Foundations of Measurement Vol. 3: Representations, Axiomatization, and Invariance Book

18. Overview
s 18.1 Nonadditive Representations (Chapter 19)
_s 18.2 Scale Types (Chapter20)
_s 18.3 Axiomatization (Chapter 21)
_s 18.4 Invariance and Meaningfulness (Chapter 22)
c 19. Nonadditive Representations
s 19.1 Introduction
s 19.2 Types of Concatenation Structures
s 19.3 Representations of PCSs
s 19.4 Completions of Total Orders and PCSs
s 19.5 Proofs
s 19.6 Connections Between Conjoint and Concatenation Structures
s 19.7 Representations of Solvable Conjoint and Concatenation Structures
s 19.8 Proofs
s 19.9 Bisymmetry and Related Properties
s Exercises
c 20. Scale Types
s 20.1 Introduction
s 20.2 Homogeneity, Uniqueness, and Scale Type
s 20.3 Proofs
s 20.4 Homogeneous Concatenation Structures
s 20.5 Proofs
s 20.6 Homogeneous Conjoint Structures
s 20.7 Proofs
s Exercises
c 21. Axiomatization
s 21.1 Axiom Systems and Representations
s 21.2 Elementary Formalization o

From the Foreword is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or unhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe.:

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