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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 ($|R|< 2^R$)?

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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 $A$

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    The point that I am not comfortable with is that Cantor proof uses the fact that N is listable. R is not listable. – 2017-01-01
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    Listable is not an issue. My sketch does not refer to a list, it refers to any claimed bijection. This one is called [Cantor's theorem](https://en.wikipedia.org/wiki/Cantor%27s_theorem) which applies to any set. – 2017-01-01