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Platonic Esahedron
5
Numbers and Measures
Abstract. Numbers are the essence of any mathematization process. They measure
entities, but also provide rigorous frameworks for analyzing the notion of infinity;
the set of natural numbers is the first example of infinity. Finiteness and infinity,
which very often appear in dialectic aporias, underly the essence of mathematics.
Numbers are essential to the notion of population, and are intrinsically related to
computation processes. In this chapter, after a brief section on sets and functions,
the essentials of numerical systems will be outlined, by emphasizing the concepts
that are most relevant for discrete structures. Then, the principle of induction, and
basic topics of arithmetic and logic will be presented. Two sections conclude the
chapter: one on series and growths (with numbers related to time, space, and matter
aggregation), the other one on basic dynamical concepts.
5.1
Sets and Functions
Sets, numbers, sequences, relations, operations, functions, and variables are the ba-
sic concepts of mathematics, intrinsically intertwined and related to the infinite, a
notion that is the essence of mathematics (“The Art of the Infinite” [179]). Here
we give a brief presentation of these concepts and some standard notation (see
[193, 174, 177, 187, 186, 169] for more extended or advanced presentations).
Sets or classes (in this context we use these terms as synonyms) are collections
of distinct objects. The specification of all the elements which belong to a set com-
pletely identifies it. If X is a set, a
X if a does
not belong to X ). If a set has a finite number of elements, it is completely described
by a list, where the appearance order of the elements is not relevant, and multiple
appearances are redundant, that is, equivalent to single appearances. A set
X means that a belongs to X ( a
{
a
,
b
}
 
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