[Note that the notation a_{0} and a_{1} are used
to represent aleph nought and aleph one respectively due to character set
limitations].

How many real numbers are there? Cantor noted that there are
Y^{X} different numbers of X digits where Y is the base used
(in this case, 10). Suppose there are precisely C real numbers that
are specified by their decimal expansions 0.abcd . . .
in which there are a_{0} digits each chosen from a set of 10
possibilities. Therefore there are 10^{a0}
possibilities. If we did the same thing in binary, we would get
2^{a0} = C. Since C must be greater than
a_{0}, we can see that almost all real numbers are
transcendental...

The continuum hypothesis is the hypothesis that C = a_{1}.
In other words, there is no set whose cardinal number lies between
that of the natural numbers and that of
the real numbers. Cantor guessed that this was true.
In 1940 Kurt Gödel showed
that Cantor's guess can never be disproved from other axioms of
mathematics. In 1963, Paul Cohen (1934-) showed that this hypothesis
was *unprovable*, in Gödel's sense of the word.