Syllabus and literature for lectures on

## "Model theory", Jan Krajicek

Course code: NMAG407

### Exam questions.

Master Thesis topics: If you contemplate the idea to write an MSc Thesis in logic, or specifically in model theory, talk to me: I put into the SIS only two topics in logic (one of them is in model theory) but there is a variety of possible ones.
Student logic seminar.

The course covers main topics of model theory with an emphasis on examples and methods important for applications of model theory in algebra, geometry and number theory.

### Syllabus

• Structures and an interpretation of a language. Tarski's truth definition.

• Embeddings and isomorphisms of structures, substructures. Elementary equivalence and the Ehrenfeucht-Fraisse game, example: the theory DLO of dense linear orderings without end-points. Preservation theorems, the diagram of a structure.

• Algebraic examples: ordered real closed fields (RCF) and algebraically closed fields (ACF_p and ACF_0), vector spaces over a fixed field, groups.

• The compactness theorem and its applications: elementary extensions, the upward Lowenheim-Skolem theorem, non-standard models of RCF and of the ring of integers. A transfer theorem from ACF_p to ACF_0. The Ax-Grothendieck theorem.

• Complete theories and kappa-categorical theories. Vaught's test. Skolemization and the downward Lowenheim-Skolem theorem.

• Quantifier elimination and its proofs for DLO and ACF. The strong minimality of ACF and the o-minimality of RCF (assuming QE for RCF).

• Strongly minimal theories and their associated (pre)geometries.

• Types, their realization and omitting. The Stone space of complete types, algebraic ex.: Zariski spectrum. Isolated types and the Omitting types theorem. MacDowell-Specker's theorem.

• kappa-saturated structures and their existence. Countable saturated structures and the size of the Stone space. Saturation of ultraproducts.

• The number of non-isomorphic models of a given cardinality. Vaught's conjecture. Morley's categoricity theorem.

### Literature:

#### Main source

• D. Marker, Model Theory: An Introduction (Springer, 2002).
(The MFF library has an access to the e-version of the book. Almost all material can be also found in Marker's lecture notes (online) listed below.)

#### Other classics

• C.C.Chang, J.H.Keisler: Model theory, NHPC 1973.

• W. Hodges: Model Theory, Cambridge Univ. Press, 1993.

• W. Hodges, Shorter Model Theory (CUP, 1997).

#### Lecture notes on the web

• D.Marker's lectures at the MSRI.

• D.Marker's course at Orsay in 2010.

• A.Pillay's lecture notes, U. Leeds.

• A.Wilkie's lecture notes, U. Manchester.