Projects in Previous Years

2nd Semester 2004/05: Surreal Numbers

Prof Dr Joel David Hamkins

If you are interested in this project, please contact the instructor by e-mail.
Teaching Goal: We will go through Knuth's charming book on the topic of surreal numbers, working on the problems that arise. Students will be expected to work through various problems posed in the book on their own. (Knuth, Donald E., Surreal Numbers, London: Addison-Wesley, 1974)
Content: The surreal numbers, invented by John Conway, unify the real numbers and the ordinals into a single elegant number system. The surreal numbers extend the ordinary real numbers to a new and much larger number system incorporating infinitesimal and infinite numbers of arbitrary degree, while retaining many of the desirable features of the real numbers and of the ordinals. The foundation is amazingly simple and elegant: one begins with nothing, and then successively takes Dedekind-style cuts in the numbers produced so far. In a grand transfinite recursion, this single idea leads to all the surreal numbers. The project will consist of our reading and working through (the early part of) Knuth's excellent book, a narrative describing two characters' exploration of the surreal numbers. This unusual book may be unique in its style of presenting mathematical concepts in a narrative format. The two characters in the book represent two mathematical philosophies or styles of doing mathematics, in how they approach the problems posed in the book.
Basic knowledge of set theory and the ordinals, particularly transfinite recursion.
Homework, mostly solving the problems posed in the book.
There are a few other expositions of the surreal numbers, which students may want to consult, and some popular accounts available on the web.
  1. Knuth, Donald E., Surreal Numbers, London: Addison-Wesley, 1974
  2. Berlekamp, E.R., J.H. Conway, R.K. Guy, Winning Ways San Francisco: Freeman 1977