Chapter 4 was changed completely due to an error in addition proof of theorem 2 was completely changed due also to an error. English throughout the article has been improved for better understanding of its contents. 7If one were to consider the possibility that; much like radicals associated with are in the same cardinality as that of the integers, it might be possible that some operations on R'' for instance might enable us to map R'' $\mapsto$R' in such a way that their cardinalities are the same, then one need only realise that for this to be the case there needs to (at the very least) corrospond some sets of finite sets of elements of R'' to every one element of R' . Given this requirement, it would take larger and larger sets of elements of R'' in order to form such a corrospondance with R' making it eventually so, that infinite measures of elements of R'' will be required in order to completely form such a mapping. This is because of the nature of the size of R'' in comparison to R'. Finally to seal the deal, we use Cantor's original method as follows: if we begin to enumerate any two $C_{inf}$ type irrational numbers, there will always be an infinite set of elements belonging to the exact same class between these. Since a subset U of such a class C can exist with a different 'cardinality' and since values of type C can exist in exaclty the same uncountable manner as that of U, between elements of U, this is all we need. This is because C is as innumerable to U as U is to the integers.
