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Focusing on significant results in set theory from the 20th century, this second edition explores the independence of the continuum hypothesis and the axiom of choice. It is tailored for graduate students and researchers across mathematics, logic, philosophy, and computer science. The updated content features expanded introductory material, new chapters, and a category theory appendix, along with recent developments and numerous exercises. This edition enhances accessibility for students in logic and set theory with additional corrections and updated background information.

Geared toward first-year graduate students, this text assumes only an acquaintance with the rudiments of set theory to explore homogeneous universal models, saturated structure, extensions of classical first-order logic in terms of generalized quantifiers and infinitary languages, and other topics. Numerous exercises appear throughout the text. 1974 edition.

The aim of this book is to provide an elementary exposition of some of the basic concepts of model theory. Model theory, which can be described briefly as the study of the relationship between formal languages and abstract structures, covers a very wide field and it is not possible to compress it into one volume. We have chosen as our theme the ultraproducts construction. We hope this book we be of use to undergraduate and practicing mathematicians