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Strong tree properties, Kurepa trees, and guessing models

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Abstract

We investigate the generalized tree properties and guessing model properties introduced by Weiß and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other prominent combinatorial principles. We introduce a weakening of Viale and Weiß’s Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almost Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular \(\theta \ge \omega _2\), there are stationarily many \(\omega _2\)-guessing models \(M \in {\mathscr {P}}_{\omega _2} H(\theta )\) that are not \(\omega _1\)-guessing models.

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Notes

  1. For concreteness, we formulate the result of \(\omega _2\), but it can be easily generalized to an arbitrary double successor of a regular cardinal.

  2. For concreteness, we focus here on \(\omega _2\), but analogous arguments work at other double successors of regular cardinals.

  3. The theorem is not stated in this form in [27], but it follows immediately from the two cited results.

  4. This is a slight abuse of notation for the sake of readability. Formally, \(\dot{D}\) is an \(\mathbb {M}\)-name for a strongly \(\mu ^+\)-\(\dot{\mathcal Y}\)-slender \((\kappa , \lambda )\)-list and, for every nice \(\mathbb {M}\)-name \(\dot{x}\) for an element of \(({\mathscr {P}}_\kappa \lambda )^{V^{\mathbb {M}}}\), \(\dot{d}_{\dot{x}}\) is an \(\mathbb {M}\)-name for the \(\dot{x}\)-th entry in \(\dot{D}\). Similar notation will be used throughout the remainder of the paper.

  5. More precisely, the function \(u \mapsto b \cap u\) defined on \(\Lambda \) is a branch through T, but it is clear that this function and b are definable from one another in all models of interest.

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Acknowledgements

Both authors were supported by the institutional support RVO:67985840. The first author was supported by GAČR grant Compactness in set theory and its applications in algebra and graph theory (23-04683S). The second author was supported by FWF/GAČR grant Compactness principles and combinatorics (19-29633L). We would like to thank Rahman Mohammadpour for discussions about an early version of this work and for pointing out the connections between our work and the question of Viale regarding \(\omega _2\)-guessing models that are not \(\omega _1\)-guessing models.

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Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Chris Lambie-Hanson & Šárka Stejskalová

  2. Department of Logic, Charles University, Celetná 20, 116 42, Prague 1, Czech Republic

    Šárka Stejskalová

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  1. Chris Lambie-Hanson
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Correspondence to Šárka Stejskalová.

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Communicated by S.-D. Friedman.

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Lambie-Hanson, C., Stejskalová, Š. Strong tree properties, Kurepa trees, and guessing models. Monatsh Math 203, 111–148 (2024). https://doi.org/10.1007/s00605-023-01922-2

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