University of Manchester

Mike Prest

Teaching

**Office Hour: **If you want to discuss something with me then send me an email and we can arrange a time. You can also ask questions by email (as well as in person). If you turn up at my office and I am free and not in the middle of something that I need to push on with, then I will be happy to talk to you/answer your questions.

My room is Alan Turing Building 1.120

Internal tel. 55875

email mprest@manchester.ac.uk

MATH43051/63051 Model Theory This course has been modified a bit since last year, the main change being the inclusion of ultraproducts and Los' Theorem.

WARNING! This is, in some sense, a hard course; to do well you have to understand the concepts! If you dislike proofs, or think that `model' refers to modelling in the sense of applied mathematics, or rely on memorisation of bookwork and/or algorithms to get through exams, then this is not the course for you.

**Outline**

Weeks 1,2,3: Ultraproducts and Los' Theorem

Week 4: Theories and their models; Downwards Lowenheim-Skolem

Weeks 5,7: Dense Linear Orders; Back-and-forth arguments; The Random Graph; 0/1 laws for finite structures

Week 8: Method of Diagrams; Upwards Lowenheim-Skolem

Weeks 9,10: Types; realising types; saturated models; the space of types

Week 11: Aleph_0 categoricity

Week 12: Consolidation and review

**Notes for the course:** here are the notes for Sections 1-3 and for the Appendix on Languages and Structures. These notes contain definitions, statements of results, proofs of those (in some cases in full detail, in other cases just the outline) plus various comments and illustrations/exercises. In the lectures I will give some more details and further examples, as well as more explanation. The material in the Appendix is kind of assumed background from Predicate Logic; `kind of' because not all of you will have done a course in Predicate Logic, so I will introduce the material that I need, but briefly and a bit informally. Sometimes the informal idea will be enough but sometimes you will have to refer to the Appendix for the precise definitions and the details.
If you spot any typos, errors, ... in the notes or elsewhere then do let me know.

**Exercises**

**Review exercises**: You can ignore these if you're happy with the ideas from Predicate Logic; they provide some practice if you're not. The first three sets are short exercises on the basics of predicate languages; the set after that contains rather longer exercises.

Exercises on Terms, Formulas and Sentences (with answers on the second page)

Exercises on Reading formulas in Structures (with answers on the second page)

Exercises on Relations and Definable Sets (with answers on the second page)

Exercises on Languages and Interpretations and with answers and the construction tree.

**Courseworks from 2016**

(Optional) Coursework 3 If you do all three courseworks, then the best two marks will count.

**Courseworks from 2015**

(Optional) 2015 Coursework 3 If you do all three courseworks, then the best two marks will count.

**Exams**
The examination for the course is 2.5 hours duration; you should answer 3 out of 4 questions (if you answer all four, your best three answers will be counted).
Past exam papers should be on the university website . And here are (partial) solutions to the January 2015 exam and some general comments and solutions to the January 2016 exam. And, finally, some general comments on the January 2017 exam.

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