Math 711 - Model Theory

This course will be a classical model theory course which will touch on all the basic themes with a modern lens.  In addition to emphasizing standard results like completeness and compactness, the framework will be broadened from the start to include the multi-sorted setting which will enable a naturual discussion of quotient objects via imaginaries. Abstract model theoretic theorems such as the Beth definability theorem and Lindstrom's characterization of first order logic will be included in a practical setting. Applications of the Henkin construction as well as the abstract role of ultraproducts will be emphasized.  The course will end with a proof of Morley's categoricity theorem in the style of Baldwin-Lachlan with some emphasis on the geometric nature of modern treatments.

Lecture times: TTh 12:30 - 2 in HH 410

Recommended text: Model theory: an introduction, by David Marker

Office hours: by appointment

Course evaluation: There will be 5 assignments (roughly one every two weeks) and a final project. The assignments will be worth 50% of the grade and the take-home final/project will also be worth 50%.

I will try to keep a running list of topics covered approximately week by week with pointers to material in Marker's book and other sources.

Week 1: Jan. 6; languages and structures, formulas, interpretations and theories - Chapter 1, Marker

Week 2: Jan. 13; homomorphisms, embeddings, elementarity, Downward-Lowenheim Skolem, ultraproducts - Chapter 2, Marker and lecture notes

Week 3: Jan. 20; atomic and elementary diagrams, upward Lowenheim-Skolem, the logic topology and types, quantifier elimination; we are somewhere between chapters 2, 3 and 4 of Marker's book

Assignment 1 and solutions

Week 4: Jan. 27: Ehrenfeucht-Fraisse games and elementarity; Lindstrom's theorem - lecture notes and a different presentation in chapter 2 of Marker

Week 5: Feb. 3: Model theoretic forcing and the Henkin construction; omitting types - lecture notes and again, a different presentation in chapter 2 of Marker 

Week 6: Feb. 10: countable saturated models, homogeneity, universality; Ryll-Nardzweski; general saturation; Skolem functions - most of this is in chapter 4 of Marker

Here is a list of possible projects for the presentations at the end of the course. You may choose from this list or pick you own project but if you do the latter, make sure it is approved by me. 

Week 7: Feb. 24: imaginaries and interpretability, T^eq - lecture notes 

Week 8: Mar. 3: partition theorems (Ramsey's theorem, Erdos-Rado), indiscernibles - most of this is in Marker's book; see Morley's presentation from his logic colloquium notes, 1967

Week 9: Mar. 10: applications with indiscernibles, omega-stable theories - Marker's book and lecture notes

Week 10: Mar. 17: Vaughtian pairs, Morley rank - Marker's book and a different presentation in the lecture notes

Week 11: Mar. 24 Finishing up and final project presentations

Week 12: Mar. 31, cont'd