For any set *X*, its power (cardinality) we shall denote by |*X*| or Card{*X*}.

Denote the set of all finite natural numbers by *N* = {1, 2, 3, ...}. Since *N* is a countable set, then |*N*| =
À_{0} .

Denote the set of all real numbers (of all proper fractions) of the segment [0,1] by *D*. Since *D* has, by the well-known Cantor's theorem, the power *C* of Continuum, then |*D*| = *C*.

Now, there are two following main formulations of Continuum Hypothesis [8].

1) The classical Cantor Continuum Hypothesis formulation:
*C*=À_{1}.

2) The generalized Continuum Hypothesis formulation, by Cohen:
"a |*P*(À_{a})| =
À_{a+1},
where *P*(À_{a})
is the power-set of any set *A* with Card{*A*} =
À_{a}.

As is known, P.J.Cohen writes in his famous monography: "... Continuum Hypothesis is a rather dramatic example of that can be called (from our today's point of view) an absolutely undecidable assertion, ..." (p.13),
and further: "Thus, *C* is greater than
À_{n},
À_{w},
À_{a},
where a =
À_{w},
and so on. " (p.282) [8]. Therefore, we shall even not try to imagine visually a set of integers of a
cardinality succeeding À_{0}, and use the following most week formulation of Continuum Hypothesis.

3) Does there exist even one set of integers, say *M*, for which it can be constructed a 1-1-correspondence with the set *D* of the continual cardinality *C*?

That is

Just such the 1-1-correspondence we shall try to construct below between the set *D* and a set *M* of integers.