The 1930s and 1940s, the early years of quantum field theory, are generally viewed as an "age of infinities", when physicists troubled by the host of divergent quantities appearing in their calculations were looking unsuccessfully for a radical revision of their formalism, attempts which all became obsolete with the development of renormalization in the late 1940s. At the same time, physicists, however, continued to work within the framework of quantum electrodynamics, even though they believed it to be essentially inadequate. I focus on one of the infinite quantities, the infinite polarization of the vacuum, in an attempt at developing a more differentiated picture of this era. No dedicated secondary literature exists on this subject, so several general questions need to be answered: How was vacuum polarization viewed in relation to the other infinite quantities of quantum electrodynamics? What attempts were made to specifically remove the infinite vacuum polarization and (how and why) did they fail? How did the work on fixing this infinity within quantum electrodynamics interact with the parallel attempts at revolutionizing the theory?