MATH 779 (Spring 2022)

MATH 779, 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: TBA
Class hours: Thursday 16:40-17:30
Friday 15:40-17:30
Classroom:
 M-215

Prerequisites:

  • If you are a graduate student, then there are no prerequisites for this course. That said, you are suggested to self-study ordinals, cardinals, transfinite induction and transfinite recursion in advance if necessary. I would like to note that a review of these topics will be done in the first week of the course anyway.
  • If you are an undergraduate student, then, in order to take this course, you must have passed MATH320 with a grade of BB or higher and must be satisfying the requirements for undergraduate students to be able to take graduate courses which you find in the e-mail sent by Burcu Yayla with title “[MATH-STUDENTS: 1730] Lisans öğrencilerinin 500 kodlu ders alımları hk.” on October 12, 2021.
  • If you are an undergraduate student, in addition to the previous requirements, you should know that, since this is a 7xx coded course, based on the department board’s decisions, you can add this course only with NI (Not Included) status.

Course description: This is an introductory graduate level-course to advanced set theory.

Since Paul Cohen’s discovery of forcing and the independence of the continuum hypothesis, which is the main technique to show that a statement is independent of ZFC and for which Cohen received a Fields medal, many natural statements have been shown to be independent of ZFC, that is, they are not provable or disprovable from the axioms of mathematics, provided that these axioms are consistent. In other words, the incompleteness phenomenon originally discovered by Gödel has many natural instances in “everyday mathematics”. For example, consider the famous Whitehead problem which asks whether or not every abelian group A with Ext¹(A, ℤ) = 0 is a free abelian group. While this may seem like an “innocent algebra problem”, it turns out that this statement cannot be proven or disproven from ZFC. As this one and many others exemplify, problems in various fields of mathematics are potentially subject to this phenomenon and hence, subject to the relevant set-theoretic principles and techniques. 

The aim of this course is to introduce the technique of forcing. Along the way, we shall also learn about various infinitary combinatorial principles and their consequences, some of which will motivate the idea behind forcing. Finally, we shall see some applications of this technique in “everyday mathematics”.

Textbook: We will follow the classic textbook of Kunen and its expanded reprint

  • Kunen, Kenneth Set theory. An introduction to independence proofs. Reprint of the 1980 original. Studies in Logic and the Foundations of Mathematics, 102. North-Holland Publishing Co., Amsterdam, 1983. xvi+313 pp. ISBN: 0-444-86839-9
  • Kunen, Kenneth Set theory. Studies in Logic (London), 34. College Publications, London, 2011. viii+401 pp. ISBN: 978-1-84890-050-9

You can also use Jech’s classic book as a reference book for general set theory background, keeping in mind that Jech’s approach to forcing is different than Kunen’s and that we shall be following Kunen’s approach.

  • Jech, Thomas Set theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xiv+769 pp.

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

Assignments and grading: There will be 6 homework assignments each out of 10(+2 bonus) points, a midterm exam and a final exam, both of which are out of 100 points. Your overall score will be calculated by the following function of three variables.

Overall score = Homework assignments x 0.5 + Midterm score x 0.3 + Final score x 0.4

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

Submission of homework assignments and bonus points: If the university buys a license for Gradescope, those students who prefer to use Gradescope over ODTÜ Class can use Gradescope to submit their homework assignments. Otherwise, you will be submitting your homework assignments through ODTÜ Class. At this point, I am not sure whether we will be able to use Gradescope; but my hope is that this will be the case.

No late submissions for homework assignments will be accepted.

LaTeX has been a fundamental tool to the mathematical community over years. A mathematician who does not use LaTeX is like a fisherman who uses a wooden stick instead of a professional rod.

For this reason, any non-empty homework submission typed in LaTeX which contains at least one attempted-solution will automatically get +2 bonus points. You can ask others who use LaTeX about how to install a TeX distribution, such as MiKTeX, and a TeX editor, such as TeXmaker.

Midterm exam and Final exam: Both your midterm and final exams will be in person exams.

Make-up policy: There will be one general make-up exam for those who missed an exam due to a reasonable excuse such as a medical report, a COVID-19 positive report or a short academic leave. Students who cannot attend an exam should contact the instructor in advance to be able to take the make-up exam. There are no make-up assignments to replace unsubmitted homework assignments.

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.