Advanced Topics in Set Theory 2003/2004; 1st Semester Institute for Logic, Language & Computation Universiteit van Amsterdam

Building on the basics of set theory, we can further investigate properties of the set-theoretic universe. Some of the techniques used are very important in several areas of mathematics. We will also look at infinite games, their set-theoretic properties and the way they connect topology, descriptive set theory, logical foundations of mathematics and game theory proper.

The course will cover selected chapters from Akihiro Kanamori's book The Higher Infinite. In particular, the course will cover the following topics:

Reprise of large cardinal material from "Axiomatic Set Theory": inaccessible cardinals, Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals; normal ultrafilters; Basic Theory of the constructible universe L; Partition Cardinals and Structures; 0^#; Descriptive Set Theory; Basics of Set-Theoretic Game Theory; Homogeneously Suslin Sets; Higher Determinacy

Topics covered so far:

• Sep 2, 2003. Basic Notions. Inaccessible, hyperinaccessible and Mahlo cardinals. (Chapters 0 and 1)
• Sep 9, 2003. Lebesgue's Measure Problem. Measurable Cardinals. Ultrapowers. Normal filters. (Chapters 0, 2, and 5)
• Sep 16, 2003. Elementary Embeddings. Mostowski's Collapsing Lemma. Scott's Trick. Properties of the ultrapower embedding of a measurable cardinal. (Chapters 0 and 5)
• Sep 23, 2003. Cancelled due to illness.
• Sep 30, 2003. Inner models and absoluteness. Constructibility. Consistency of AC and CH. (Chapters 0 and 3)
• Oct 7, 2003. Relative constructibility. The constructible universe and measurable cardinals. Partition relations and partition cardinals. Infinitary languages. Strongly compact and weakly compact languages and cardinals. (Chapters 3, 4 and 7)
• Oct 14, 2003. More on strongly and weakly compact cardinals. More partition relations. Erdös cardinals. Ramsey cardinals. (Chapters 4 and 7)
• Oct 21, 2003. Exam week. No lecture.
• Oct 28, 2003. Square bracket partition relations. Rowbottom's Skolemization Theorem. Rowbottom cardinals. Large cardinals and the reals in L. (Chapter 8)
• Nov 4, 2003. Jónsson cardinals. The theory of 0^#. Baire space. (Chapter 8 and 9)
• Nov 11, 2003. Topology of Baire space. Trees. Characterization of closure by trees. Perfect set property. Bernstein construction.
• Nov 18, 2003. Borel sets. sigma-algebras and sigma-ideals. The Borel hierarchy. Universal sets. Regularity properties: Marczewski-Burstin algebras. (Chapter 12)
• Nov 25, 2003. Projective Sets. Tree representation of Sigma¹₁ sets. Suslin property. Shoenfield Tree. WO. Boundedness Lemma. (Chapter 12 and 13)
• Dec 2, 2003. The complexity of the wellorder of L. Descriptive Set Theory in L. (Chapter 13)
• Dec 9, 2003. Gale-Stewart games. Determinacy. The Gale-Stewart theorem. The perfect (asymmetric) game. Determinacy and the perfect set property. The Axiom of Determinacy. Homogeneous trees. Martin's Pi¹₁ Determinacy Theorem. (Chapter 27 & 31)
• Dec 16, 2003. Exam week. No lecture.

Homework:

1. Homework Set #1 (Warm-up exercises). Deadline: September 16th, 2003.
2. Homework Set #2 (Measurable cardinals and constructibility). Deadline: September 30th, 2003.
3. Homework Set #3 (Constructibility and partition relations). Deadline: October 28th, 2003.
4. Homework Set #4 (Partition relations and cardinals). Deadline: November 11th, 2003.
5. Homework Set #5 (Borel sets). Deadline: November 18th, 2003.
6. Homework Set #6 (Set theory of the reals). Deadline: November 25th, 2003.
7. Homework Set #7 (Suslin property). Deadline: December 2nd, 2003.
8. Homework Set #8 (Determinacy). No Deadline: This homework set doesn't count for the grade.