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

\end{displaymath} (1)

\end{displaymath} (2)

where $\aleph_0$ is Aleph-0. It is also true that
\end{displaymath} (3)

\end{displaymath} (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

© 1996-9 Eric W. Weisstein