Ch 4 Difference Measurement Difference Measurement In Ch

Ch 4: Difference Measurement

Difference Measurement • In Ch 3 we saw the kind of representation you can get with a concatenation operation on an ordered set A • “The question arises whether similarly tight representations ever exist when there is no concatenation operation. ” 136

Difference Measurement • Extensive measurement: consider a set of movable rods • Difference measurement: consider fixed points on a line. Consider a set of intervals between points • We can construct standard sequences in A with an auxiliary, uncalibrated rod to lay off equal intervals 136

Difference Measurement • Denoting elements of A by a, b, e, d, we denote intervals in A by ab, cd, etc. • We distinguish between ab and ba. • Comparison with a set of movable rods generates an ordering on the intervals in A. • ab ≿ cd if some rod does not exceed ab but exceeds or matches cd. 137

Axiomatization of Difference Measurement • Holder (1901) showed how the measurement of intervals between points on a line can be reduced to extensive measurement. • Standard sequences of equally spaced elements a 1, a 2, a 3, . . . , where the intervals a 1 a 2 ∼ a 2 a 3 ∼. . . • Equivalent intervals are identified with a single element, their equivalence class 143

Otto Ludwig Hölder

Positive Difference Structures 145

Positive Difference Structures 145

Positive Difference Structures Interpret A as the set of endpoints of intervals. A* is the set of positive intervals, and is a subset of A x A. 147

Positive Difference Structures Transitivity 147

Positive Difference Structures Axiom 3 guarantees that there are no null intervals. Note it also follows that A* is not reflexive or symmetric. 147

Positive Difference Structures Weak monotonicity: this is needed to guarantee that concatenation of non-adjacent intervals gets the right results 147

Positive Difference Structures Archimedean axiom: ana 1 = (n-1)a 2 a 1 147

Positive Difference Structures Archimedean axiom: ana 1 = (n-1)a 2 a 1 147

Positive Difference Structures 147

Positive Difference Structures 147

Algebraic Difference Structures We now allow for negative and null intervals, so we don’t need A*. 151

Algebraic Difference Structures Axioms 2 and 3 of Definition 1 are here replaced by Axiom 2. It is a pretty intuitive axiom 151

Algebraic Difference Structures Axioms 3 -5 are correspond to axioms 4 -6 of Definition 1 151

Algebraic Difference Structures 151

Cross Modality Difference Structures Solvability axiom: The first part says that any element in Ai x Ai can be matched with an element in A 1 x A 1. The second part is just the normal solvability property for A 1. But because of the first part, it follows that all the Ai have the solvability property. This is also why the Archimedean axiom is formulated for A 1. 165

Finite, Equally Spaced Difference Structures 167

Absolute-Difference Structures Axiom 3: Betweenness is well behaved i) If b is between a and c, and if c is between b and d, then c and b are between a and d. ii) If b is between a and c is between a and d, then ad exceeds bd 172

Absolute-Difference Structures Weak Monotonicity: If b is between a and c and b’ is between a’ and c’, and ab ∼a’b’, then bc ≿ b’c’ iff ac ≿ a’c’ 172

Absolute-Difference Structures Solvability: if ab ≿ cd, then there is some d’ that is between a and b such that ad’ ∼ cd 172

Absolute-Difference Structures Archimedean: ai is between a 1 and ai+1 for all i, and successive intervals are non-null. aia 1 is strictly bounded. 172

Absolute-Difference Structures 173

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