The Birth of Model Theory: Löwenheim's Theorem in the Frame of the Theory of Relatives
What do the theorems of Gödel–Deligne, Chevalley–Tarski, Ax–Grothendieck, Tarski–Seidenberg, and Weil–Hrushovski have in common? And what do they have to do with the book under review? Each of these theorems was proven by techniques in a particular mathematical area and by model theoretic methods. In fact, these model theoretic methods often show a pattern that extends across these areas. What are model theoretic methods? Model theory is the activity of a “selfconscious” mathematician. This mathematician distinguishes an object language (syntax) and a class of structures for this language and “definable” subsets of those structures (semantics). Semantics provides an interpretation of inscriptions in the formal language in the appropriate structures. At its most basic level this allows the recognition that syntactic transformations can clarify the description of the same set of numbers. Thus, x − 3x < −6 is rewritten as x < −2 or x > 3; both formulas define the same set of points if they are interpreted in the real numbers. After clarifying these fundamental notions, we give an anachronistic survey of three themes of twentieth century model theory: the study of a) properties of first order logic, b) specific first order theories, and c) classification of first order theories. In this survey we will highlight the increasing interactions between “pure” model theory and the analysis of topics in core mathematics. Then we return to the book at hand and see how Löwenheim’s 1915 paper [Löw67] set the stage for these developments. On the syntactic side, first order logic contains several logical symbols: equality; the quantifiers ∀, ∃; a sequence of variables, vi; and the logical connectives ∧,¬,∨. A vocabulary τ for first order logic consists of a collection of relation and function symbols that is appropriate for the area of mathematics being formalized. Terms of the language based on τ are built up inductively from constant and variables using the function and relation symbols of the language. An atomic formula has the form R(t1, . . . tn) where R is a relation symbol with n-arguments and the ti are terms. The first order language associated with τ is the least set of formulas containing the atomic τ -formulas and closed under the Boolean operations and quantification over individuals. Formulas in which each variable is bound by a quantifier are called τ -sentences. On the semantical side, a structure for this vocabulary consists of a domain (also denoted A) and a relation on A for each relation symbol with n arguments (a function from A into A for each function symbol with n arguments) and with equality interpreted as identity. In a structure A, each variable-free term denotes an element of A. Truth in a structure is also defined inductively: if there are no quantifiers, a formula is true in the structure A if the interpretation of the terms lies in the relation which is the interpretation of the formula. The truth of Boolean combination and or quantified formulas is defined in the natural way; e.g.,