MATH 320 (Spring 2023)

MATH 320 – Set Theory
Course Syllabus

Instructor: Asst. Prof. Burak Kaya
E-mail: burakk@metu.edu.tr
Website: http://blog.metu.edu.tr/burakk
Office: M-126
Phone number: +90 (312) 210 2996
Office hours: Announced at this link
Class hours: Tuesday 13:40-15:30 and Thursday 10:40-11:30
Classroom: M-105. (The Zoom link for the live lectures will be posted to the course page on ODTU Class.)

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 some 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.

Lectures: Following the decision of the University Senate on 20 February 2023, lectures will be conducted online until April 2023 and may be conducted in hybrid mode afterwards, depending on the decision of the Council of Higher Education. Here is how I am planning to conduct the online lectures:

  • Most classrooms in our department have a built-in camera system that can record the blackboards at a good video quality. Past experience shows that it is more efficient for me to use blackboards instead of tablets while lecturing. Consequently, I will conduct the online live lectures on the blackboard and broadcast these online live lectures over Zoom.

The video recordings of each week’s lectures will be publicly uploaded to YouTube afterwards, so students who cannot attend the live lectures may watch the recordings later.

Supplementary lecture videos: My lectures for this course from Spring 2018 were recorded as a part of the METU OCW project. You can access the YouTube playlist for these lectures from this link. You should keep in mind that the video quality and resolution of a few videos in this playlist are not as good as they should have been. Consequently, you are strongly suggested to watch the recordings of the live lectures instead of these.

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

Grading: There will be two midterm exams (each out of 100 points), one final exam (out of 100 points) and a bonus oral exam (out of 20 points). Each midterm will have %30 weight, the final exam will have %40 weight and the bonus oral exam will have %15 bonus weight in your overall score that is to be used for letter grades. More specifically, your total score will be calculated using the following function of four variables.

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

Your letter grades will be given based on your total score.

I will use Gradescope to grade your midterm and final exams, unless you prefer otherwise in which case you should let me know in advance.

In the case that the Council of Higher Education allows universities to switch to hybrid mode instead of online, all exams will be conducted in person.  Otherwise, I will restructure midterms as take-home exams and the final exam as a Zoom-recorded online exam. Therefore, how exams will be conducted is going to be ultimately determined by the decision of the Council of Higher Education, possibly in April.

The oral exams will be conducted as 20 minute interviews that will take place

  • in my office if we switch to hybrid mode, or
  • over Zoom if we do not switch to hybrid mode.

In the first case, your answers in the oral exam will be either recorded using a voice recorder (with your permission) or written down on paper by you as you answer the questions. In the second case, your whole interview will be recorded through Zoom.

I would like to note that I have been conducting oral exams for this elective course long before the COVID-19 pandemic; so it has been a tradition for me. In these oral exams, contrary to usual midterms and finals where you are expected to solve a relatively complicated problem or prove some implication, I usually ask definitions, basic concepts, some basic examples and fundamental facts. This type of oral exam cannot be handled by memorizing the course material without understanding the material. Please study the material well from the beginning of the semester, otherwise it may be impossible to digest all this material in couple of weeks while you prepare for the oral exam. We will decide on the oral exam schedule later during the semester.

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.

Information for students with disabilities: Students who experience difficulties due to their disabilities and wish to obtain academic adjustments and/or auxiliary aids must contact ODTU Disability Support Office and/or course instructor and the advisor of students with disabilities at academic departments as soon as possible. For detailed information, please visit the website of Disability Support Office.

Weekly course plan: Below is the tentative course plan that I usually use for in-person education. You should keep in mind that I may add/remove topics or change the order of topics depending on our progress each week.

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.