Strong unfoldability, shrewdness and combinatorial consequences

Lücke, Philipp

doi:10.1090/proc/15943

Strong unfoldability, shrewdness and combinatorial consequences
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Proc. Amer. Math. Soc. 150 (2022), 4005-4020 Request permission

Abstract:

We show that the notions of strongly unfoldable cardinals, introduced by Villaveces [J. Symbolic Logic 63 (1998), pp. 1116–1136] in his model-theoretic studies of models of set theory, and shrewd cardinals, introduced by Rathjen [Bull. Symbolic Logic 1 (1995), pp. 468–485] in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of ordinal anticipating Laver functions for strong unfoldability. With the help of these functions, we show that the principle $\Diamond _\kappa (\mathrm {Reg})$ holds at every strongly unfoldable cardinal $\kappa$ with the property that there exists a subset $z$ of $\kappa$ such that every subset of $\kappa$ is ordinal definable from $z$. While a result of Džamonja and Hamkins [Ann. Pure Appl. Logic 144 (2006), pp. 83–95] shows that $\Diamond _\kappa (\mathrm {Reg})$ can consistently fail at a strongly unfoldable cardinal $\kappa$, this implication can be used to prove that various canonical extensions of the axioms of $\mathrm {ZFC}$ are either compatible with the assumption that $\Diamond _\kappa (\mathrm {Reg})$ holds at every strongly unfoldable cardinal $\kappa$ or outright imply this statement. Finally, we will also use our methods to contribute to the study of strong chain conditions of partials orders and their productivity.

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