June 29th, 2016, 07:42 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100  Natural Numbers and the Continuum
Definition Reverse binary/decimal representation of the natural numbers: a0x2^0 + a1x2^1 + a2x2^2 + a3x2^3 + ....., an=0,1 a0x10^0 + a1x10^1 + a2x10^2 + a3x10^3 + ....., an=0,1,.....,9 ALL n. 01523 <> 32510 (reversed <> standard) From the definition it follows there is a 1:1 correspondence between the natural numbers and the real numbers (continuum) by adding or removing a "decimal" point. 015 <> .015 (0x10^0 + 1x10^1 + 5x10^2 <> 0x10^1 + 1x10^2 + 5x10^3) 1101 <> .1101 00101.. <> .00101.. 33333..... <> .33333 Finite and repeat natural numbers correspond to the rational numbers, and nonrepeating natural numbers correspond to irrational numbers. Thus there is a 1:1 correspondence between the natural numbers and the continuum. In decimal notation, corresponding to each point in [0,1) is an integer point on the line. The representation of all the points on the line by natural numbers is obvious. Corollary i) The reals are countable. ii) Cantor's Diagonal Argument is wrong because any countably infinite binary sequence corresponds to a natural number, by above. The Argument fails because: a) The countably infinite sequence of countably infinite binary digits is not square (2^n sequences per n digits, ALL n), or b) The constructed sequence whose nth digit is the complement of the nth digit of the nth sequence in the list can only be at the end of the list, and so doesn't exist because the list has no end. [Note by moderator: a natural number is finite (the summation you give doesn't converge unless it contains only a finite number of nonzero terms), so 33333... isn't a natural number. Also, if the "constructed sequence" cannot be in the list used, that confirms that an allinclusive list is impossible, so Cantor's "not enumerable" conclusion is correct.] Last edited by skipjack; June 29th, 2016 at 09:10 PM. 
June 29th, 2016, 10:23 AM  #2  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra  Quote:
Just saying "for all $n$" like some magic mantra doesn't make what you say any less incorrect. Instead of repeating your incoherent nonsense over and over again, try learning what that phrase actually means. Last edited by skipjack; June 29th, 2016 at 09:12 PM.  
June 29th, 2016, 05:57 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,806 Thanks: 1045 Math Focus: Elementary mathematics and beyond  v8archie, please address other members respectfully, regardless of how impertinent their ideas may be. As this thread is redundant I am closing it. 

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