Inclusion modulo nonstationary

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. 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. 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. 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. 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. 5.

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

  6. 6.

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

  7. 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. 8.

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

  9. 9.

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

  10. 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. 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. 12.

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

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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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Correspondence to Assaf Rinot.

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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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Keywords

  • Universal order
  • Nonstationary ideal
  • Diamond sharp
  • Local club condensation
  • Higher Baire space

Mathematics Subject Classification

  • Primary 03E35
  • Secondary 03E45
  • 54H05