This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis.
1.      Counting the fractional binary numbers
2.      Fractional binary numbers on the real line
3.      Countability of BF
4.      Set of all binary numbers, B
5.      On Cantor's diagonal argument
6.      On Cantor's theorem
7.      On infinite digital expansion of irrational number
8.      On the continuum hypothesis

Please read the article at 
Cardinality of the set of binary-expressed real numbers 
http://pengkuanonmaths.blogspot.com/2015/12/cardinality-of-set-of-binary-expressed.html
or 
https://www.academia.edu/19403597/Cardinality_of_the_set_of_binary-expressed_real_numbers