## Continuum

The nondenumerable set of Real Numbers, denoted . It satisfies

 (1)

and
 (2)

where is Aleph-0. It is also true that
 (3)

However,
 (4)

is a Set larger than the continuum. Paradoxically, there are exactly as many points on a Line (or Line Segment) as in a Plane, a 3-D Space, or finite Hyperspace, since all these Sets can be put into a One-to-One correspondence with each other.

The Continuum Hypothesis, first proposed by Georg Cantor, holds that the Cardinal Number of the continuum is the same as that of Aleph-1. The surprising truth is that this proposition is Undecidable, since neither it nor its converse contradicts the tenets of Set Theory.