# Annex X.3 -
SPECIFIC APPROACHES OF SOME MATHEMATIC OBJECTS WITHIN THE OBJECTUAL
PHILOSOPHY

## X.3.1 The set of the real numbers

The establishment
of the historical motivation which was the basis for assigning the
epithet “real” to the numbers which belong to that
particular set is not the topic of the present paper, but it is
possible that this denomination would have occurred from the need to
differentiate the class of the common numbers which are currently
used, from another class of numbers which has appeared among time on
the mathematics “scene”, that is the class of so-called
“imaginary numbers”. As regards the class model, the
numbers which are considered “real” are different from
the “imaginary” ones only by the rule which sets the sign
of the result *R* of the product between two terms *Ta* and
*Tb*, depending on their signs (both terms belonging to the same
class of numbers). The two rules are:

*direct rule*, which was
applied to the “real” numbers, according to which:

*reverse rule*, which was
applied to the “imaginary” numbers, according to which:

If we let aside the
sign of the numbers and the above-mentioned rules, both numbers
classes comply with the condition imposed __by definition__ to
both sets, which requires them to be sets containing exclusively
numbers __with an infinity of figures (digits)__. According to the
objectual philosophy, in which the meaning of the word **real****
**is a totally different one, being strictly related to the concept
of realizability (achievability) of an object or process, the result
is that both sets of numbers from mathematics, exclusively contain
virtual objects which can be symbolically rather than effectively
achieved.

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