We describe a number of related questions at the interface of set theory and homology theory, centering on (1) the additivity of strong homology, and (2) the cohomology of the ordinals. In the first, the question is, at heart: To how general a category of topological spaces may classical homology theory be continuously extended? And in the tension between various potential senses of continuity lie a number of delicate set-theoretic questions. These questions led to the consideration of the Cech cohomology of the ordinals; the surprise was that this is a meaningful thing to consider at all. It very much is, describing or suggesting at once (i) distinctive combinatorial principles associated to the nth infinite cardinal, for each n, holding in ZFC, (ii) rich connections between cofinality and dimension, and (iii) higher-dimensional extensions of the method of minimal walks.