## X.23.2
The support sets of the variables classes

As we have seen in
chapter 9, each class of abstract objects represents a set of objects
which have the same model, set which is called *support
set* of that class. We have also noticed
that the number of support elements (the set’s cardinal) is
direct proportional with the abstraction level, starting with the
basic level - concrete object - which has only one element as its
support.

As regards the case
mentioned in the previous section, concerning the set {R} from
mathematics, it is well-known that the support set of the class is
infinite, both for the generic set {R} and for its particular subsets
{Q},{N},{Z} etc., fact which makes
impossible the comparison between the support sets of each distinct
variable class. This comes from fact that according to the current
mathematics, the different levels of infinite cannot be compared,
although logically, we realize that a subset has fewer elements than
the set which includes it. This kind of absurdity disappears in case
of the set of real *realizable*
numbers(introduced
in the annex X.3), set which contains a finite number of singular
values in a finite interval. In this case, a however large but finite
interval of
shall
contain a finite number of singular values which may belong to any of
the sets
,,,
all of them with a finite number of elements. Thus, the cardinal of
each specific set of numerical values is finite and a comparison
between them may be carried out.

Comment
X.23.2.1: Because the sets
,,etc.
are subsets of
,
this means that as regards a specific non-determination interval
of the set
,
all the other subsets shall have the same
as well.

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