Beginning Model Theory: The Completeness Theorem and Some Consequences

An essential feature of mathematical logic distinguishing it from other branches of mathematics is the explicit study of the language in which mathematics may be described. This book is intended as an introduction to that aspect of logic which concerns the relationship between, on the one hand, mathematical theories, described in a formal language, and on the other, mathematical structures realizing those theories. The text is based on a series of lectures given in Oxford each year from 1970 to 1973. The course was intended for second-year undergraduates reading Mathematics and Philosophy who had already had an introductory logic course. As a consequence, we assume here familiarity with some elementary concepts and results. For instance, the reader should ideally have studied the propositional (sentential) calculus, including the notions of a propositional language, a tautology, and a formal system of axioms and rules for generating all tautologies as theorems. Acquaintance with the formalization of statements involving quantifiers is desirable though not strictly essential. Mathematical examples are given for which some familiarity with elementary abstract algebra is necessary. A summary of some set-theoretic concepts used in the text is given in the introduction.