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| = À₀ .

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=À₁.

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 À₀, 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

We shall indicate below just such the set M, construct such the 1-1-corerespondence, and prove that the set M has the continual cardinality C.