ABSTRACT
The only least upper bounds considered in Chapter 11 were those of sets containing just a value and its negative. This chapter takes the least upper bound of any finite set of values to be values. This widening of scope to what I call “least upper bound models” requires no new axioms but vastly expands the potential for defining new concepts and proving new theorems. Cartesian models are shown to be least upper bound models. Operations on value sets are defined, including the negative operation, addition and subtraction, and multiplication by integers—all of which differ from but have illuminating relations to the analogous operations for individual values. The negative operation on value sets enables us to characterize a duality between least upper and greatest lower bounds that yields “translations” of statements about one into statements about the other. The weird idea of self-negative value sets is introduced. Values are shown to have positive and negative parts. A coherent picture of the orthantic structure of least upper bound models gradually emerges. We witness the repeated appearance of multidimensional “boxes” and geometrically strange triangle equalities (sic). Each of these structures has an axiological significance that is explained in the text. In sum, we see that the nine axioms—all chosen for their axiological importance and relying only on basic arithmetical and bounding operations—describe remarkably intricate yet fully comprehensible nonlinear value structures.
