Make your own free website on Tripod.com


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 [8].

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 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 Àn, Àw, Àa, where a = Àw, 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 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

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

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



NEXT

VisMath HOME