Description
Book SynopsisLowenheim's theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory - that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with L
Trade Review"A first-rate contribution to the history and philosophy of logic, this is scholarship at its best. It is, to my knowledge, the first book in the history of logic that focuses completely on a single result. Very original in approach and conception, it goes against the grain of much recent scholarship. Given the complexity of the subject, Badesa could not have done a better job of being clear and making the presentation accessible."
—Paolo Mancosu, University of California, Berkeley"
The Birth of Model Theory represents a long overdue, in-depth analysis and exposition of one of the most important results in mathematical logic. There are hardly any informed, sustained treatments of Löwenheim's work to be found in the literature. This well-written book should fill this gap."
—Richard Zach, University of Calgary"This book will be extremely useful to those seeking to make sense of Löwenheim's work and those seeking to put it into its historical context. Calixto Badesa draws well-supported conclusions that contradict the entire modern body of scholarship on the topic."
—Shaughan Lavine, University of ArizonaTable of ContentsPreface ix Chapter 1. Algebra of Classes and Propositional Calculus 1 1.1 Boole 1 1.2 Jevons 10 1.3 Peirce 12 1.4 Schroder 17 Chapter 2. The Theory of Relatives 31 2.1 Introduction 31 2.2 Basic concepts of the theory of relatives 33 2.3 Basic postulates of the theory of relatives 40 2.4 Theory of relatives and model theory 51 2.5 First-order logic of relatives 66 Chapter 3. Changing the Order of Quantifiers 73 3.1 Schroder's proposal 73 3.2 Lowenheim's approach 81 3.3 The problem of expansions 87 3.4 Skolem functions 94 Chapter 4. The Lowenheim Normal Form 107 4.1 The Lowenheim normal form of an equation 107 4.2 Comments on Lowenheim's method 113 4.3 Conclusions 122 Chapter 5. Preliminaries to Lowenheim's Theorem 129 5.1 Indices and elements 129 5.2 Types of indices 132 5.3 Assignments 135 5.4 Types of equations 138 Chapter 6. Lowenheim's Theorem 143 6.1 The problem 143 6.2 An analysis of Lowenheim's proof 148 6.3 Reconstructing the proof 191 Appendix. First-Order Logic with Fleeing Indices 207 A.1 Introduction 207 A.2 Syntax 207 A.3 Semantics 211 A.4 The Lowenheim normal form 217 A.5 Lowenheim's theorem 220 References 227 Index 237
