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Inclusion modulo nonstationary

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Abstract

A classical theorem of Hechler asserts that the structure \(\left( \omega ^\omega ,\le ^*\right) \) is universal in the sense that for any \(\sigma \)-directed poset \({\mathbb {P}}\) with no maximal element, there is a ccc forcing extension in which \(\left( \omega ^\omega ,\le ^*\right) \) contains a cofinal order-isomorphic copy of \({\mathbb {P}}\). In this paper, we prove the following consistency result concerning the universality of the higher analogue \(\left( \kappa ^\kappa ,\le ^S\right) \): assuming \(\textsf {GCH }\), for every regular uncountable cardinal \(\kappa \), there is a cofinality-preserving \(\textsf {GCH }\)-preserving forcing extension in which for every analytic quasi-order \({\mathbb {Q}}\) over \(\kappa ^\kappa \) and every stationary subset S of \(\kappa \), there is a Lipschitz map reducing \({\mathbb {Q}}\) to \((\kappa ^\kappa ,\le ^S)\).

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Notes

  1. A comparison of the generalization considered here with the one obtained by replacing the ideal of finite sets with the ideal of bounded sets may be found in [4, §8].

  2. Here, \(L[E]||\alpha \) stands for \(\langle J_{\alpha }^{E}, \in , E\mathbin \upharpoonright \omega \alpha , E_{\omega \alpha } \rangle \), following the notation from [27]. For the definition of weakly iterable, see [27, p. 311].

  3. For all the small \(\alpha \in S'{\setminus } S\) such that \(M_{\alpha ^+}\ne H_{\alpha ^+}\), simply let \(N'_\alpha :=N_{\min (S)}\).

  4. Note that \(\beta \) is not needed to define \(\textsf {LCC }(\alpha ,\beta )\) in the structure \(\langle M_{\beta },{\in }, \vec {M} \mathbin \upharpoonright \beta \rangle \). Indeed, by \(\textsf {LCC }(\alpha ,\beta )\) we mean \(\psi _{1}(\alpha )\) as in Remark 2.12.

  5. In particular, \(\langle M_{\beta },{\in }\rangle \models \alpha \text { is uncountable}\).

  6. In particular, \(\delta >\kappa \).

  7. Recalling Definition 2.23, this means that \(\langle M_{\delta _{n+1}},{\in }, \vec {M}\mathbin \upharpoonright \delta _{n+1}\rangle \models ``\vec {{\mathfrak {B}}_{n}} \text { is the } {<_\varTheta }\text {-least }\vec {{\mathfrak {B}}}\text { such that }(\psi _{a} \wedge \psi _{b} \wedge \psi _{c} \wedge \psi _{d} \wedge \psi _{e})(\vec {{\mathfrak {B}}},\vec {{\mathcal {F}}}_{A_0,\kappa },\delta _n,\vec {M}\mathbin \upharpoonright (\delta _n+1))\)”.

  8. Notice that the argument of this claim also showed that D is stationary.

  9. For the definition of acceptable J-structure, see [27, p. 4].

  10. For any \(\alpha \) such that \(\eta _\alpha \) is not a function from \(\alpha \) to \(\alpha \), simply replace \(\eta _\alpha \) by the constant function from \(\alpha \) to \(\{0\}\).

  11. \(N_{\alpha }\) is transitive and rud-closed (in fact, p.r.-closed), so that \(N_{\alpha } \models \textsf {GJ }\) (see [18, §Other remarks on GJ]). Now, by [18, §The cure in \(\textsf {GJ }\), proposition 10.31], \(\mathbf {Sat}\) is \(\varDelta _{1}^{\textsf {GJ }}\).

  12. Note that in this case, \({\mathbb {P}}\) is moreover \(({<}\kappa )\)-directed-closed and has the \(\kappa ^{+}\)-cc.

References

  1. Abraham, U., Magidor, M.: Cardinal arithmetic. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, 2, 3, pp. 1149–1227. Springer, Dordrecht (2010)

    Chapter  Google Scholar 

  2. Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, 2, 3, pp. 395–489. Springer, Dordrecht (2010)

    Chapter  Google Scholar 

  3. Brodsky, A.M., Rinot, A.: Reduced powers of Souslin trees. Forum Math. Sigma 5, e2–82 (2017)

    Article  MathSciNet  Google Scholar 

  4. Cummings, J., Shelah, S.: Cardinal invariants above the continuum. Ann. Pure Appl. Log. 75(3), 251–268 (1995)

    Article  MathSciNet  Google Scholar 

  5. Devlin, K.J.: The combinatorial principle \(\diamondsuit ^{\sharp }\). J. Symb. Log. 47(4), 888–899 (1982)

    Article  MathSciNet  Google Scholar 

  6. Drake, F.R.: Set theory—an introduction to large cardinals. 76, xii+351 (1974)

  7. Fernandes, G.: On local club condensation. In preparation (2020)

  8. Fernandes, G., Miguel, M., Assaf, R.: Fake reflection (2020). http://assafrinot.com/paper/40, Submitted February, 2020

