## 2. SOME FORMULATIONS OF CONTINUUM HYPOTHESIS

Introduce the following notations.

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 .

1) The classical Cantor Continuum Hypothesis formulation: C=À1.

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

As is known, P.J.Cohen completes his monography  by the following estimation of the Continuum Cardinality: "Thus, C is greater than Àn, Àw, Àa, where a = Àw, and so on. " (p.282) . 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

\$ M [|M| = C] ?, where M is some set of integers.          (1)

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.

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