Cohomological localizations and set-theoretical reflection

Homological localizations of spaces and spectra have been a fundamental tool in algebraic topology since the decade of 1970. However, it is still unknown whether the existence of cohomological localizations can be proved in ZFC or not. Although this is apparently a homotopy theoretical problem, it turns out to be closely related with the phenomenon of set theoretical reflection. We prove that the existence of homological localizations follows from the Löwenheim-Skolem theorem combined with Adams' original argument, and explain why adapting the same proof for cohomology theories requires to assume a stronger form of the reflection principle, hence the existence of uncountable regular limit cardinals.