  Intersection models of generic extensions obtained from a commutative
projection systems of notions of forcing has recently regained interest,
especially in the study of descriptive set theory. Here, we show that it
provides a fruitful framework that opens the door to solving some open problems
concerning compactness principles of small cardinals. To exemplify, from
suitable assumptions, we construct intersection models satisfying ZFC and any
of the following:
  1. There is a weakly compact cardinal $\kappa$ carrying an indecomposable
ultrafilter, yet $\kappa$ is not measurable. This answers a question of Ketonen
from the late 1970's.
  2. For proper class many cardinals $\lambda$, the least $\lambda$-strongly
compact cardinal is singular. This answers a question of Bagaria and Magidor
who asked for merely two such cardinals.
  3. There is a strongly inaccessible cardinal whose $C$-sequence number is a
singular cardinal. This answers a question of Lambie-Hanson and the first
author.
