Math 160B (Spring 2025)
Summary
Math 160B is the second course in the two-part series Math 160AB in mathematical logic. It continues where Math 160A left off in the fall.
Some topics we will definitely cover include:—
- Ordinals, recursion theory and transfinite induction.
- Some associated "informal" set theory (cardinals, etc.)
- ZF / ZFC set theory as a first-order theory.
- Some more model theory, and in particular the use of ultrafilters.
Contacts
The instructor is Freddie Manners (email fmanners); my office is AP&M 7343. There is no grader or TA for this course.
Class and section
Lectures are held on Mondays, Wednesdays and Fridays, 1100–1150. They will be held in-person in APM-2402. The course will not be generally available as a podcast, so lecture attendance is required. I will take lecture attendance; there is no grade penalty for not attending, but as a courtesy, whenever possible, let me know if you will be unable to attend.
The course enrollment currently is very small, and so the class is not assigned a TA or a grader. This gives us the freedom to structure homework and section rather differently than usual. I plan to do the following:–
- I will set longer homework assignments due roughly bi-weekly, with five problem sets overall.
- In lieu of normal section and homework grading, I will meet students in in-person "supervisions" in groups of one or two, for one hour per group, shortly after the homework due date to go over their submitted homework.
- There is a substantial part of course grade assigned to homework. However, I plan to assign each problem set a holistic grade, rather than the more usual add-the-points system. In particular, I will award credit based on whether I judge – based on the written work and the in-person meeting – that the student showed understanding of, and engaged seriously with, the homework problems. I anticipate giving full credit to assignments that did not solve every part of every problem.
- I will suggest student pairings, which can be modified by mutual agreement.
Note there is NO class on Friday May 23 or on Monday May 26 (Memorial Day).
Exams
There will be two evening midterm exams, on Thursday April 24 (Week 4) and Thursday May 15 (Week 7), 1900–2030 (i.e., 7pm to 8:30pm), in HSS-2154.
The final exam is on Friday June 13, at 1130–1430. It is also in-person, in APM-2402.
You should ensure at the start of the quarter that you can attend these exams. In general, alternative modalities, dates or times (in particular, remote exams) will not be offered.
(Note: it is standard practice in the Mathematics Department to use multiple versions of exams where necessary.)
Exams will be closed-book. Use of any outside resources or communication in exams is, of course, strictly forbidden.
Homework
Homework is due every other Tuesday at 23:59 (11:59pm), starting on April 8 (Week 2). Deadlines may change as circumstances require.
You may collaborate in discussing and thinking about homework problems; indeed this is often beneficial. However, if you do, you must list your collaborators on your assignment when you turn it in. Moreover, you must not simply show another student a written solution to a problem, and when you write your own solutions, you \textbf{must do so alone and in your own words}, without looking at another solution. You may use outside resources, including the internet, other textbooks or AI tools, to help you understand the subject material. However, you may not attempt to find a solution to a specific homework problem using such resources. In particular, you may not post any homework problem as a question on the internet, or to an AI system.
It is guaranteed that every exam will contain at least one problem which is a close variant of a homework problem.
Grading
Your combined grade for the course is: 25% homework + 20% midterm 1 + 20% midterm 2 + 35% final exam.
Given the size of the class, grade cut-offs will be determined as appropriate and without reference to a fixed scale.
Resources
In addition to this website, the course has a Canvas page, a Gradescope page and a Piazza site. The sign-up codes for Gradescope and Piazza are listed on the Canvas home page.
Provisional schedule
A very rough provisional course schedule is given below. It can and probably will change.
| Week | Mon | Wed | Fri | Topic |
|---|---|---|---|---|
| 1 | L1 | L2 | L3 | Introduction; naive set theory. Ordinals. |
| 2 | L4 | L5 | L6 | More ordinals. Induction and recursion. |
| 3 | L7 | L8 | L9 | Ordinal arithmetic. Hartogs' Lemma. Transfinite Induction. The Burali--Forti paradox. |
| 4 | L10 | Review | L11 | Posets. Zorn's Lemma. Bourbaki--Witt. Applications. |
| 5 | L12 | L13 | L14 | ZF and ZFC set theory. Aximos. ∈-induction and ∈-recursion. |
| 6 | L15 | L16 | L17 | Transitive sets and Mostowski collapse. The von Neumann Universe. |
| 7 | L18 | Review | L19 | Cardinals. Filters and ultrafilters. |
| 8 | L20 | L21 | X | The ultrafilter existence lemma. Ultrafilters and the compactness theorem. |
| 9 | X | L22 | L23 | Further topics (if time). Fast-growing hierarchies. Other hierarchies. |
| 10 | L24 | L25 | Review | Further topics (if time). Revisiting the completeness theorem. |
Office hours
Regular office hours and locations are listed in the table below. However, please check the calendar below for any one-off changes or cancellations.
| Instructor / TA | Location | Regular hours |
|---|---|---|
| Freddie Manners | APM-7343 | Mondays 1-2pm, Fridays 2-3pm |
Course calendar
A link to add this calendar is available on the course's Canvas page.