### X.3.1.3 Conclusions

1. The abstract objects which belong to a certain class have all the same model; since the integers, as well as the rational and irrational numbers belong to the class of the real numbers, this means that all these categories (subsets of {R} are also made-up from AAV, an infinity of digits is therefore required for the representation of an actual value. In this respect, the proof is that both the irrational and the transcendental numbers have an infinite number (officially acknowledged) of decimals, the continuity of the real numbers axis requiring that the other members (singular values) to also own the same infinite number of decimals for their representation. An accurate representation of the integers such as AAV would mean (for instance, according to the decimal syntax), an integer followed by an infinity of zero digits after the decimal separator. The mutual convention of letting aside the series of zero digits into the current representations of the integers does not also mean the renunciation at the principle that any number belonging to{R}is identical in terms of model features with any other number belonging to this set, all of them being AAV.

2. The necessity of an infinity of digits for a complete representation of an AAV makes that such numbers to not be realizable neither under a material nor an abstract form (because it would require an ISS with infinite sizes), these numbers are therefore virtual, unable to exist as concrete values (class instances). Thus, it is worth mentioning that despite of its denomination as set of the real numbers, this set is exclusively made-up from virtual numbers.

3. The realizability of a numerical value is strictly related to the necessity that this value to contain a finite quantitative information amount; consequently, any value which is materially or abstract realizable must have a non determination interval. Obviously, this interval is gradually decreased, as the technological and scientific progress is increasing, but it will never be null.

4. The virtual object named “the set of the real numbers”, from the official mathematics, represents an asymptotic model, a limit impossible to be reached for the realizable objects, but towards which all the numerical values tend to, as the progress in the scientific field keeps rise-up. It is an example of extreme idealization of a notion: the numerical value (existential attribute, scalar).

5. The objectual philosophy is structured on objects, with a permanent control on the types of objects used for modelling the human knowledge. It is natural that their associated objects and processes, meant to model the material systems, to belong to the class of materially realizable objects. The concrete abstract objects with which the real IPS are able to operate within the abstract processes, must belong to the class of abstract realizable objects, whose quantitative values must be finite, regardless of the IPS’s performance degree. On the other hand, as we have seen in chapter 9, the absolute reality has attributes with infinite quantitative values, values which belong to the virtual (mathematic) set{R}. In other words, the continuous set {R}, just as the absolute reality, are virtual, asymptotic objects, towards which any kind of knowledge tends to, but at the same time, we must be aware that they are intangible in relation to any realizable knowledge.