Model Theory: The Algebraic Basics

This textbook is a gently-paced, comprehensive introduction to model theory suitable for students of philosophy, linguistics, computer science, or mathematics who specialise in logic. The book assumes no preliminary knowledge of logic or algebra beyond the barest rudiments of set theory. After a thorough discussion of the elements of model theory (languages, structures, morphisms), the reader is led into a study of key model-theoretic properties (quantifier elimination, model-completeness), ideas (types, Morley rank) and classic applications (to groups, vector spaces, algebraically closed and real closed fields). The book also discusses applications of model-theoretic ideas to philosophy and mathematical social science. More than 300 exercises with full solutions are designed to consolidate understanding and build confidence. A set-theoretic appendix covers in detail the theory of ordinal and cardinal numbers needed in the second half of the book.

1 Some concrete structures.- 2 Languages and Structures.- 3 Theories and models.- 4 Morphisms, substructures and extensions.- 5 Ultraproducts.- 6 Essential field theory.- 7 Complete theories.- 8 Model-completeness and quantifier elimination.- 9 Types.- 10 Algebraically closed fields and algebraic geometry.- 11 Real closed fields.- 12 Fraisse limits and measurement scales.- Further Reading.- Appendix A. Set-theoretic background.- Appendix B. Solutions to Exercises.- Bibliography.- Index.

Davide Rizza is currently Head of Philosophy and Associate Professor at the University of East Anglia, School of Politics, Philosophy and Area Studies. He received his PhD in 2009 from the University of Sheffield and joined the University of East Anglia in the same year. His research work focusses primarily on philosophy of mathematics and the history of logic. His contributions to international journals like Philosophy of Science, The Philosophical Quarterly, Erkenntnis, Synthese, span a broad range of topics including the foundations of measurement, the foundations of probability, the impact of logical and geometrical methods on mathematical modelling in social science.