Shelah developed his theory of possible cofinalities - known as pcf theory - in order to analyze the combinatorial properties of singular cardinals without the use of additional set theoretic hypotheses. The culmination of this analysis is his proof that \({\aleph_\omega}^{\aleph_0} < \max (2^{\aleph_0}, \aleph_{\omega_4})\). This is part of a series of talks which will develop the basic aspects of Shelah's pcf theory with the goal of proving this celebrated inequality.