Instructor: Dr Benedikt Löwe
Time: Tuesday 3-5
Course language: English
Intended Audience: MoL students, Mathematics students in their fourth
Prerequisites: This course assumes knowledge comparable to the course
"Axiomatic Set Theory".
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;
Descriptive Set Theory;
Basics of Set-Theoretic Game Theory;
Homogeneously Suslin Sets;
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 Set #1 (Warm-up
exercises). Deadline: September 16th, 2003.
- Homework Set #2 (Measurable
cardinals and constructibility). Deadline: September 30th, 2003.
- Homework Set #3
(Constructibility and partition relations). Deadline: October 28th,
- Homework Set #4
(Partition relations and cardinals). Deadline: November 11th,
- Homework Set #5
(Borel sets). Deadline: November 18th,
- Homework Set #6
(Set theory of the reals). Deadline: November 25th,
- Homework Set #7 (Suslin
property). Deadline: December 2nd, 2003.
- Homework Set #8 (Determinacy).
No Deadline: This homework set doesn't count for the grade.
Last update : Dcember 11th, 2003