Model Theory
MATH 6830, Spring 2020
10:10-11:00 MWF in 206 Malott
office hours: TuTh 11:30-noon; W 11:10-noon

This course will provide an introduction to model theory with an emphasis on the algebraic aspects of the subject. After reviewing some preliminaries from first order logic (semantics, syntax, compactness, completeness), we will cover the fundamentals of the subject: the Lowenheim-Skolem theorem, categoricity, the Tarski-Vaught test, back and forth arguments, quantifier elimination, the space of types and the basics of stability, homogeneity and saturation, and indiscernibility. These techniques will be put to use to prove the Ax-Grothendick theorem for polynomial maps from \(\mathbb{C}^n\) to \(\mathbb{C}^n\) and the 0-1 law for graphs. We will prove quantifier elimination for the theories of algebraically closed, real closed, and differentially closed fields and use this to study the definable subsets of \(\mathbb{R}^n\) and \(\mathbb{C}^n\). The culmination of the course will be to prove Morley's Categoricity Theorem.

Students are assumed to have a solid undergraduate education in mathematics (in particular, they should be comfortable with basic concepts and definitions from topology, algebra and analysis - compactness, groups, rings, fields, etc.). Prior exposure to mathematical logic will be helpful but not essential.

The lectures in the course will be based on Marker's text:

Those enrolling in the course for a grade are expected to complete regular homework assignments. These assignments will be made in class approximately once a week and posted on the web.

Week 12: Homework 7 (due 5/4 at 11:59PM): 5.5.3, 5.5.4, 5.5.5.

Week 11:

Week 10:

Week 9: Homework 6 (due 4/15): 4.5.8 (all parts)

The great COVID-19 hiatus - no classes 3/16-4/3

Week 8: Homework 5 (due 4/8): 3.4.28, 3.4.35, 4.5.1; prove that RCF proves Rolle's Theorem for polynomials.

Week 7:

Week 6: Homework 4 (due 3/4): 3.4.10, 3.4.17 (all parts).

Week 5:

Week 4: Homework 3 (due 2/17): 3.4.3, 3.4.4, 3.4.6 from the text; prove that the theory of any Fraïssé limit has quantifier elimination.

Week 3: Homework 2 (due 2/10).

Week 2: Homework 1 (due 2/3): 2.5.19 from the text; show that if \(D\) is a nonprincipal ultrafilter on the prime numbers, then \(\prod_{p} \overline{\mathbb{F}_p} / D\) is isomorphic to \(\mathbb{C}\).

Week 1: