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May 23rd, 2016, 07:34 AM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
Countably Infinite Reversed Binary Sequence Definitions: Binary Number, Binary Sequence, Reversed Binary Number, Reversed Binary Sequences $\displaystyle BN=p_{n}2^{n}+p_{n1}2^{n1}+...p_{0}2^{0}\\ BS=p_{n}p_{n1}....p_{2}p_{1}p_{0}\\ RBN=p_{0}2^{0}+p_{1}2^{1}+p_{2}2^{2}+...p_{n}2^{n} \\ RBS=p_{0}p_{1}p_{2}...p_{n}\\ p_{i}=0,1\\ BN \equiv BS \equiv RBN \equiv RBS\\ $ $\displaystyle CIRBS$, countably infinite reversed binary sequence, is a RBS to which a countably infinite number of zeros have been added. Note n+countably infinite = countably infinite, The definitions hold for every finite n, but not for n=$\displaystyle \infty$ (undefined). $\displaystyle CIRBS \equiv RBS \equiv RBN \equiv BN \equiv$ the same natural number. Example: $\displaystyle 4=1x2^{2}+0x2^{1}+0x2^{0}\\ 4=0x2^{0}+0x2^1+1x2^{2}+0x2^{3}+0x2^{4}+.....\\$ 4=100=001000000000..... Underlining represents a CIRBS, but is generally left out when understood by context. That every CIRBS is a natural number can also be shown by (underlining implied) 1) 000000....... 2) 010000....... 3) 110000....... 4) 001000....... ...................... The importance of CIRBS is that it disproves Cantor's Diagonal Argument: The natural numbers are countable. It is interesting to note that CIRBS establish a 1:1 correspondence between the real numbers between 0 and 1 and the natural numbers: .111000000..... $\displaystyle \equiv$ 111000000..... thus proving the real numbers are countable. Last edited by skipjack; May 23rd, 2016 at 04:30 PM. 
May 23rd, 2016, 08:43 AM  #2  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,615 Thanks: 2604 Math Focus: Mainly analysis and algebra  Quote:
I will note, however, that your CIBRS are not what Cantor means by Infinite Binary Sequences because all of yours end with an infinite string of zeros. His don't. Quote:
Note that, to establish a bijection between two sets it is necessary to do more than provide a single example. Last edited by skipjack; May 23rd, 2016 at 04:37 PM.  
May 23rd, 2016, 08:50 AM  #3 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
It is also interesting to note that, in binary number notation: lim .111111111111111111.... = 1 but .111111111111111..... is never 1, in the same sense that lim 1/n = 0 but 1/n is never 0. 
May 23rd, 2016, 08:56 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,615 Thanks: 2604 Math Focus: Mainly analysis and algebra  This is true (except that .111111... is equivalent to a limit expression), but almost certainly doesn't mean what you think it means.
Last edited by skipjack; May 23rd, 2016 at 04:42 PM. 
May 23rd, 2016, 05:49 PM  #5 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
By the same token, the fractional part of pi has a unique CIRDS (natural number) associated with it: .14159..... equiv 14159....... CIRDS Countably Infinite Reversed Decimal Sequence. It has all worked out so beautifully. In two weeks it will be buried and disappear. 
May 23rd, 2016, 06:04 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,615 Thanks: 2604 Math Focus: Mainly analysis and algebra  $(\pi3)$ has infinitely many nonzero digits so "$n=\infty$" and your definitions do not hold. If you are in any doubt, simply note that $(\pi3)$ does not have infinitely many zeros added.

May 23rd, 2016, 06:57 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 20,370 Thanks: 2007  You know that's incorrect... each natural number has a finite number of digits (when expressed in the usual way, without a decimal separator and without redundant zeros), and you've used that fact many times in your examples.


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