  9. Friedman, S.-D., Holy, P.: Condensation and large cardinals. Fundam. Math. 215(2), 133–166 (2011)

    Article  MathSciNet  Google Scholar 

  10. Friedman, S.-D., Hyttinen, T., Kulikov, V.: Generalized Descriptive Set Theory and Classification Theory, vol. 230. American Mathematical Society, Providence (2014)

    MATH  Google Scholar 

  11. Galvin, F., Hajnal, A.: Inequalities for cardinal powers. Ann. Math. 2(101), 491–498 (1975)

    Article  MathSciNet  Google Scholar 

  12. Gregory, J.: Higher Souslin trees and the generalized continuum hypothesis. J. Symb. Log. 41(3), 663–671 (1976)

    Article  MathSciNet  Google Scholar 

  13. Hechler, S.H.: On the existence of certain cofinal subsets of \(^{\omega }\omega \). In: Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), pp. 155–173. Amer. Math. Soc., Providence, R.I. (1974)

  14. Holy, P., Welch, P., Liuzhen, W.: Local club condensation and L-likeness. J. Symb. Log. 80(4), 1361–1378 (2015)

    Article  MathSciNet  Google Scholar 

  15. Hyttinen, T., Kulikov, V., Moreno, M.: On \(\Sigma _1^1\)-completeness of quasi-orders on \(\kappa ^\kappa \). Fundam. Math. (2020). https://doi.org/10.4064/fm679-1-2020

  16. Hyttinen, T., Kulikov, V.: Bore\(l^*\) sets in the generalized baire space and infinitary languages. In: van Ditmarsch, H., Sandu, G. (eds.) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics, pp. 395–412. Springer, Cham (2018)

    Chapter  Google Scholar 

  17. Lücke, P.: \(\Sigma ^1_1\)-definability at uncountable regular cardinals. J. Symb. Log. 77(3), 1011–1046 (2012)

    Article  Google Scholar 

  18. Mathias, A.R.D.: Weak systems of Gandy, Jensen and Devlin. In: Bagaria, J., Todorcevic, S. (eds.) Set Theory, Trends Math., pp. 149–224. Birkhäuser, Basel (2006)

    Chapter  Google Scholar 

  19. Mekler, A., Väänänen, J.: Trees and \(\Pi ^1_1\)-subsets of \(^{\omega _1}\omega _1\). J. Symb. Log. 58(3), 1052–1070 (1993)

    Article  Google Scholar 

  20. Schimmerling, E., Zeman, M.: Square in core models. Bull. Symb. Log. 7(3), 305–314 (2001)

    Article  MathSciNet  Google Scholar 

  21. Schimmerling, E., Zeman, M.: Characterization of \(\square _\kappa \) in core models. J. Math. Log. 4(1), 1–72 (2004)

    Article  MathSciNet  Google Scholar 

  22. Schindler, R., Zeman, M.: Fine structure. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, 2, 3, pp. 605–656. Springer, Dordrecht (2010)

    Chapter  Google Scholar 

  23. Shelah, S.: Models with second order properties. IV. A general method and eliminating diamonds. Ann. Pure Appl. Log. 25, 183–212 (1983)

    Article  MathSciNet  Google Scholar 

  24. Shelah, S.: The Erdos-Rado arrow for singular cardinals. Can. Math. Bull. 52(1), 127–131 (2009)

    Article  MathSciNet  Google Scholar 

  25. Todorčević, S., Väänänen, J.: Trees and Ehrenfeucht-Fraïssé games. Ann. Pure Appl. Log. 100(1–3), 69–97 (1999)

    Article  Google Scholar 

  26. Todorčević, S.: Partition Problems in Topology. Contemporary Mathematics, vol. 84. American Mathematical Society, Providence, RI (1989)

    Book  Google Scholar 

  27. 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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Acknowledgements

This research was partially supported by the European Research Council (Grant Agreement ERC-2018-StG 802756). The third author was also partially supported by the Israel Science Foundation (Grant Agreement 2066/18). The main results of this paper were presented by the second author at the 4th Arctic Set Theory workshop, Kilpisjärvi, January 2019, by the third author at the 50 Years of Set Theory in Toronto conference, Toronto, May 2019, and by the first author at the Berkeley conference on inner model theory, Berkeley, July 2019. We thank the organizers for the invitations. The authors express their gratitude to the referee for a careful, thoughtful and valuable report.

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  1. Department of Mathematics, Bar-Ilan University, 5290002, Ramat-Gan, Israel

    Gabriel Fernandes, Miguel Moreno & Assaf Rinot

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  1. Gabriel Fernandes
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  2. Miguel Moreno
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  3. Assaf Rinot
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Correspondence to Assaf Rinot.

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Communicated by Adrian Constantin.

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Fernandes, G., Moreno, M. & Rinot, A. Inclusion modulo nonstationary. Monatsh Math 192, 827–851 (2020). https://doi.org/10.1007/s00605-020-01431-6

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