![The Birth of Model Theory : Lowenheim's Theorem in the Frame of the Theory of Relatives - Calixto Badesa](https://www.booktopia.com.au/covers/big/9781400826186/7734/the-birth-of-model-theory.jpg)
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The Birth of Model Theory
Lowenheim's Theorem in the Frame of the Theory of Relatives
By: Calixto Badesa
eBook | 10 January 2009
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256 Pages
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Löwenheim'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 Löwenheim's theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem--a logician raised in the algebraic tradition, like Löwenheim--appears to have attributed to him. In The Birth of Model Theory, Calixto Badesa provides both the first sustained, book-length analysis of Löwenheim's proof and a detailed description of the theoretical framework--and, in particular, of the algebraic tradition--that made the theorem possible.
Badesa's three main conclusions amount to a completely new interpretation of the proof, one that sharply contradicts the core of modern scholarship on the topic. First, Löwenheim did not use an infinitary language to prove his theorem; second, the functional interpretation of Löwenheim's normal form is anachronistic, and inappropriate for reconstructing the proof; and third, Löwenheim did not aim to prove the theorem's weakest version but the stronger version Skolem attributed to him. This book will be of considerable interest to historians of logic, logicians, philosophers of logic, and philosophers of mathematics.
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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
ISBN: 9781400826186
ISBN-10: 1400826187
Published: 10th January 2009
Format: ePUB
Language: English
Number of Pages: 256
Audience: College, Tertiary and University
Publisher: Princeton University Press
Edition Type: Course Book
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