Non-additivity of strong homology

A counter-example is constructed that shows that neither higher inverse limits of pro-groups nor strong homology of topological spaces are additive. The previous counter-example by S. Mardešić and A. Prasolov depended on the Continuum Hypothesis. The approach developed in this paper is applied also to that example in order to calculate the cardinality of the corresponding strong homology groups. It appeared that the above cardinality depends essentially on a set-theoretic model: it is hypercontinuum under a weaker version of the Continuum Hypothesis, and zero under the Proper Forcing Axiom.