Syllabus
Syllabus is provisional and may change based on the background and interests of the students.
1. Applications of transfinite induction and recursion, including constructions of geometrically interesting subsets of Euclidean space
2. Applications of the Axiom of Choice and its relatives: de Bruijn-Erdős theorem, Nielsen-Schreier theorem, existence of non-measurable sets, existence of algebraic closures
3. Applications of ultrafilters and ultraproducts: Arrow's Impossibility Theorem, Ax-Grothendieck theorem
4. Further applications to abelian group theory: Constructions of almost free nonfree groups, slender groups
5. Infinite games
Annotation
Set theory and mathematical logic are often studied through the lens of their role in providing a rigorous underpinning for mathematics and metamathematics. In this course, we will consider a slightly less well-known role played by set theory and logic in modern mathematics, surveying a number of instances in which tools and techniques from set theory and logic are used to actually prove new theorems in various fields of pure mathematics, including algebra, analysis, combinatorics, geometry, and voting theory.