December 3rd, 2015, 04:41 AM  #1 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  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 
December 3rd, 2015, 08:44 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
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? 
December 4th, 2015, 09:43 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
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".

December 8th, 2015, 10:01 AM  #4  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
 
December 8th, 2015, 10:05 AM  #5  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
I thought the set of binary numbers in the unit interval contains irrationals. It turned out no. Because when the number of digits goes to infinity, the digits are not actually infinite many.  
December 8th, 2015, 11:06 AM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,511 Thanks: 2514 Math Focus: Mainly analysis and algebra  Quote:
Quote:
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. Last edited by v8archie; December 8th, 2015 at 11:20 AM.  
September 16th, 2018, 04:44 AM  #7 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  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 
September 17th, 2018, 03:13 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,974 Thanks: 1850 
13/22 = 0.590909090..., whereas 0.509090909... is the expansion of 28/55.

September 17th, 2018, 12:47 PM  #9 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  
September 18th, 2018, 01:22 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 19,974 Thanks: 1850 
Your analysis of Cantor's theorem doesn't seem to quote any part of Cantor's proof of it.


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binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set 
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