Winter 2020: Set Theory
We explore the basic concepts of set theory, starting with the construction of the real numbers, assuming we know what rational numbers are. Then we continue by questioning this assumption, which eventually leads us to figure out a way to define the natural numbers. This in turn leads us to formulating fundamental axioms underlying all of mathematics, the axioms of set theory. We then learn how to work with them, and how to extend the notion of natural number to include infinite numbers called ordinals.
What you need for this course is to be familiar with proofs in Calculus, as taught in MATH 1XA3 or MATH 3A03, as well as basic mathematical notations, and to bring with you an enthusiasm for abstract reasoning. Proof writing is an essential feature of this course.
Announcements
Apr 22: The course grade is not avaiable on Avenue, but it will be available on Mosaic. Thanks for listening, and have a good summer!
Apr 17: Quite a few of you did not deem it necessary to explain many of your arguments in your answers to the final exam, even though I thought I was clear I expected you to do so. This led to some low scores. You may submit rewrites by Tuesday, April 21 at 11:30pm, for a maximum of 80% of the full score per resubmitted problem. These rewrites should be submitted as part of the Final Exam assignment, and their file name should clearly indicate they are rewrites (unless you want me to miss them…).
Apr 9: The Final Exam is now posted on Avenue as an assignment. It becomes visible to you tonight at midnight, and it is due on Thursday, April 16, at 11:30pm.
Apr 3: There is overwhelming support for a take-home final exam. So I will post it here on Thursday, April 9, and it will be due one week later, submitted on Avenue. In the meantime, thanks for your patience with my online lectures!
Mar 31: I’m thinking of holding a Final Exam on Avenue 2 Learn, scheduled at the original final exam time. I would post the exam as a quiz on Avenue, and you will have two hours to complete the exam (on paper, say) and upload (a scan of) it. Obviously, you could use your notes and book, so the questions would take this into account. Please let me know by email if you have any objections to this; the alternative is a take-home exam (more time, harder questions).
Mar 25: Class cancelled due to technical difficulties. I’m trying to find a way to run these lectures from home…
Mar 18: Our first Webex lecture was successful, and the recording is now posted on Avenue to Learn. Please let me know if you cannot access it.
Mar 17: I have just scheduled a Webex meeting for tomorrow’s lecture. You should receive an email invitation that lets you join the meeting during our usual class time. The meeting will be recorded and, if successful, I will post it on Avenue to Learn afterwards.
Mar 17: I’m still working on moving lectures to online delivery. In the meantime, please keep reading the sections added to the list below, and start working on Homework 4…
Mar 16: It is unlikely that everything will be ready by this Tuesday to deliver my lecture online; for one thing, my plan to do so needs to be officially approved before I can do so. So I will post reading assignments below, one for Tuesday and probably one for Wednesday. Please continue to check this page for updates; and remember that I can still be contacted by email.
Mar 13: Starting with the next homework assignment, you will submit your answers on A2L, say in the form of a scanned file. Please make sure that your scans are clearly legible; otherwise, I will return them unmarked. We will of course continue with our kritik assignments as usual. I will decide what to do about the final exam soon…
Mar 13: Following McMaster announcement to cancel all in-class lectures and exams for the remainder of this semester, I am activating this class on Avenue to Learn. You should already be able to see it in your A2L page; if you do not (maybe because you are not registered for this class, or for similar reasons), please let me know by email so I can add you to the class list. The page of reference for material covered will continue to be this one rather than the one on A2L; the latter is only used for homework submission and posting of recorded lectures (yes, I will record my lectures and post them).
Jan 8: I mentioned in class that $\sqrt 2$ and $e$ are irrational numbers. You can find proofs in these slides for MATH 1XA3.
Jan 8: If you are signing up with Kritik.io for more than one course, you can get a discount. Just contact them through their site after you signed up for both classes.
Homework
To be handed in at the beginning of class on the indicated due date.
| Date due | Assignment | Remarks |
| Jan 28 | Homework 1 | Updated version! Problem 1b is simplified… |
| Feb 11 | Homework 2 Exercises 3.17, 3.23(d), 3.28(a) and 3.35 | For each exercise, you may use all previous exercises from the book as references. |
| Mar 10 | Homework 3 | Note the new deadline… |
| Mar 24 | Homework 4: Exercises 6.6, 6.13(d), 6.22, 6.27 | Please hand in on Avenue to Learn |
Kritik assignments
Please do not include your name in your kritik assignment answers! The evaluation process should be as anonymous as possible.
| Assignment | Date available | Due date for your answer | Due date for your peer evaluations | Due date for your feedback |
| Activity 1 | Saturday, Jan 11 at 12am | Tuesday, Jan 14 at 11:55pm | Thursday, Jan 16 at 11:55pm | Friday, Jan 17 at 11:55pm |
| Activity 2 | Saturday, Jan 25 | Tuesday, Jan 28 at 11:55pm | Thursday, Jan 30 at 11:55pm | Friday, Jan 31 at 11:55pm |
| Activity 3 | Feb 8 at 12am | Feb 11 at 11:55pm | Feb 13 at 11:55pm | Feb 14 at 11:55pm |
| Activity 4 | Mar 7 at 12am | Mar 12 at 11:55pm | Mar 14 at 11:55pm | Mar 15 at 11:55pm |
| Activity 5 | Mar 28 at 12am | Mar 31 at 11:55pm | Apr 2 at 11:55pm | Apr 3 at 11:55pm |
Covered so far
| Week | Tuesday | Wednesday | Friday | Required reading |
| Jan 6 – 10 | The real numbers as we know them | Dedekind real numbers: definition and examples | Dedekind real numbers: ordering | Section 2.2 |
| Jan 13 – 17 | Dedekind real numbers: arithmetic operations | Dedekind real numbers: multiplication | Rational numbers | Sections 2.2 and 2.4 |
| Jan 20 – 24 | Rational numbers | Peano’s axioms for the natural numbers | Construction of the natural numbers | Sections 2.4 , 3.1 and 3.2 |
| Jan 27 – 31 | The Peano axioms | Defining addition, multiplication and exponentiation | Arithmetic | Sections 3.2 and 3.3 |
| Feb 3 – 7 | class cancelled | Finite sets | Language of set theory | Sections 3.4, 4.1 and 4.2 |
| Feb 10 – 14 | Zermelo-Fraenkel axioms 1 – 3 | Zermelo-Fraenkel axioms 4 – 6 | Zermelo-Fraenkel axioms 4 – 6 | Sections 4.3 and 4.4 |
| Feb 24 – 28 | Zermelo-Fraenkel axioms 7 – 9 | Recursion principle | Axiom of Choice | Sections 4.5, 5.1 and 5.2 |
| Mar 2 – 6 | Axiom of Choice | Infinite Products | Zorn’s Lemma | Sections 5.2, 5.3 and 5.4 |
| Mar 9 – 13 | Equinumerosity | Schröder-Bernstein Theorem | Some infinitary arithmetic | Sections 6.1 – 6.3 |
| Mar 16 – 20 | Countable sets Cantor’s Theorem Reading assignment! Note: Theorem 6.6 uses AC! Make sure you understand Exercise 6.36. | Uncountable sets and the real numbers (up to Theorem 6.9) | Strict partially ordered sets | Sections 6.4, 6.5 and 7.2 |
| Mar 23 – 27 | Order sum | Cancelled | Order product | Section 7.3 |
| Mar 30 – Apr 3 | Well-orderings Make sure you understand Exercise 7.49 | Initial segments | Comparing well-ordered sets | Section 7.4 |
| Apr 6 – 7 | Ordinals | Section 8.2 |
General information
| Instructor: | Patrick Speissegger |
| Office: | HH 322 |
| Telephone: | extension 23430 |
| E-mail: | speisseg@math.mcmaster.ca |
| Office Hours: | Tuesdays 13:00-14:00 in HH 322 |
| Lectures: | TWF 9:30-10:20 in HH 104 |
| Course materials (required): | Classic Set Theory for guided independent study, by Derek Goldrei, Chapman & Hall/CRC 1996, ISBN 0-412-60610-0 Kritik.io : in place of midterm tests, we will have a weekly Kritik assignment. You will get an invitation by email to sign up with Kritik for this course; the cost is about $15 for the semester. |
| Assessment: | Homework = 20% Weekly Kritik assignment = 20% Final Exam = 40% Participation = 20% These percentages are subject to adjustment at the instructor’s discretion. |
| Final exam: | check on Mosaic |
| Missed academic work: | In the event of an absence for medical or other reasons, students should review and follow the Academic Regulation in the Undergraduate Calendar Requests for Relief for Missed Academic Term Work. |