**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: |
Monday 16:40-17:30 and Wednesday 15:40-17:30 |

Classroom: |
The Zoom links 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 **Z**ermelo-**F**raenkel set theory with **C**hoice. 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

- Introduction to Set Theory, Third Edition, Revised and Expanded

by Karel Hrbacek and Thomas Jech, ISBN: 0-8247-7915-0.

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:** Lectures will be **synchronous**; we will have **live lectures** over the Zoom software every week during the lecture hours announced above. The Zoom link to attend the lectures will be announced on ODTU Class. You should keep in mind that each live lecture **will be recorded** and these recordings will be only shared with students taking the course, over ODTU Class. Therefore, any student attending the lectures is automatically assumed to have given consent his/her voice or image to be recorded for the purposes of this course only.

**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. That said, students who wish to watch lectures carried on a blackboard instead of a tablet may want to watch these videos as well.

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

**Grading:** There will be two open-book take-home midterms (each one out of 80), a webcam proctored final exam (out of 100) and a mandatory oral exam (out of 20). Your total score will be calculated by the following function of four variables.

Total score=(Midterm 1 + Midterm 2) x 0.25 + Final x 0.40 + Oral Exam

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

**Exam structures:**

**Midterms:**The two open-book take-home midterm assignments will be submitted either over Gradescope or over ODTU Class. Whether we will use Gradescope or ODTU Class is going to be announced later during the semester, depending on whether my Gradescope license continues. For each midterm, we will choose a week together via a questionnaire on ODTU Class. On the week of the midterm, I will post the midterm**at exactly 09:00 a.m. Friday of that week**and the submission deadline is going to be**exactly 12:00 p.m. Monday of the following week**. So you will have exactly**75 hours to submit**your midterm. For midterms, you can discuss the questions with your friends and share ideas with each other. However, submissions should be written down individually, there are no group submissions.**No late submission is allowed**. You should**not just copy-paste**your friend’s solutions or some solution you found on the internet. Whatever you are going to submit for your midterms, you are expected have understood well. Because in the final exam and oral exam, you will not have your friends or your internet resources with you. Moreover, I have been teaching mathematics long enough to be able to easily spot the solutions that are not done by the student. For this reason, please try to**understand the solutions**, even if you get additional help.**Final exam:**Your final exam will be a**webcam proctored**exam which will be conducted over Zoom. Due to regulations regarding the storage of exams, the final exam**will be recorded**. The final exam is**not**open-book. Those students who do not have the necessary equipment (computer, webcam, internet infrastructure etc.) should contact Burcu Yayla from the department’s secreterial office to learn where they can obtain these. Our university may provide technical support to those students who cannot afford these.**Oral exam:**Towards the end of the semester, you will have an approximately 20 minute oral exam conducted over Zoom in the form of an interview. For this exam, you will need to have a webcam and a properly working microphone, just like your final exam. 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. That said, I am expecting our oral exams to start on the week of June 13, 2020 and continue through the final exam weeks, depending on the number of students taking the course.

You can reach my past exam questions here. Keep in mind that these questions were prepared for the traditional in-person exams and hence may differ from your take-home midterm questions in depth and style.

**Make-up policy:** Since your midterm assignments are take-home assignments with 75 hours to work on, there will be no make-ups for midterms. In an extraordinary situation which prevents you from doing/submitting your midterm during all of these 75 hours, you should **contact me as soon as possible** so that we can think of a solution.

For the final exam, if you have technical difficulties (internet connection interruption, webcam failure etc.) during the exam, then you can take a make-up exam. The make-up exam for the final is going to be a webcam proctored exam, just like the final itself. For the oral exam, if you have technical difficulties during the exam, we can simply postpone the remaining part of your exam to a later date.

**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**. Some 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. |