Continuum

The nondenumerable set of Real Numbers, denoted $C$. It satisfies

$$\aleph_0+C=C$$ (1)

and

$$C^r=C,$$ (2)

where $\aleph_0$ is Aleph-0. It is also true that

$${\aleph_0}^{\aleph_0}=C.$$ (3)

However,

$$C^C=F$$ (4)

is a Set larger than the continuum. Paradoxically, there are exactly as many points $C$ 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.

See also Aleph-0, Aleph-1, Continuum Hypothesis, Denumerable Set