Introduction to Model theory

These notes aim at giving the basic definitions and results from model theory. My intention in writing them, is that they should provide the reader with many examples, even with some proofs, and contain most of the definitions. Differences in vocabulary can be quite an obstacle to mutual understanding, and having the terminology written up somewhere should prove helpful. An index is included at the end. People interested in reading more should consult standard model theory books. For instance: C.C. Chang, H.J. Keisler, Model Theory, North-Holland Publishing Company, Amsterdam 1973; W. Hodges, A shorter model theory, Cambridge University Press, 1997. The notes are organised as follows. Chapter 1 gives the very basic definitions of languages, structures and satisfaction. Chapter 2 gives more definitions and some important theorems, such as the compactness theorem. Section 3 introduces ultraproducts and their properties. In section 4, we give a short proof of the decidability of the theory of the field of real numbers.