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5. NEW KIND OF
TRANSFINITE INTEGERS

Consider a real number, x Î [0,1] :

          x = 0.a1 a2 a3 ... ai ... Þ 

x = åk = 1¥  ai2-i
" i ((ai =0) Ú (ai =1)).    (1)

There are only two following cases of interest.

CASE 1. x is a rational number with a 0-tail, i.e. $ k " i > k (ai =0).

By virtue of Y , this x is corresponded by an integer x- possessing the following property:

y(x) = 
x
 
= ...
a
 

k
...
a
 

2

a
 

1

a
 

0
= åk = 1¥
a
 

i
2-i = åk = 1¥
a
 

i
2-i < ¥
so  x- is a finite natural number.

Designate: Q0 is the set of all rational fractions x Î [0,1] with the 0-tail;

N is the set of all finite natural numbers.

Thus, we have

THEOREM 1. Q0 is y -equivalent to N, i.e. Card {Q0} = Card {N}.

CASE 2. x is an irrational number, i.e. " k $ i > k (ai =1).

By virtue of Y , this x is corresponded by an integer x- possessing the following property:

y(x) = 
x
 
= ...
a
 

k
...
a
 

2

a
 

1

a
 

0
= åk = 1¥
a
 

i
2-i
(1)

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 well-known 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. Ord { x- } = w .

PROOF.

1.1. 


x
 
= åk = 1¥
a
 

i
2-i ³ åk = 1¥ ai2-i
  - it's evident.

1.2. x-is a transfinite integer, since it is a countable sum of finite natural numbers, and consequently "nÎ N (x->n).

1.3. 


x
 
³ w
since w is the least transfinite integer, by Cantor.

2.1. 


x
 
= åi = 1¥ ai2i £ å1¥2i
since $ i³ 1  (a-i = 0) in x-.

2.2. 

åk=0¥ 2i = lim åi=0k2i
by the classical mathematical analysis definition of the ¥ -sum limit.

2.3. 

 lim åi=0¥2i= lim {21,22,...,2k
since " k³ 1
åi=1¥ 2i £ 2k+1

2.4. lim { 21 , 22, ... , 2k, ... } £ lim { 1, 2, ... , k, ... } = w , 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-.

3.1. From 1.3 and 2.5 we have 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-) = À0 .

Further, by Y, we have the following property of  x- ÎX -.

Property 1. For any two real numbers x1 Î D and x2 Î D,

y: (x1 ¹ x2) (
x
 

1
¹
x
 

2
)

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 an transfinite ordinal w -type number.

Further, denote the set of all irrational numbers of the segment [0,1] by Dir. Then, by virtue of Theorem 2 and Property 1, we have

THEOREM 3: Dir is y -equivalent to X -, i.e. Card {X -} = Card {Dir} = C .

Thus, this Theorem solves the Continuum Problem in its week formulation 3 (see above).

Further, we have

Property 2. For any n-level of the tree TL , all the transfinite integers x- ÎX - with confinal heads (from the n-level to the w-level of TL ) form a finite set, say Gn , consisting of 2n transfinite integers. G is a "Galaxy" [10,11] since the difference between any two transfinite integers of  Gis 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 0-tails) into the unordered set X - of all transfinite integers x-.

Remark, however, that the set X - can be ordered (not well-ordered !) by means of the natural order given on D: for example, if x1 < x2 then x-1x-2.



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