The authors treat the principles of the axiomatic method from the set theoretic point of view. Starting with arbitrary sets of axioms in propositional language, the work progresses through finite first order formulae, i.e. ordinary predicate calculus, and into infinite first order and finite higher order formulae. Principal topics include the notions of consequence and definability. The book concludes with two appendices: one explaining in general terms how model theory has led to results on particular axiomatic systems currently studied in mathematics (and to the precise formulation of ideas which previously played only a heuristic role); the other is intended for readers interested in the philosophical foundations of mathematics. The main differences from the traditional style of text are: a sharp separation between the mathematical developments and the formulation of foundational questions (allowing the reader to concentrate specifically on considerations which are essential either to foundations or to mathematical practice); a restriction of the main text to model theory without introducing notions from proof theory or from recursion theory; the introduction of the language of ordinary first order predicate logic as well as so-called higher order languages and languages with infinitely long formulas (a practice which avoids the mistake of giving a privileged role to first order logic in model theory); and in contrast to most introductions to mathematical logic, the inclusion of invariant definitions in the main text. Since the book has been designed as a genuine textbook, not as a compendium of the authors' recent research, the style is systematic, compact, and includes many applications in the form of exercises. Philosophically, the authors present a coherent point of view with explicit formulation of set theoretic and formalist foundations. The book is a revised English version of the widely acclaimed French original.

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