The course has two main parts. In the first part, we will cover descriptive set theory, which is the study of definable sets in Polish spaces. One goal is to see to what extend certain regularity properties (measurability, perfect set property, etc) hold for subsets of such spaces. We will learn that as long as we restrict ourselves to Borel sets, all common regularity properties hold. However, as we move past Borel sets into the projective hierarchy, we find that these properties cannot be established anymore.
This will lead us to the second part, in which we will take a closer look at why this is the case. After developing set theory axiomatically (via the well-known Zermelo-Fraenkel axioms, ZF), we will encounter a set-theoretic universe (Gödel's L) in which regularity properties fail at very low levels of the projective hierarchy. Finally, we will learn about the method of forcing, which allows us to construct new set theoretic universes from given ones, and in which we can can control (to a rather surprising extent) which set-theoretic properties hold in the new universe. As a crowning achievement of this method, one can show that the Continuum Hypothesis is independent of ZF, and also construct set-theoretic universes in which the common regularity properties hold for all sets.
- Perfect subsets of the real line
- Polish spaces
- Urysohn space
- Trees
- Borel sets
- Borel sets as clopen sets
- Measure and category
- Axiom of Choice
- Structure of the Borel hierarchy
- Continuous images of Borel sets
- Analytic sets
- Regularity properties of analytic sets
- Co-analytic sets and the Projective hierarchy
- Axioms of ZFC
- Models of set theory
- The constructible universe L
- Regularity properties in L
- Forcing
- Independence of GCH
- Solovay's model in which every set of reals is measurable