Phil 350A: Model Theory

Quarter: Fall
Instructor: Eric Pacuit
Instructor's Office Location: Room 92B, Building 90
Office Hours: Monday and Wednesday, 2:00PM - 3:00PM
Meeting Times: Monday and Wednesday, 11:00AM - 12:15PM
First Class: Monday, September 22, 2008
Location: Building 110, Rm 111A

Final Presentation

Details about the final presentation have been added. See below.

Overview

Course Description: Model theory is a branch of mathematical logic that studies properties of mathematical structures expressible in a formal language (eg., first-order logic). Model theorists have traditionally focused on two main themes: 1. start with concrete mathematical structures (such as the field for real numbers) and develop techniques to obtain new information about the structure and the definable sets, and 2. start with theories (sets of formulas) and prove general structural theorems about their models. This course will provide an introduction to the main techniques and results of the subject while emphasizing examples and applications to other areas.

Course Material: Topics to be covered include the compactness theorem, preservation theorems, quantifier elimination, Ehrenfeucth-Fraisse games, realizing and omitting types, saturated and homogeneous models, indiscernibles and partition theorems, and ultra-power constructions. Time permitting, we will also discuss a selection of the following more advanced topics: Morley's categoricity theorem, Lindstrom's Theorem characterizing first-order logic, nonstandard analysis and/or generalized quantifiers.

Prerequisites: The course assumes familiarity with the syntax and semantics of first-order logic and basic model-theoretic results (eg. Godel's completeness theorem). The formal prerequisites are Phil 150 and Phil 151/251, or equivalent.

Reading Material

The course is based on the following textbook.

W. Hodges, A Shorter Model Theory, Cambridge University Press, 1997
- You can purchase the book at the book store or through Amazon.
- You can find the errata for the book here.

We will also make use of the following texts (the books will be placed on reserve at Tanner Library).

D. Marker, Model Theory: An Introduction, Springer, 2002
K. Doets, Basic Model Theory, CSLI Lecture Notes, 1996

The following texts are classics (but still worth reading!). They will be placed on reserve at Tanner Library.

C. C. Chang and H. J. Keisler, Model Theory, Elsevier, 1973
J. Bell and A. Slomson, Models and Ultraproducts: An Introduction, Dover Publications, 1974
H. J. Keisler, Fundamentals of Model Theory, in The Handbook of Mathematical Logic, J. Barwise (ed.), 1984

The following texts are recommended for background reading:

Mathematical logic: H. Enderton, A Mathematical Introduction to Logic, Academic Press, 2001; R. Smullyan, First-Order Logic, Dover, 1995 (first edition: 1968); and H.-D. Ebbinghaus, J. Flum, and W. Thomas, Mathematical Logic, Springer, 1995
Algebra: S. Lang, Algebra, Springer, 2002

The following are recommended for further investigation:

Finite Model Theory: E. Gradel et al., Finite Model Theory and its Applications, Springer, 2007
Modal Logic: V. Goranko and M. Otto, Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic, P. Blackburn, J. van Benthem, F. Wolter (eds.), Kluwer, 2007
Non-standard Analysis: R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Springer, 1998
Abstract Model Theory: J. Vaananen, Barwise: Abstract model theory and generalized quantifiers, Bulletin of Symbolic Logic, vol. 10:1, 2004
Craig's Interpolation Theorem: Special issue of Synthese dedicated to this important theorem.

Check out the following online resources:

Stanford Encyclopedia of Philosophy entry on Model Theory by Wilfred Hodges
Stanford Encyclopedia of Philosophy entry on First-Order Model Theory by Wilfred Hodges
Modnet: European research training network in model theory
Short collection of notes on model theory by D. Zambella

Schedule

Below is a tentative outline for the course which will be updated as the course proceeds. The reading refers to sections from A Shorter Model Theory by Wilfred Hodges. Note that we will cover material from Chapter 4 as needed.

Tentative Schedule
Date	Lecture Topic	Reading	Notes
9/22	Introduction, Structures and Isomorphisms	1.1-1.2
9/24	Structures and Languages	1.3-1.4
9/29	Classifying Classes of Structures, Basic Concepts	1.5, 2.1 - 2.2	Problem Set
10/1	Basic Concepts, Classifying Maps	2.2 - 2.3
10/6	Hintikka Sets, Preservation Theorems	2.3 - 2.5
10/8	Quantifier Elminiation	2.6-2.7	Problem Set Due
10/13	More on Quantifier Elimination	2.7
10/15	Skolemization, Back-and-Forth	3.1-3.2
10/20	Back-and-Forth Equivalence	3.2
10/22	EF Games	3.3 & Vaananen's book
10/27	Compactness and Types	5.1-5.2
10/29	Types and Stone Space	5.2 - 5.3
11/3	More on Stone Space and Amalgamation	5.3, Marker Section 4.1
11/5	Amalgamation Theorems	5.4-5.5
11/10	Indiscernibles and Partition Theorems	5.6, additional material
11/12	Fraisse's Construction, Omitting Types	6.1 - 6.2
11/17	Saturated and Homogeneous Models	8.1 - 8.2
11/19	Ultrapower Constructions	8.5, TBA
11/24	No Classes (Thanksgiving)
11/26	No Classes (Thanksgiving)
12/1	Lindstrom's Theorem	TBA
12/3	Additional Topics	TBA

Grading

Regular homeworks will be assigned and graded (roughly, every 1-2 weeks depending on the material we cover). The final grade will be based on your homework and a final presentation on a topic of your choosing. Details about the presentation will be provided later in the quarter.

Final Presentation

The final grade will be based in part on a final presentation. You may select any of the following topics:

(Cole) Prove the Lowenheim-Skolem Theroem following Section 5.3 from Vaananen's book
(Wes) Prove the Scott Isomorphism Theorem following pgs. 57 - 60 from Marker.
(Thomas) Prove the Parris-Harrington Theorem, see section 5.4 in Marker
Discuss the paper Finite information logic by Rohit Parikh and Jouko Vaananen
Discuss the paper Modal logic and invariance by Johan van Benthem and Denis Bonnay
(Nal) Discuss the paper Lindstrom theorems for fragments of first order logic by Balder ten Cate, Johan van Benthem and Jouko Vaananen.
Discuss Craig's Interpolation Theorem: follow Harmonious logic: Craig’s interpolation theorem and its descendants by Sol Feferman
Discuss the paper Barwise: Abstract model theory and generalized quantifiers by Jouko Vaananen
(Osprey) Give a report on the book The Birth of Model Theory by Calixto Badesa

If there is a topic not on this list that you are interested in, please contact me. You should let me know which topic you will present in the next couple weeks (before November 3). Each presentation should take 30-40 minutes including questions. The presentations will take place during exam week.

Notes/Handouts

Problem Set (pdf)
Due Wednesday, October 8