First order definability and the Zermelo-Fraenkel axioms; cardinal arithmetic, ordered sets, well-ordered sets (axiom of choice), transfinite induction, the filter of closed unbounded sets (Fodor, Ulm and Solovay~s theorems), Delta systems, basic results in partition calculus (e.g., Ramsey~s Theorem and the Erdos-Rado Theorem); small to medium large cardinals; applications to general topology (e.g., Alexandroff~s conjecture), and the basic ideas of descriptive set theory. The independence of Suslin conjecture from the usual axioms. Godel~s axiom of constructibility. Time permitting, the Galvin-Hajnal-Shelah inequality will be proved.