The Model Theory of Mathematics

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. We call a set of sentences in a formal language a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory.Model theory recognizes and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang & Keisler (1990)universal algebra + logic = model theory.Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997):model theory = algebraic geometry − fields,although model theorists are also interested in the study of fields. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis.In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. The most prominent professional organization in the field of model theory is the Association for Symbolic Logic.This book is designed to be a general overview of the topic and provide you with the structured knowledge to familiarize yourself with the topic at the most affordable price possible. The level of discussion is that of Wikipedia. The accuracy and knowledge is of an international viewpoint as the edited articles represent the inputs of many knowledgeable individuals and some of the most currently available general knowledge on the topic, based on the date of publication.