Abstract: A number of unsolved problems in algebraic topology, such as the existence of cohomological localizations, involve a subtle distinction between homology and cohomology. It was proved in 2005 that the solutions to some of those problems require the use of large-cardinal axioms from set theory. More recent results show that the clue resides in the Levy complexity of the definition of classes of acyclic spaces. This will be a survey talk avoiding the technical machinery behind these claims.
