Math 102: Axiomatic Set Theory

TuTh 9:35–10:50a.m.

Monroe B34

Valentina Harizanov

• Mathematics Department

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.