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

Abstract

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 $SR(\kappa^{++})$ hold, and (ii) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ = \mathfrak{t}(\kappa) < \mathfrak{u}(\kappa)< 2^\kappa$ and $SR(\kappa^{++}), TP(\kappa^{++})$ and $\neg wKH(\kappa^+)$ hold. The cardinals $\mathfrak{u}(\kappa)$ and $2^\kappa$ can have any reasonable values in these models. We obtain these results by combining the forcing construction from Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results for compactness principles. Apart from $\mathfrak{u}(\kappa)$ and $\mathfrak{t}(\kappa)$ we also compute the values of $\mathfrak{b}(\kappa)$, $\mathfrak{d}(\kappa)$, $\mathfrak{s}(\kappa)$, $\mathfrak{r}(\kappa)$, $\mathfrak{a}(\kappa)$, $\mathrm{cov}(M_\kappa)$, $\mathrm{add}(M_\kappa)$, $\mathrm{non}(M_\kappa)$, $\mathrm{cof}(M_\kappa)$ which will all be 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 $\kappa^+$. Finally, we observe that (i) and (ii) hold also for the traditional invariants on $\kappa = \omega$, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property $DSS(\omega_2)$, which implies the negation of the approachability property $\neg AP(\omega_2)$.