Math 582,
Introduction to Set Theory
Winter 1999 Peter Selinger |

**Course Description:** One of the great discoveries of modern
mathematics was that essentially every mathematical concept may be
defined in terms of sets and membership. Thus set theory plays a
special role as a foundation for the whole of mathematics. One of the
goals of this course is to develop some understanding of how set
theory plays this role. The analysis of common mathematical concepts
(e.g. function, ordering, infinity) in set-theoretic terms leads to a
deeper understanding of these concepts. At the same time, you will be
introduced to many new concepts (e.g. transfinite ordinal and cardinal
numbers, the axiom of choice) which play a major role in many branches
of mathematics. The development of set theory will be largely
axiomatic with the emphasis on proving the main results from the
axioms.

**Topics include:** the intuitive concept of sets, and the
paradoxes arising from it; type theory and the cumulative hierarchy of
sets; the Zermelo-Fraenkel axioms for sets theory; set-theoretic
representations of the fundamental concepts of mathematics (such as
functions, numbers), and of the basic proof principles for these
concepts (e.g. mathematical induction); infinite cardinal and ordinal
numbers and their arithmetic; transfinite induction; the axioms of
choice and its equivalent formulations (e.g. Zorn's Lemma). If time
permits, we may discuss some advanced topics, such as Gödel's
incompleteness theorems or the independence of the continuum
hypothesis from the axioms of set theory.

**Prerequisites:** The official prerequisite, "Math 412 or 451 or
equivalent experience with abstract mathematics," means that students
should be comfortable with writing mathematical proofs. No specific
knowledge of set theory will be presupposed.

**Course Work:** We will have two in-class (1 hour) midterms,
probably on

**Grading:** Grades will be based on exam performance. Each midterm
counts 30% and the final 50%. Out of these 110%, the lowest 10% will
be dropped. In borderline cases, homework will be the tie-breaker.

**Textbook:** Herbert B. Enderton. *Elements of Set
Theory*. Academic Press.

**Office Hours:** Mon 3-4 in the Math Lab (B860 EH), Wed 3-4 in my
office (3847 EH). Phone: 763-0292. Email: selinger@mathstat.dal.ca.

**Course Homepage:** Updated information, homework sets, any
handouts, etc., will be available from
http://www.mathstat.dal.ca/~selinger/courses/582W99/

To Peter Selinger's Homepage:

selinger@mathstat.dal.ca / PGP key