My Math Forum Cantor's Diagonal Argument and Binary Sequences

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 May 23rd, 2016, 07:34 AM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 Countably Infinite Reversed Binary Sequence Definitions: Binary Number, Binary Sequence, Reversed Binary Number, Reversed Binary Sequences $\displaystyle BN=p_{n}2^{n}+p_{n-1}2^{n-1}+...p_{0}2^{0}\\ BS=p_{n}p_{n-1}....p_{2}p_{1}p_{0}\\ RBN=p_{0}2^{0}+p_{1}2^{1}+p_{2}2^{2}+...p_{n}2^{n} \\ RBS=p_{0}p_{1}p_{2}...p_{n}\\ p_{i}=0,1\\ BN \equiv BS \equiv RBN \equiv RBS\\$ $\displaystyle CIRBS$, 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
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 Originally Posted by zylo The importance of CIRBS is that it disproves Cantors Diagonal Argument: The natural numbers are countable.
No it doesn't. Since you've provided no evidence to support your argument (in this post) I will leave it at that for now.

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:
 Originally Posted by zylo 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.
This is trivially false, since all of your real numbers are, by your own definition, of finite length (excluding the trailing zeros). Thus your "real" numbers contain no irrationals, and no rational whose denominator is not a power of two. So, according to you neither $(\pi-3)$, nor $\frac13$, nor $\frac15$ is a real number.

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: 125 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
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 Originally Posted by zylo 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.
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: 125 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
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 Originally Posted by zylo $\displaystyle CIRBS$, countably infinite reversed binary sequence, is a RBS to which a countably infinite number of zeros have been added. ... The definitions hold for every finite n, but not for n=$\displaystyle \infty$ (undefined).
$(\pi-3)$ has infinitely many non-zero digits so "$n=\infty$" and your definitions do not hold. If you are in any doubt, simply note that $(\pi-3)$ does not have infinitely many zeros added.

May 23rd, 2016, 06:57 PM   #7
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Quote:
 Originally Posted by zylo By the same token, the fractional part of pi has a unique CIRDS (natural number) associated with it: .14159..... equiv 14159.......
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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