Class Information

Course Title Math 335: Set Theory

Instructor Kameryn Williams

Website http://kamerynjw.net/teaching/2024/math335/

Email kwilliams [at] simons-rock (dot) edu

Primary out of class contact course website or email me

Class Hours and Room MWF 2:40–3:35 FSH-201

Office Hours MT 3:30–5:00, W 2:00–2:30

Office 2T Hall College Center

Textbook none

Course Description Set theory lies at the foundation of modern mathematics, and it involves the study of infinity and the behavior of infinite objects. The class will blend theory and connections with other parts of mathematics to understand the place of set theory in a wider context. Beginning from the Zermelo-Fraenkel axioms, the goal is to develop the theory of ordinal and cardinal numbers and their arithmetic, with applications to analysis, topology, and combinatorics. We will discuss different sizes of infinite sets, infinite trees, mathematical games, Baire space, and the combinatorics of uncountable sets.

Prerequisite MATH 220 or permission of the instructor.

Learning Outcomes

This is an introductory course is set theory. We will cover ordinals, transfinite recursion, cardinals, the cumulative hierarchy, axioms for set theory, and trees. Upon leaving this class you should be able to reason mathematically about infinite sets, ordinals and cardinals, use the proof technique of transfinite recursion, explain the role of axioms in mathematics and what it means for a statement to be independent of a collection of axioms, and apply these concepts in other areas of mathematics.

Grading Policy

Grading for this class is done a specifications model. What this means is, rather than averaging together numerical scores to determine your overall grade, your grade is determined based on you meeting specifications in multiple categories. There are three categories: Project, Portfolio, and Participation. Project and Portfolio are graded on a high pass/pass/fail scale, while Participation is graded on a pass/fail scale.

Here’s the specifications to earn each grade:

A: High pass in Project and Portfolio, pass in Participation.
B: Not meet the requirements for an A, and pass in Project, Portfolio, and Participation.
C: Not meet the requirements for a B, and pass in Portfolio and Participation.
D: Not meet the requirements for a C, and pass in Participation.
F: Not meet the requirements for a D.

I reserve the right to use +’s or -‘s for borderline cases.

To be explicit about what this means: if you want a B or better, you need to do the project and the portfolio. If you want a C or better, you need to do the portfolio. Or phrased in the converse: if you don’t want to do the project, then you can make that choice but the highest grade you can get is a C.

Part of the point of this class is to give you pratice at working as a mathematician. In mathematics anything worth doing is worth doing well. Just like an incomplete proof wouldn’t be acceptable in a published paper in a research journal, it doesn’t count as ‘partial credit’ toward your grade. This means your work is held to a high standard, and what you present for a final grade should be good quality. It should be mathematically correct, clearly presented, and typset using LaTeX.

On the flip side, there is ample opportunity to get feedback when doing research mathematics, and this class is structured to provide you the same. For both the project and the portfolio you have multiple opportunities to get comments from me. (Indeed, you have to do so to earn a high pass.) And we will have regular problem sessions during class time for you to give each other feedback.

Here’s the overview of how each category is graded. Details about the project and portfolio are later in the syllabus.

Project
- High Pass. Do all check-ins and submit the paper draft, give a good quality presentation, and submit a good quality paper.
- Pass. Don’t meet the requirements for a high pass, miss at most one check-in, and either give a good quality presentation or submit a good quality paper.
- Fail. Don’t meet the requirements for a pass.
Portfolio
- High Pass. Submit all partial portfolios, and submit a final portfolio with 30 good quality problems, with at least 5 from each of chapter 0 through 4.
- Pass. Don’t meet the requirements for a high pass, miss at most one partial portfolio submission, and submit a final portfolio with 24 good quality problems, with at least 4 from each of chapter 0 through 4.
- Fail. Don’t meet the requirements for a pass.
Participation
- Pass. Give a good faith effort at engaging in the course material. You should attend lectures, take part in problem sessions, talk to me out of class when you have questions, and so on.
- Fail. Don’t meet the requirements for a pass. If you are in danger of failing your Participation grade I will reach out to you.

I reserve the right to modify the grading requirements for an individual student based on special circumstance. For example, if you have an emergency and have to miss class for a week that shouldn’t be a mark against your Participation grade. But you have to communicate with me. If anything comes up that seriously impairs your ability to engage in the class please let me know as soon as possible.

Project

As part of this class you will investigate a topic that builds on the material in this class, but isn’t itself something we cover in lecture. The purpose of this is twofold: to give some experience learning and writing mathematics on your own, and so that the class can see a wider range of topics than will be covered in lecture. Your topic can be anything you like, subject to approval by me. For the project you will both prepare a 20–30 minute presentation to present to the class and write a short (5 to 10 pages) paper. Your final work is due at the end of the semester, but along the way there are check-ins so we can ensure you’re staying on track and for me to help guide you. The check-ins are done out of class at my office hours.

Your presentation can either use slides or be done at the blackboard. Since making slides for math is a time-consuming process, I advise you to give a blackboard talk. You’ll have around 20 to 30 minutes, depending on the exact number of students who do the project. Given the limited time, you should not give detailed proofs. Instead, you should introduce the big ideas, state the main theorems, and give the audience a suggestion of the methods used to prove them. Anyone who wants to see more detail can read your paper.

