Strong unfoldability, shrewdness and combinatorial consequences

Abstract

We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen 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 D\v{z}amonja and Hamkins 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 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.