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

TL;DR

This paper constructs models with large inaccessible cardinals where generalized cardinal invariants like tower and ultrafilter numbers are precisely controlled, extending classical results to higher cardinals with compactness properties.

Contribution

It introduces new forcing techniques combining Mitchell and Brooke-Taylor et al.'s methods to manipulate cardinal invariants at inaccessible cardinals with compactness at , including their values and related combinatorial properties.

Findings

Existence of models with specified invariant values at inaccessible cardinals.

Demonstration of consistency of certain invariants with compactness and stationary reflection.

Extension of classical invariants results to higher cardinals with compactness properties.

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 $\neg AP(\omega_2)$.