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 completes his monography [8] by the following estimation of the Continuum Cardinality:
"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 weak formulation of Continuum Hypothesis.

3) Whether there exists a set of integers, say *M*, such that a 1-1-correspondence between the set *M* and the set *D* of all real numbers (proper fractions, geometrical points) of the segment [0,1] can be realized?

That is