Determinacy and Set Theory

Part of the UCLA Logic Center 2010 Summer School for Undergraduates
When: Monday through Friday, 3PM-5PM. The first day is July 5 and the last is July 23.
Where: The lectures will be held in MS 5117. Problem solving sessions will be held daily in 6617C.
Instructor: Justin Palumbo
Office: MS 6617F
Office hours: I will be available by appointment. Just let me know.

Here is the original abstract for the course.

Principles of determinacy state the existence of winning strategies in certain two-player games of infinite length. They were initially considered in the '30s and '50s in connection with various topological properties; for example the famous Banach-Mazur game relates determinacy to the Baire property and other games relate determinacy to the perfect set property and Lebesgue measurability. Despite their somewhat humble beginnings, these innocuous-looking principles are today understood to have surprisingly deep connections with set theory at the highest and lowest levels of the infinite. At the high end they are intricately connected to large cardinal axioms, and at the low end they can be used to develop a deep structure theory for definable sets of reals. In this course we will explore the effects that determinacy hypotheses have on the structure of the set theoretic universe and especially on the real line, while simultaneously introducing students to basic techniques and tenets of descriptive set theory. We will also investigate the extent to which determinacy can be realized just on the basis of the classical mathematical axioms.

Day 1 (pdf): Sequential rock-paper-scissors: some motivational remarks (not in pdf). Axiom of choice, and weakenings. Trees, Koenig's Lemma.
Relevant Problems

Day 2 (pdf): The Baire and Cantor spaces and some topological properties. The Baire category theorem.
Relevant Problems

Day 3 (pdf): Ulam games, the Axiom of Determinacy. AD and AC can't coexist. Gale-Stewart Theorem. Determinacy of complements of determined pointclasses.
Relevant Problems

Day 4 (pdf): Polish spaces and Borel sets. Baire space can be continuously surjected onto any Polish space. Closure properties of Borel sets.
Relevant Problems

Day 5 (pdf): The hierarchy theorem for the Borel hierarchy. Wolfe's theorem.
Relevant Problems

Day 6 (pdf): Perfect sets and a related game.
Relevant Problems

Day 7 (pdf): Property of Baire. Kolmogorov 0-1 law for category. A related game.
Relevant Problems

Day 8 (pdf): Wadge degrees, Lipschitz degrees, Wadge's lemma and some consequences.
Relevant Problems

Day 9 (pdf): Well-foundedness of the Wadge degrees. And now for something completely different: Baire space is homeomorphic to the irrationals.
Relevant Problems

Day 10 (pdf): Projective hierarchy and its relationship to the Borel hierarchy.
Relevant Problems

Day 11 (pdf): Analysis of Co-analytic sets: Kleene-Brouwer order, Well-orderings, Boundedness Theorem
Relevant Problems

Day 12 (pdf): Cardinals, Ultrafilters, Ramsey's Theorem, Weakly Compact Cardinals.
Relevant Problems

Day 13 (pdf): Analytic Determinacy from a Measurable Cardinal.

Day 14 (pdf): If AD holds then omega_1 is measurable.