Math 102: Axiomatic Set Theory
TuTh 9:35–10:50a.m.
Monroe B34
Valentina Harizanov
Monroe Hall, Room 240
Tel: (202) 994–6235, Fax: (202) 994–6760
·
Professor
Valentina Harizanov
Office: Goverment Hall, Room 220
Tel: (202) 994–6595
E-mail: harizanv@gwu.edu
Web
page: http://home.gwu.edu/~harizanv/
·
Office
Hours
Tu 11:00a.m.–12:00noon
Th 1:00–2:00p.m.
At other times by appointment.
Any time by e-mail.
- Description. This course will provide an
introduction to modern set theory. Set theory was founded by Cantor, who
invented infinite numbers. Cantor defined a set as a “collection into a
whole of definite, distinct objects of our intuition or our thought.” However,
this definition allows the existence of some unusual sets that lead to
paradoxes. These paradoxes show
that there are properties that do not define sets, leaving set theorists
with the task of determining which ones do define sets. Unfortunately, Gödel’s results indicate that a complete answer to this question
is not even possible. Therefore, axiomatic set theory attempts a less
lofty goal. It formulates some of the relatively simple properties of sets,
used by mathematicians, as axioms. Within this axiomatic system,
practically all notions of contemporary mathematics can be defined and
their properties can be derived. In this sense the axiomatic set theory
serves as a foundation of mathematics.
- Topics.
Zermelo-Frankel axioms, infinite sets, countable sets,
development of real numbers, cardinal numbers, ordinal numbers, ordinal
and cardinal arithmetic, transfinite induction, partially ordered sets,
the axiom of choice and its equivalents (such as Zorn’s lemma).
- Required background. Math 32 or permission of instructor. Familiarity
with proofs.
- Grading.
Take-home assignments (48%), midterm exam (22%), take-home final exam (22%),
attendance and class participation (8%).
- Textbook. Introduction to Set Theory by Karel Hrbacek and Thomas Jech, Third
Edition, CRC Press.
Optional additional book: Classic Set Theory (A Guided
Independent Study) by Derek Goldrei, Chapman and
Hall.