According to a
convention, the existential (quantitative) information amount
achieved as a result of an experiment by means of which the
uncertainty of an attribute value shall be reduced from the initial
value
to the final value
(in which, obviously
**
)
is given by the following relation:

(X.3.1.1.1)

According to a
convention as well, the measuring unit of the existential information
amount is *the bit, *this being related to the decrease by half
of the final non determination as compared to the initial one (for
that reason we have the logarithmation under basis two). By looking
at the relation X.3.1.1.1, we shall observe that in order to maintain
the finite information amount by starting from a finite and known
initial range
,
it is required that the non determination range (uncertainty)
to be as low as it can be, __but not zero__.

By considering
these aspects, but also the ones about the information processing
systems which were mentioned in chapter 8, let us make a brief review
on the axis (set) of the real numbers {R}within the mathematics
field. One of the basic properties of such a *continuous *set of
numbers (which can be represented through an infinite axis named the
*axis of the real numbers*), is that any finite interval,
regardless how small it is,
which belongs to this axis, contains an infinity of *singular
values *(numbers which are associated with dimensionless points
placed on this axis). By taking into account this axis as a
distribution, if the finite range
is considered as a support interval, and the number of possible
singular values
within this range corresponds with the distributed amount, one may
observe that the distribution *density *of the number of real
singular values arranged on the axis with the same name:

(X.3.1.1.2)

__it is infinite__ (because
is
infinite, whereas
is finite).

The fact that an interval of finite values contains an infinity of singular values, this also means that the non determination interval:

(X.3.1.1.3)

of a singular value placed on this axis
is null, otherwise speaking, the information amount contained by such
a number is an infinite one^{8}6.
Due to this reason, this kind of numerical values are named *absolute
accurate values* (AAV) according to the objectual philosophy. The
major problem of these values which contain an infinite information
amount is that they can be only symbolically^{8}7
displayed (such as, for instance, the symbol
,
or any other literal symbol for the values along the real axis), but
they cannot be displayed as class instances (actual numerical values)
because the representation of an absolute accurate numerical value
would require an infinite-sized ISS (a number with an infinity of
digits).

According to the
aspects mentioned so far, it results that the AAV which compose the
so-called *axis of real numbers* are not even *abstract
realizable* (therefore, they are *virtual*), because the
abstract realizability requires that the abstract object to be
contained within a finite ISS, and that it must be compatible in
terms of size with the maximum accepted size of ISS which IPS can
operate with.

The denomination of
“real numbers” was assigned to these AAV when an analysis
of the information amount contained into these numbers was not
possible. Since the values of the real objects attributes which were
experimentally determined could be assigned with any value, a set
which was able to include all the possible values had to be found.
However, a significant remark must be added: the real numbers coming
from measurements or calculations __were not absolute accurate
values__, but they were approximations (truncations) of such
values, with a no determination range
,
and this interval was due either to the determination (measure)
error, or to the limitation of the number of ISS digits to acceptable
values (which were processed much easier by the human brain or by the
computers).

These numbers (the
approximate ones), named *normal numerical values* in this
paper, which can be indeed real, (because they are either *abstract
realizable* or *materially realizable* as values of the
attributes belonging to some real, material objects), do not comply
with the definition of the numbers along the axis {R}, because a
finite interval ***x*
__does not contain an infinite number of values__ any longer, but
a finite number:

(X.3.1.1.4)

Comment
X.3.1.1.1: For most of the industrial mechanical works, it is enough
to maintain an accuracy in the determination of 1
for dimension sizes, this fact implying that the numbers which
represent these dimensions must have no more than three decimals
after the decimal separator (in the industry based on the metric
system, its size is usually expressed in millimeters). In this case,
a real interval of length values
*=
1 *mm shall include only one thousand possible distinct values
(not an infinity). Even the most high-tech AIPS (computers) which may
be found nowadays are not able to currently operate with numbers
higher than tens of digits, and in case of a single value (such as,
for instance, the computed value of **),
numbers with tens of thousand digits may result, but an absolute
accurate value cannot be reached.

86
According to the paper entitled “*Mică Enciclopedie
Matematică* – *Editura Tehnică, București*, 1980”,
at p. 80 we find the following assertion: “The set of real
numbers is made-up from the set of positive and negative decimal
fractions with an infinity of digits”

87 In Annex X.3.6, there will also be presented a version of “realizable” AAV as the relative accurate values (RAV), but in this case we are dealing with a symbolic representation, the brackets which frame the decimal period being only a symbol (a substitute) for an infinity of digits.

Copyright © 2006-2011 Aurel Rusu. All rights reserved.