Question 884187
By the Cantor–Bernstein–Schroeder theorem that c = 2^|N|, where c is the cardinality of the real number set, |N| cardinality of the natural number set, this implies that, given c, there exists a real number R|c being, namely, the largest real number in R. 

However, since (~R and R) is empty, the R|c does not exist in ~R. Therefore there is no largest real number in ~R.

Reference:

Cardinality of the continuum, http://en.wikipedia.org/wiki/Cardinality_of_the_continuum