**Phil 350A: Model Theory**

[Overview | Reading Material | Schedule | Grading | Student Presentation | Notes/Handouts]

**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.
We will also make use of the following texts (the books will be placed on reserve at Tanner Library).
The following texts are classics (but still worth reading!). They will be placed on reserve at Tanner Library.
The following texts are recommended for background reading:
The following are recommended for further investigation:
Check out the following online resources:

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

- 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

: 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*Mathematical logic*: S. Lang, Algebra, Springer, 2002*Algebra*

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

- 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.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: Craigâ€™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

**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**