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:
-
Model theory: an introduction, D. Marker.
This course will serve as the main reference for the algebraically oriented part of the course.
(Beware of the typos! Here is a list of
errata.)
-
Those with a limited background in logic may find Lou van den Dries's
notes on logic
helpful.
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.
- Monday (4/27): Partition theorems
- Wednesday (4/29): Order indiscernibles and Ehrenfreucht-Mostowski models
- Friday (5/1): Order indiscernibles and \(\omega\)-stability
Week 11:
- Monday (4/20): quantifier elimination for differentially closed fields
- Wednesday (4/22): Vaughtian pairs
- Friday (4/24): Vaught's two cardinal theorem
Week 10:
- Monday (4/13):
- Wednesday (4/15): universal models
- Friday (4/17): uses of saturation
Week 9: Homework 6 (due 4/15): 4.5.8 (all parts)
- Monday (4/6): omitting types; prime and atomic models
- Wednesday (4/8): \(\aleph_0\)-homogeneous models; \(\omega\)-stability
- Friday (4/10): homogeneous and saturated models
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.
- Monday (3/9): types, elementary extensions, and automorphisms
- Wednesday (3/11): Stone spaces, isolated types, scattered compact spaces
- Friday (3/13): some examples: types in DLO and ACF
Week 7:
- Monday (3/2): quantifier elimination for RCF
- Wednesday (3/4): semi-algebraic sets and functions: continuity and selections
- Friday (3/6): cell decomposition in RCFs
Week 6: Homework 4 (due 3/4): 3.4.10, 3.4.17 (all parts).
- Monday (2/24): February break
- Wednesday (2/26): real closed fields, part 2.
- Friday (2/28): uniqueness of real closures of ordered fields.
Week 5:
- Monday (2/17): quantifier elimination for ACF; Zariski closed and
constructible sets
- Wednesday (2/19): Hilbert's Nullstellensatz; definable functions in ACF are quasirational
- Friday (2/21): formally real fields; real closed fields, part 1.
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.
- Monday (2/10): quantifier elimination for DLO; criteria for quantifier elimination.
- Wednesday (2/12): the theory of Divisible Abelian Groups (DAG) and its quantifier elimination.
- Friday (2/14): quantifier elimination for Ordered Divisible Abelian Groups (ODAG).
Week 3: Homework 2 (due 2/10).
- Monday (2/3): Ehrenfeucht-Fraïssé Games.
- Wednesday (2/5): Completeness of the theory of Discrete Linear Orders.
- Friday (2/7): Scott-Karp Analysis.
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 \(\mathbb{C}\).
- Monday (1/27): the upward Löwenheim-Skolem theorem; categoricity and Vaught's Test for complete theories;
the Ax-Grothendick Theorem.
- Wednesday (1/29): the downward Löwenheim-Skolem theorem;
\(\aleph_0\)-categoricity and the back and forth argument.
- Friday (1/31): The Random Graph and the 0-1 law.
Week 1:
- Wednesday (1/22): overview of the course.
- Friday (1/24): Henkin's construction.