## 2012-13

#### Course content

We will start with constructing the reals and the natural numbers, to see how these fundamental mathematical concepts can be understood using sets. Then we'll turn to the axiomatic set-up for set theory, and see how very few axioms are needed to build the complicated set-theoretic objects we are used to dealing with. Returning to the natural numbers, we'll prove that definition by recursion gives a set which is a function, and hence show how to define arithmetic on the natural numbers. The notion of equinumerous gives us a sense of sizes of infinity, with and without an ordering (ordinals and cardinals). We see a need for the Axiom of Choice, and we'll discuss some it's surprising consequences in other parts of mathematics. Depending on time available, and the interest of the class, we can look at Ramsey's theorems, non-well-founded set theory or large cardinals. We will finish with a discussion of consistency and the continuum hypothesis.

#### Textbook

"Introduction to Set Theory" Karel Hrbacek and Thomas Jech, 3rd edition, CRC Press

Another useful reference: "Classic set theory for guided independent study" Derek Goldrei, Chapman and Hall

#### Course structure

The course meets three times per week: M 9:3--10:20, T 10:30-11:20 and Th 9:30-10:20. I plan to lecture about once per week; the other two hours will have students presenting lemmas and exercises that move the theory forward. In general, I expect to introduce a topic on a Thursday, giving you four days to prepare presentations for Monday and Tuesday. Topics will be assigned (and posted on the calendar below) each Thursday. There will also be further homework problems posted, and anyone might be asked to present one of these. For each subsection of the text that we cover, two students will be assigned to write up the class solutions to the exercises. In the last week of class, there will be a take-home final exam. Graduate students registered for 6LT3 will also be required to write a short discussion on an advanced topic in set theory.

#### Assessment

Class participation and presentations: 50%
Written exercises: 25%
Final exam: 25%

Announcements

28 March 2013 Tour of the Russell archive is on for Thursday April 4 at 9:30. Let's meet just inside Mills Library.

23 March 2013 There is no final exam for this course, but there will be a final homework set. So far, we have seen the fundamentals of axiomatic set theory. For the last homework assignment, I would like you to read one of the later chapters of the textbook, which give an idea of what advanced set theory and contemporary research in set theory is all about. So choose one of the following reading and problem sets. Your final submission is due by noon on  Wednesday, April 10 (last day of classes).
Chapter 11: read 11.1, problems 1.3, 1.7
read 11.3 problems 3.2, 3.6, 3.7

Chapter 12: read 12.1 problems 1.1, 1.4

Chapter 14: read 14.1 problems 1.3, 1.4