Set theory has proven useful in the study of derived limits. These functors are widely studied for their applications in algebraic topology, and their behavior is to some extent independent from ZFC. As already shown by Bergfalk and Lambie-Hanson in the case of ordinals, the derived limits associated with some set-theoretic objects tend not to vanish in \(\mathbb{L}\). This corresponds to some form of incompactness. Here I present a similar result for \({}^\kappa \omega\) that uses diamonds and special Aronszajn trees.
This is a work in progress with Jeffrey Bergfalk.