Additive and Polynomial Representations

¿PrefaceMathematical BackgroundSelecting Among the ChaptersAcknowledgmentsNotational Conventions1. Introduction 1.1 Three Basic Procedures of Fundamental Measurement 1.1.1 Ordinal Measurement 1.1.2 Counting of Units 1.1.3 Solving Inequalities 1.2 The Problem of Foundations 1.2.1 Qualitative Assumptions: Axioms 1.2.2 Homomorphisms of Relational Structures: Representation Theorems 1.2.3 Uniqueness Theorems 1.2.4 Measurement Axioms as Empirical Laws 1.2.5 Other Aspects of the Problem of Foundations 1.3 Illustrations of Measurement Structures 1.3.1 Finite Weak Orders 1.3.2 Finite, Equally Spaced, Additive Conjoint Structures 1.4 Choosing an Axiom System 1.4.1 Necessary Axioms 1.4.2 Nonnecessary Axioms 1.4.3 Necessary and Sufficient Axiom Systems 1.4.4 Archimedean Axioms 1.4.5 Consistency, Completeness, and Independence 1.5 Empirical Testing of a Theory of Measurement 1.5.1 Error of Measurement 1.5.2 Selection of Objects in Tests of Axioms 1.6 Roles of Theories of Measurement in the Sciences 1.7 Plan of the Book Exercises2. Construction of Numerical Functions 2.1 Real-Valued Functions on Simply Ordered Sets 2.2 Additive Functions on Ordered Algebraic Structures 2.2.1 Archimedean Ordered Semigroups 2.2.2 Proof of Theorem 4 (Outline) 2.2.3 Preliminary Lemmas 2.2.4 Proof of Theorems 4 and 4' (Details) 2.2.5 Archimedean Ordered Groups 2.2.6 Note on Hölder's Theorem 2.2.7 Archimedean Ordered Semirings 2.3 Finite Sets of Homogeneous Linear Inequalities 2.3.1 Intuitive Explanation of the Solution Criterion 2.3.2 Vector Formulation and Preliminary Lemmas 2.3.3 Proof of Theorem 7 2.3.4 Topological Proof of Theorem 7 Exercises3. Extensive Measurement 3.1 Introduction 3.2 Necessary and Sufficient Conditions 3.2.1 Closed Extensive Structures 3.2.2 The Periodic Case 3.3 Proofs 3.3.1 Consistency and Independence of the Axioms of Definition 1 3.3.2 Preliminary Lemmas 3.3.3 Theorem 1 3.4 Sufficient Conditions when the Concatenation Operation is not Closed 3.4.1 Formulation of the Non-Archimedean Axioms 3.4.2 Formulation of the Archimedean Axiom 3.4.3 The Axiom System and Representation Theorem 3.5 Proofs 3.5.1 Consistency and Independence of the Axioms of Definition 3 3.5.2 Preliminary Lemmas 3.5.3 Theorem 3 3.6 Empirical Interpretations in Physics 3.6.1 Length 3.6.2 Mass 3.6.3 Time Duration 3.6.4 Resistance 3.6.5 Velocity 3.7 Essential Maxima in Extensive Structures 3.7.1 Nonadditive Representations 3.7.2 Simultaneous Axiomatization of Length and Velocity 3.8 Proofs 3.8.1 Consistency and Independence of the Axioms of Definition 5 3.8.2 Theorem 6 3.8.3 Theorem 7 3.9 Alternative Numerical Representations 3.10 Constructive Methods 3.10.1 Extensive Multiples 3.10.2 Standard Sequences 3.11 Proofs 3.11.1 Theorem 8 3.11.2 Preliminary Lemmas 3.11.3 Theorem 9 3.12 Conditionally Connected Extensive Structures 3.12.1 Thermodynamic Motivation 3.12.2 Formulation of the Axioms 3.12.3 The Axiom System and Representation Theorem 3.12.4 Statistical Entropy 3.13 Proofs 3.13.1 Preliminary Lemmas 3.13.2 A Group-Theoretic Result 3.13.3 Theorem 10 3.13.4 Theorem 11 3.14 Extensive Measurement in the Social Sciences 3.14.1 The Measurement of Risk 3.14.2 Proof of Theorem 13 3.15 Limitations of Extensive Measurement Exercises4. Difference Measurement 4.1 Introduction 4.1.