Cardinality of the set of binaryexpressed real numbers 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 binaryexpressed real numbers PengKuan on Maths: Cardinality of the set of binaryexpressed real numbers or https://www.academia.edu/19403597/Ca...d_real_numbers 
Your proof in section 1 is incorrect  you can't assume that set cardinalities will behave like exponents in the limit, despite the suggestive notation. (The result is correct, though.) Your claim in section 2, explored in section 3, is incorrect. The set you actually work with is the dyadic rationals, but you conflate them with the set of fractional binary numbers (your $\mathbf{B_F}$, i.e., the set of real numbers between 0 and 1, which you choose to represent in binary). Section 4 is incorrect by virtue of relying on the incorrect result from sections 2 and 3. Section 5 consists of a proof by assertion. Section 6, like section 4, relies on incorrect results and hence is also wrong. I did not bother checking further. Review: Is this a mathematical breakthrough? 
Surely you understand that the "number of real numbers" in a given interval does NOT depend upon how the numbers are expressed? So I don't understand your reference to "binary expressed numbers" and "binary numbers". 
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There is no such thing as "a binary number". Any real number is expressible in decimal, binary or any other base you care to name (as long as you allow me to have infinite strings of numbers). Rather neatly, if you choose any rational base, the set of numbers that you can't express without recourse to infinite strings of numbers is the same (here I am making use of notations like $13 \div 22 ={13 \over 22}= 0.5\dot0\dot9=0.5090909\ldots$ all of which are finite representations as is the equivalent binary fraction $0.\dot1000001001010011110\dot0$). So your use of binary is completely redundant and flawed. 
Analysis of the proof of Cantor's theorem Analysis of the proof of Cantor's theorem Cantor's theorem states that the power set of ℕ is uncountable. This article carefully analyzes this proof to clarify its logical reasoning Please read the article at PDF Analysis of the proof of Cantor's theorem PengKuan on Maths: Analysis of the proof of Cantor's theorem or Word https://www.academia.edu/37356452/An...antors_theorem 
13/22 = 0.590909090..., whereas 0.509090909... is the expansion of 28/55. 
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Your analysis of Cantor's theorem doesn't seem to quote any part of Cantor's proof of it. 
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