5. NEW KIND OF


There are only two following cases of interest.
CASE 1. x is a rational number with a 0tail, i.e. $ k " i > k (a_{i} =0).
By virtue of Y , this x is corresponded by an integer x^{} possessing the following property:

Designate: Q_{0} is the set of all rational fractions x Î [0,1] with the 0tail;
N_{ }is the set of all finite natural numbers.
Thus, we have
THEOREM 1. Q_{0} is y equivalent to N, i.e. Card {Q_{0}} = Card {N} = À_{0}
CASE 2. x is an irrational number, i.e. " k $ i > k (a_{i} =1).
By virtue of Y , this x is corresponded by an integer x^{} possessing the following property:

so we have a new mathematical object, with the same ontological status as the irrational number. It is quite appropriate here, to remind of the wellknown G.Cantor's words: "...the transfinite numbers exist in the same sense as the finite irrational numbers."
Consider the main properties of this new mathematical objects.
THEOREM 2. For all x^{}: Ord { x^{} } = w .
PROOF.
1.1. x^{}is a transfinite integer, since it is a countable sum of finite natural numbers, and consequently "nÎ N (x^{}>n).
1.2.

2.1.

2.2.

2.3.


2.4. lim { 2^{0}, 2^{1}, 2^{2}, ... , 2^{k}, ... } £ lim { 1, 2, ... , k, ... } = w , when k tends to ¥, since the first is a subsequence of the second and by Cantor's (and F.Hausdorff [9] ) definition of the least transfinite integer w .
2.5. x^{}£ w.
3.1. From 1.2 and 2.5 we have w£ x^{} £ w and consequently Ord {x^{}} = w.
The Theorem is proved.
Denote the set of all transfinite integers x^{} as X ^{}. Then we get
Corollary 1. For any x^{}ÎX ^{} Card(x^{}) = À_{0} .
Further, by Y, we have the following property of all such x^{} ÎX ^{}.
Property 1. For any two real numbers x_{1} Î D and x_{2} Î D,

that is any two transfinite integers x^{}_{1} and x^{}_{2} are different mathematical objects in the same sense as any two real numbers x^{}_{1} and x^{}_{2} are different in a sense of Classical Mathematics.
From this Property 1 and the definition of an element x^{}ÎX ^{}, we have
Corollary 2. Every infinite subsequence of the series of natural numbers is an individual mathematical object representing a transfinite ordinal w type number.
Further, denote the set of all irrational numbers of the segment [0,1] by D_{ir}. Then, by virtue of Theorem 2 and Property 1, we have
THEOREM 3: D_{ir }is y equivalent to X ^{}, i.e. Card {X ^{}} = Card {D_{ir}} = C .
Thus, this Theorem solves the Continuum Problem in its weak formulation 3 (see above).
Further, we have
Property 2. For any nlevel of the tree T_{L} , all the transfinite integers x^{} ÎX ^{} with confinal heads (from the nlevel to the wlevel of T_{L} ) form a finite set, say G_{n} , consisting of 2^{n} transfinite integers. Thus, G_{n } is a "Galaxy" [10,11] since the difference between any two transfinite integers of G_{n }is a finite integer.
Lastly, the mirror mapping Y possesses the following interesting but rather unique
Property 3. The mirror mapping Y transforms the ordered continual set D (without the countable set of all rational fractions with 0tails) into the unordered set X ^{} of all transfinite integers x^{}.
Remark, however, that the set X ^{} can be ordered (not wellordered !) by means of the natural order given on D: for example, if x_{1} < x_{2} then x^{}_{1} < x^{}_{2}.