5. NEW KIND OF
TRANSFINITE INTEGERS
Consider a real number, x Î
[0,1] :
x = 0.a1 a2 a3 ...
ai ... Þ
"
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
|
i |
... |
a
|
2 |
|
a
|
1 |
|
a
|
0 |
= åi= 0¥ |
a
|
i |
2i = åi
= 0k |
a
|
i |
2i < ¥ |
|
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} = À0
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
|
i |
... |
a
|
2 |
|
a
|
1 |
|
a
|
0 |
= åi = 0¥ |
a
|
i |
2i |
|
(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. 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.
since
w is the least transfinite integer,
by Cantor.
2.1.
|
x
|
= åi =
0¥ a-i2i
£ åi=0¥2i |
|
since
not all a-i in x- are equal to 1.
2.2.
when k tends to ¥, by the classical mathematical analysis definition of the ¥
-sum limit.
2.3.
| lim åi=0k2i=
lim {21,22,...,2k} |
|
when k tends to ¥, since
" k³
1
2.4. lim { 20, 21, 22, ... , 2k,
... } £ 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-) = À0
.
Further, by Y, we have the following property
of all such x- ÎX -.
Property 1. For any two real numbers x1 Î
D and x2 Î D,
| y: (x1
¹ x2) Þ ( |
x
|
1 |
¹ |
x
|
2 |
) |
|
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 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 weak 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. Thus, Gn
is a "Galaxy" [10,11] since the difference between any two transfinite
integers of Gn 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 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-1
< x-2.
|