Topics in Set Theory

Axiomatics. The formal axiomatic system of ordinary set theory (ZFC). Models of set theory. Absoluteness. Simple independence results. Transfinite recursion. Ranks. Reflection principles. Constructibility. [4] Infinitary combinatorics. Cofinality. Stationary sets. Fodor’s lemma. Solovay’s theorem. Cardinal exponentiation. Beth and Gimel functions. Generalized Continuum Hypothesis. Singular Cardinals Hypothesis. Prediction principles (diamonds, squares, black boxes). Partial orders. Aronszajn and Suslin trees. Martin’s Axiom. Suslin’s Hypothesis. [6] Forcing. Generic extensions. The forcing theorems. Examples. Adding reals; collapsing cardinals. Introduction to iterated forcing. Internal forcing axioms. Proper forcing. [4] Large cardinals. Introduction to large cardinals. Ultrapowers. Scott’s theorem. [2] Partition relations and possible cofinality theory. Partition relations. Model-theoretic methods. Ramsey’s theorem; Erdős-Rado theorem. Kunen’s theorem. Walks on ordinals. Todorcevic’s theorem. Introduction to pcf theory. [4] Applications. Selection from algebra, analysis, geometry, and topology. [4]