MMath Mathematics

Model theory deals with those properties of mathematical structures which can be expressed using formulae of a formal predicate language. One theme is the investigation of the class of those structures which are the models of a set of sentences from predicate logic. Another theme is the analysis of definability in individual structures and the use of elementary extensions to produce non-standard elements.  An example of the latter is producing infinitesimals in extensions of the set of real numbers, an infinitesimal being an element x satisfying x>0 and x<1/n for every positive integer n.  The Compactness Theorem says that, since this (infinite) set of conditions is finitely satisfied in the field of real numbers, there is an elementary (=nice) extension of the reals which contains such an element. 

We will introduce and use the ultraproduct construction, which is a way of producing, from a family of structures, an “average” structure (and it can be used to give a neat proof of the Compactness Theorem).  Back-and-forth is an inductive method for building up maps between structures.  We’ll use it to investigate the Random Graph (a countably infinite graph which contains a copy of every countable graph).

Types are descriptions, using the formal language, of elements and potential elements.  We will see how these can be used to try to classify the models of a theory.  A particularly nice case is when a theory is countably categorical, meaning that it has just one countable model up to isomorphism; we will characterise these theories in various ways (one being that the automorphism group of a countable model has only finitely many orbits on n-tuples). Compactness of the space of types is a key ingredient in the proof of these characterisations.

Some acquaintance with algebraic structures (groups and fields – these will be used as examples) and with predicate languages and their structures, such as may be obtained from the above courses, will be assumed (though there will be a brief recap at the beginning of the course unit).  It would be helpful, though not necessary, to have seen some basic model theory (such as in MATH3301).

Students are not permitted to take, for credit, MATH43051 in an undergraduate programme and then MATH63051 in a postgraduate programme at the University of Manchester, as the courses are identical.

In this course, the student will learn how various important kinds of mathematical objects, such as (algebraically closed) fields, groups and dense linear orders, can be studied using first-order logic. The student will see how these kinds of mathematical objects can be viewed as first-order structures and, in doing so, how tools from first-order logic give us a number of properties of such mathematical objects. The course will also investigate certain applications of these properties to computer science.

On successful completion of this course unit students will be able to: 

• describe the ultraproduct construction and apply it.
• produce proofs of results about filters, ultraproducts and related concepts.
• state and apply Los' Theorem.
• apply the back-and-forth technique.
• prove results relating theories and their models.
• prove results about 0/1 laws and analyse examples.
• analyse examples model-theoretically, for example determining definable sets, types, automorphisms; categoricity and other properties.
• produce examples and counterexamples with specified model-theoretic properties.
• prove basic results about definable sets and types and use the technique of realising types.

1. Review of predicate logic and examples of structures. [3 lectures]
2. Back-and-Forth technique; the Random Graph; 0/1 laws. [3 lectures]
3. Ultraproducts and Los’ Theorem; definable sets. [4 lectures]
4. The space of types; saturated structures. [5 lectures]
5. Countable categoricity. [3 lectures]
6. The models of a theory; examples. [4 lectures]

• Mid-semester coursework: two take home tests weighting 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

I will provide full course notes but there are quite a few texts on model theory around.  For example, those below, but they are aimed at graduate students so don’t expect to move quickly when reading them.   There are also sets of lecture notes on the web.  So you can browse around and see what you like/what’s helpful.

• David Marker, Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002. viii+342 pp.
• C.C.Chang and H.J.Keisler, Model Theory, various editions (though it goes overboard on ultraproducts).
• Wilfrid Hodges, Model theory. Encyclopedia of Mathematics and its Applications, 42. Cambridge University Press, Cambridge, 1993. xiv+772 pp.  “A Shorter Model Theory” is the cut-down, student, version

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further update.