The Continuum Hypothesis

[Note that the notation a₀ and a₁ 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 a₀ 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 a₀, we can see that almost all real numbers are transcendental...

The continuum hypothesis is the hypothesis that C = a₁. 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.