Generalized cardinal invariants for an inaccessible $κ$ with compactness at $κ^{++}$

Radek Honzik, Sarka Stejskalova

Published 2023-08-25Version 1

We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal $\kappa$ and compactness principles at $\kappa^{++}$. We show that if the existence of a supercompact cardinal $\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent, then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and $\mathfrak{u}(\kappa)$ are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ < \mathfrak{t}(\kappa)= \mathfrak{u}(\kappa)< 2^\kappa$ and stationary reflection and the disjoint stationary sequence property at $\kappa^{++}$ hold. (ii) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ = \mathfrak{t}(\kappa) < \mathfrak{u}(\kappa)< 2^\kappa$ and the principles in (i) hold plus the tree property at $\kappa^{++}$ and the negation of the weak Kurepa Hypothesis at $\kappa^+$. The cardinals $\mathfrak{u}(\kappa)$ and $2^\kappa$ can have any reasonable values in these models. We compute several other values of cardinal invariants, such $\mathfrak{a}(\kappa)$ and $\mathfrak{b}(\kappa)$, and the invariants of the meager ideal (they are all equal to $\mathfrak{u}(\kappa)$). In (ii), we compute $\mathfrak{p}(\kappa) = \mathfrak{t}(\kappa) = \kappa^+$ by observing that the $\kappa^+$-distributive quotient of the Mitchell forcing adds a tower of size $\kappa^+$. As a corollary of the construction, we obtain that (i) and (ii) are also true for $\kappa = \omega$ (starting with a weakly compact cardinal in the ground model).