MATH 563, Model Theory I, Spring 1998
  MWF 3:35--4:25 PM, 113 Osmond, Schedule Number 386685, 3 credits
  Stephen G. Simpson, simpson@math.psu.edu
  http://www.math.psu.edu/simpson/


Textbook: None.  Lecture notes will be provided by the
instructor.

Prerequisites: This course is suitable for all mathematics
graduate students, especially those who are interested in algebra or
mathematical logic.

Course Description: An important branch of mathematical logic
is model theory, the study of first-order theories and the classes of
models defined by such theories.  This course will include numerous
applications of model theory to algebra, especially ordered fields and
differential fields.  All of the necessary background in mathematical
logic and algebra will be presented in detail.  Among the high points
of the course will be (1) the solution of Hilbert's 17th problem on
positive definite forms, (2) the proof that every differential field
of characteristic  has a unique differential closure.

Course Outline:

1. sentences and models
2. complete theories 
3. the compactness theorem
4. decidability
5. elementary extensions
6. algebraically closed fields
7. saturated models
8. elimination of quantifiers
9. real closed fields
10. prime models (countable case)
11. differential fields of characteristic 
12. totally transcendental theories
13. prime models (uncountable case)