In “Strong Homology is not Additive” (1988), authors Sibe Marde\v{s}i\'c and Andrei Prasolov computed strong homology groups $\bar{H}_p$ for $X^{(k)}=\coprod_{i\in\omega} Y^{(k)}$, where $Y^{(k)}$ is the $k$-dimensional Hawaiian earring. Most striking among their findings was that $\bar{H}_p(X^{(p+1)})=\text{lim}^1\mathbb{A}$ (where $\mathbb{A}$ is the pro-group of $A_f=\bigoplus_{i\in\omega}\mathbb{Z}^{f(i)}\;(f\in\ww)$) - and that the value of $\text{lim}^1\mathbb{A}$ is a combinatorial question independent of ZFC.

At stake in their computation was the additivity of a homology theory for a broad class of spaces; theirs, moreover, was the second derived functor of note (cf. Shelah's solution to the Whitehead problem) found to depend essentially on set theoretic assumptions.

This is a talk, in other words, a little bit about algebra, a little bit about topology, and a little bit about set theory; the intrigue is that they should have any common business at all.