell, W., Prikry, K.: Precipitous ideals. J. Symb. Log. 45(1), 1–8 (1980)

    Article  MathSciNet  Google Scholar 

  11. Jensen, R., Steel, J.: \(K\) without the measurable. J. Symb. Log. 78(3), 708–734 (2013)

    Article  MathSciNet  Google Scholar 

  12. Krueger, J.: Internal approachability and reflection. J. Math. Log. 8(1), 23–39 (2008). https://doi.org/10.1142/S0219061308000701

    Article  MathSciNet  MATH  Google Scholar 

  13. Larson, P., Shelah, S.: Bounding by canonical functions, with CH. J. Math. Log. 3(2), 193–215 (2003)

    Article  MathSciNet  Google Scholar 

  14. Sakai, H.: Semiproper ideals. Fund. Math. 186(3), 251–267 (2005). https://doi.org/10.4064/fm186-3-4

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.apal.2011.04.002

    Article  MATH  Google Scholar 

  16. Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)

    Book  Google Scholar 

  17. Steel, J.R.: The Core Model Iterability Problem, vol. 8. Springer, Berlin (1996)

    MATH  Google Scholar 

  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. 385–398

  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). https://doi.org/10.1002/malq.201110037

    Article  MathSciNet  Google Scholar 

  20. Torres-Pérez, V., Wu, L.: Strong Chang’s conjecture and the tree property at \(\omega _2\). Topol. Appl. 196, 999–1004 (2015)

    Article  Google Scholar 

  21. Usuba, T.: Bounded dagger principles. MLQ Math. Log. Q. 60(4–5), 266–272 (2014). https://doi.org/10.1002/malq.201300019

    Article  MathSciNet  MATH  Google Scholar 

  22. Usuba, T.: Reflection principles for \(\omega _2\) and the semi-stationary reflection principle. J. Math. Soc. Jpn. 68(3), 1081–1098 (2016)

    Article  MathSciNet  Google Scholar 

  23. Veličković, B.: Forcing axioms and stationary sets. Adv. Math. 94(2), 256–284 (1992). https://doi.org/10.1016/0001-8708(92)90038-M

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1515/9783110213171

    Book  Google Scholar 

  25. Zeman, M.: Inner Models and Large Cardinals, de Gruyter Series in Logic and its Applications, vol. 5. Walter de Gruyter & Co., Berlin (2002)

    Book  Google Scholar 

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  1. Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA, 23284, USA

    Sean D. Cox

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Correspondence to Sean D. Cox.

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Communicated by A. Constantin.

The author gratefully acknowledges support from the VCU Presidential Research Quest Fund, and Simons Foundation Grant 318467.

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Cox, S.D. Chang’s conjecture and semiproperness of nonreasonable posets. Monatsh Math 187, 617–633 (2018). https://doi.org/10.1007/s00605-018-1182-y

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