Foundations of Measurement

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.:See Table of Contents and MAQ.

18. Overviews 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 Representationss 19.1 Introductions 19.2 Types of Concatenation Structuress 19.3 Representations of PCSss 19.4 Completions of Total Orders and PCSss 19.5 Proofss 19.6 Connections Between Conjoint and Concatenation Structuress 19.7 Representations of Solvable Conjoint and Concatenation Structuress 19.8 Proofss 19.9 Bisymmetry and Related Propertiess Exercisesc 20. Scale Typess 20.1 Introductions 20.2 Homogeneity, Uniqueness, and Scale Types 20.3 Proofss 20.4 Homogeneous Concatenation Structuress 20.5 Proofss 20.6 Homogeneous Conjoint Structuress 20.7 Proofss Exercisesc 21. Axiomatizations 21.1 Axiom Systems and Representationss 21.2 Elementary Formalization o