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: TW 11 - 12:30; see the Fields website for zoom details.

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. 11; languages and structures, formulas, interpretations and theories - Chapter 1, Marker

Week 2: Jan. 18; homomorphisms, embeddings, elementarity, Downward-Lowenheim Skolem, ultraproducts - Chapter 2, Marker and lecture notes

Week 3: Jan. 25; 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

Week 4: Feb. 1: Ehrenfeucht-Fraisse games and elementarity; Lindstrom's theorem - lecture notes and a different presentation in chapter 2 of Marker

Week 5: Feb. 8: Model theoretic forcing and the Henkin construction; omitting types - lecture notes and again, a different presentation in chapter 2 of Marker

Week 6: Feb. 22: countable saturated models, homogeneity, universality; Ryll-Nardzweski; general saturation; Skolem functions - most of this is in chapter 4 of Marker

Week 7: Mar. 1: imaginaries and interpretability, T^eq - lecture notes

Week 8: Mar. 8: 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

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: TW 11 - 12:30; see the Fields website for zoom details.

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. 11; languages and structures, formulas, interpretations and theories - Chapter 1, Marker

Week 2: Jan. 18; homomorphisms, embeddings, elementarity, Downward-Lowenheim Skolem, ultraproducts - Chapter 2, Marker and lecture notes

Week 3: Jan. 25; 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

Week 4: Feb. 1: Ehrenfeucht-Fraisse games and elementarity; Lindstrom's theorem - lecture notes and a different presentation in chapter 2 of Marker

Week 5: Feb. 8: Model theoretic forcing and the Henkin construction; omitting types - lecture notes and again, a different presentation in chapter 2 of Marker

Week 6: Feb. 22: countable saturated models, homogeneity, universality; Ryll-Nardzweski; general saturation; Skolem functions - most of this is in chapter 4 of Marker

Week 7: Mar. 1: imaginaries and interpretability, T^eq - lecture notes

Week 8: Mar. 8: 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. 15: applications with
indiscernibles, omega-stable theories - Marker's book and lecture notes

Week 10: Mar. 22: Vaughtian pairs, Morley rank - Marker's book and a different presentation in the lecture notes

Week 10: Mar. 22: Vaughtian pairs, Morley rank - Marker's book and a different presentation in the lecture notes

Week 11: Mar. 29: Morley's categoricity
theorem