The paper should look like a professional mathematical paper, as you would see in a journal or on the arXiv. You should have sections—introduction, etc.—adhere to the definition/theorem/proof structure of written mathematics, and include a bibliography. Your paper must be typeset in LaTeX.

Timeline:

First check-in, choose topic by 3/15
Second check-in by 4/15
Draft of paper due 4/29
Presentations 5/6 and 5/8
Final paper submission 5/13

Portfolio

Each chapter (with the exception of chapter $\omega$) has a problem set, consisting of 10 problems. Over the course of the semester you will create a portfolio consisting of your solutions to problems from these sets.

Treat your portfolio like a polished, professional piece of work. Your final submissions should be both mathematically correct and well explained. You should adhere to the standards of writing mathematics: clearly state what you’re going to prove before you give the proof, define any terms you introduce, and cite any outside work you are relying on. For your audience, think of a fellow student in the class. You don’t need to explain content from lecture or the class notes, and the level of detail should be enough that they can follow what you are doing. Only work that meets this high standard will be considered for determining the Portfolio portion of your grade.

Your portfolio must be typeset in LaTeX. I will not accept handwritten work.

The final submission of your portfolio is at the end of the semester. Along the way there are five partial portfolios that are due, at the ends of chapters 0 through 4. These are opportunities for you to get feedback on your work before the final submission. What you submit for your partial portfolios doesn’t have to yet meet the high standards for the final submission. But please clearly separate the completed work you want me to check for correctness versus in-progress work you want feedback on.

(Note that, in addition to the problem sets, in the class notes there are exercises for each section. These are not part of your portfolio. Instead, they are more straightforward problems you should solve as part of reading and following the class material.)

Timeline:

1st partial portfolio 2/12
2nd partial portfolio 3/4
3rd partial portfolio 3/15
4th partial portfolio 4/8
5th partial portfolio 4/22
Final portfolio submission 5/10

Class Content and (No) Textbook

There is no required textbook for this class. Lecture notes divided into chapters will be posted to the course website.

Chapter 0 is an introduction to the subject by way of looking at countable sets.
Chapters 1 and 2 cover, respectively, the core set theoretical objects of ordinals and cardinals.
Chapter 3 is about the structure of the set theoretic universe and the axioms of set theory.
Chapter 4 introduces you to a current area of research interest, studying trees on uncountable cardinals.
Chapter $\omega$ will be about a topic chosen by the class.

If you are interested in a different look at the same material, I recommend Ernest Schimmerling, A Course in Set Theory (ISBN: 978-1107400481) and Karel Hrbáček and Thomas Jech, Introduction to Set Theory, 3rd ed. (ISBN: 978-0824779153).

Academic Honesty

You are expected to know and uphold the college’s policies on academic honesty as described in the Student Handbook. For the specifics of what that means for this class I want us to talk together to decide.

Accessibility and Inclusion

Everyone has equal access to this class, regardless of disability, background, or identity. Mathematics has a reputation for being removed from social concerns and identities. Whether or not this is true for the content of mathematics, it is certainly false for the process of learning mathematics. Our classroom is to be a welcoming one, where everyone feels able to participate and learn. As learners it is your obligation to treat others with respect and generosity, and be willing to exchange ideas with others.

Students with disabilities are legally entitled to reasonable accommodations to ensure equal access to education. I am happy to work with you to ensure reasonable accommodations. Because the accommodations offered are usually forward-looking modifications, it is important to get them set up as soon as possible.

Anyone who feels they may need accommodation based on the impact of a disability should contact Jeannie Altshuler, Director of Accessibility and Academic Support, in the Win Commons (jaltshuler@simons-rock.edu; 413-528-7383).

The Americans with Disabilities Act defines a disability as a medical condition that substantially limits one or more major life activities—including things like walking, sleeping, taking care of yourself, learning, and regulating your emotions—or major bodily functions. If you have a medical condition—including mental health conditions—that significantly interferes with your schoolwork, you probably qualify. You do not need to disclose your condition to your instructors to receive accommodations.

Other Campus Resources

The Wellness Center Health and Counseling Services, as the name says, offers health and counseling services.

The Win Student Resources Commons offers academic support, tutoring, accessibility, and career advice.

Schedule

Chapter 0: Countable sets
- 1/29–2/9
- Problem session 2/9
- Partial portfolio due 2/12
Chapter 1: Ordinals and transfinite recursion
- 2/12–3/1
- Problem session 3/1
- Partial portfolio due 3/4
Chapter 2: Cardinals
- 3/4–3/15
- Problem session 3/15
- Partial portfolio due 3/15
- First project check-in, choose topic by 3/15
Spring Break 3/18–3/22
Chapter 3: The cumulative hierarchy and axioms
- 3/25–4/5
- Problem session 4/5
- Partial portfolio due 4/8
Chapter 4: Trees
- 4/8–4/19
- Second project check-in by 4/15
- Problem session 4/19
- Partial portfolio due 4/22
Chapter $\omega$: Selected topic
- 4/22–5/1
- Draft of paper due 4/29
- Problem session 5/1
Student Presentations
- 5/6–5/8
- Portfolio due 5/10
- Paper due 5/13

Notice of Changes

This syllabus is subject to change. If this happens, you will be informed of any additions or changes.

(3/29) Updated portfolio grading requirements.