MATH320 (Spring 2019)

MATH 320 – Set Theory
Course Syllabus

Instructor: Asst. Prof. Burak Kaya
Office: M-126
Phone number: +90 (312) 210 2996
Office hours: Announced at this link
Class hours: Tuesday 10:40-11:30 and Friday 13:40-15:30
Classroom: M-102 on Tuesday and M-105 on Friday

Prerequisites:   Officially none; unofficially MATH 111.

Course description: This is an introductory course to axiomatic set theory. We shall learn the axiomatic system ZFC, the Zermelo-Fraenkel set theory with Choice. The main objectives of this course are

  • to understand how ZFC provides a foundation for (virtually all) mathematics,
  • to learn about set-theoretic techniques that are frequently used in mathematics,
  • to learn about set theory as a field of mathematics on its own,

Textbook: There will be no “official” textbook for this course. We shall be mainly following my lecture notes, which you can find at this link. Besides my lecture notes, you can use the following excellent textbook

a copy of which is available in the library. However, you should keep in mind that some notational conventions and details in my lecture notes may differ from the book. Those students who wish to have supplementary resources in Turkish can also read the old issues of Matematik Dünyası that covered axiomatic set theory.

Lecture videos: My lectures for this course from Spring 2018 were recorded as a part of the METU OCW project. You can access the lecture videos from this YouTube channel. While the lecture videos are online, you are strongly suggested to attend the lectures (and interact with the instructor, which is something that you cannot do with online lectures.) I personally believe that the lecture videos should be used to go over the material or make up for the lectures that you missed.

Attendance: Attendance is not mandatory, however, is strongly suggested.

Grading: There will be two midterms (each one out of 100), a final exam (out of 100) and an oral exam (out of 20). Your overall score will be calculated by the following function of four variables.

(Midterm 1 + Midterm 2) x 0.30 + Final x 0.40 + Oral Exam x 0.75

Your final letter grades will be given based on your overall score (out of 115) with respect to the official university grade catalog.

Make-up policy: No make-ups will be given without an official report. This policy is non-negotiable. If you are going to miss an exam because of extraordinary conditions, which may not be documented via reports, you should contact me in advance.

Academic dishonesty policy: You are expected to be familiar with the university’s academic integrity guide for students. No form of academic dishonesty is tolerated. If you are caught cheating, then you will fail the course and official disciplinary action may be pursued.

Weekly course plan: Below is the tentative course plan. You should keep in mind that I may add/remove topics or change the order of topics depending on our progress each week. The topics of the last week are optional and if we fall behind the schedule, we may use the two lecture hours of the last week to make up.

Week 0 Some historical remarks. Language of set theory and the axioms of ZFC. Some elementary operations on sets.
Week 1 Ordered pairs, relations and functions. Products and sequences.
Week 2 Equivalence relations and partitions.
Week 3 Order relations. Well-orders and well-founded relations. Natural numbers.
Week 4 Natural numbers. Induction, recursion and arithmetic on natural numbers.
Week 5 Equinumerosity. Finite sets.
Week 6 Infinite sets. Cantor’s theorem. Cantor-Schröder-Bernstein theorem.
Week 7 Construction of various number systems.
Week 8 Ordinal numbers. The structure of the class of ordinals numbers.
Week 9 The structure of the class of ordinals numbers. Transfinite induction and transfinite recursion on ordinal numbers.
Week 10 Arithmetic of ordinal numbers. Cantor normal form of ordinals.
Week 11 Some equivalent forms and consequences of the Axiom of Choice. Cardinal numbers. Arithmetic of cardinal numbers.
Week 12 Arithmetic of cardinal numbers. Continuum Hypothesis and Generalized Continuum Hypothesis.
Week 13 Cofinality. König’s theorem and its consequences. The von Neumann hierarchy. Epsilon-induction and epsilon-recursion.