Cardinaly of the power set of $R$ Cantor established that the set of natural numbers is countable while the set of real numbers is not countable (i.e. $|N| < 2^N$). In other words, there is no bijection between  $N$ and $R$.

Why can't we have a proof that there is no bijection between the $R$ and its power set ($|R|< 2^R$)? 


We can and do.  Cantor's proof works generally between any set and its power set to show that the power set has strictly greater cardinality.  Take any purported bijection $f:A\leftrightarrow P(A)$.  For each element $a \in A$ ask whether it is in $f(a)$  The subset $B=\{a \in A|a \not \in f(A)\}$ is not the image of any element of $A$
