• cardinal invariants
• logic
• proper forcing axiom
• set theory
• strong homology

The main purpose of this thesis is to introduce Strong Homology and show how a strong homology group associated to a certain space, the coproduct of countably many k-dimensional hawaiian earrings, depends on some set-theoretic assumptions that go beyond the axioms of ZFC. More precisely, the group corresponds to the value of the first derived functor of the inverse limit on a certain pro-group; its being trivial is shown in turn to be equivalent to a certain condition on families of functions which take values in the naturals, and therefore to be strongly related to the study of cardinal invariants of the Continuum. Assuming the Continuum Hypothesis, one has straightforwardly that the aforementioned coproduct gives a counterexample to the additivity of Strong Homology, while the same proof cannot work under the assumption of the Proper Forcing Axiom. The last chapter is dedicated to some recent generalizations such as the characterization of higher derived limits and some consistency results on their triviality.