This course will be given in May and June 2017 at Peking University, Beijing. Its aim is to present the main tools and the basic results in Model Theory. It is recommended to students interested in Mathematical Logic and particularly to those who wish to attend Zoé Chatzidakis lectures on Model theory of valued fields and applications. No specific knowledge in Logic is required, but we expect some familiarity with elementary Mathematics: naive set theory, standard algebraic structures, linear algebra, and elementary analysis. Thus the course is suitable for third or fourth year undergraduate students.  

In the first part of the course we will introduce the basic notions of first order Logic: languages, formulas, theories, structures and models; morphisms, submodels, elementary equivalence, elementary extensions, ultra-products, complete theories; inference rules. We will prove the main theorems: compactness, completeness, Löwenheim-Skolem, Łoś … We will need some notions from axiomatic set theory, mainly ordinal and cardinal numbers.If we have enough time, we will give a general presentation of set theory. Otherwise we will just give the needed results.

The second part of the course will be devoted to some further topics in Model Theory, chosen among the following: diagrams, interpolation and definability theorems, Fraïssé's back and forth method, quantfier elimination, preservation theorems, categoricity, saturated models, omitting types theorem. We shall give many examples of theories and structures: groups, rings, fields, ordered structures, Peano arithmetic…

Bibliography:

• R. Cori & D. Lascar, Mathematical Logic. A Course with Exercises, Oxford University Press, 2 volumes, 2000 & 2001.

• W. Hodges, Model Theory, Cambridge Univesity Press, 1993.

• W. Hodges, A Shorter Model Theory, Cambridge Univesity Press, 1997.

• D. Marker, Model Theory: An Introduction, Springer, 2002.

Timetable:

May 2  15:00–17:00

May 3  10:00—12:00

May 4  8:00—10:00

May 9   15:00–17:00

May 10  10:00—12:00

May 11   8:00—10:00

May 12  10:00—13:00

May 16  15:00–17:00

May 17  10:00—12:00

May 18   8:00—10:00

May 19  10:00—13:00

May 23  15:00–17:00

May 25   8:00—10:00

June 1   8:00—10:00

June 2   13:00—15:00

June 6   15:00–17:00

June 8    8:00—10:00

June 13  15:00–17:00

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