 Some real numbers, though, are uncomputable, as Turing proved. Turing’s proof pointed to specific examples of uncomputable real numbers, but it is easy to see in a general way that there must be real numbers that cannot be computed by any Turing machine, since there are more real numbers than there are Turing-machine programs. There can be no more Turing-machine programs than there are whole numbers, since the programs can be counted: 1st program, 2nd program, and so on; but, as Georg Cantor proved in 1874,there are vastly more real numbers than whole numbers (Cantor 1874).