Cantor's Diagonal Slash Disproved

[137]

Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

 

However I have been able to visually disprove Cantor's Diagonal argument simply by reordering the sequence .. like such below...

 

 

 

As you can clearly see, in such a sequence every Natural number is listed (& in order) and will eventually get enumerated, corresponding one-to-one to all natural numbers.

 

By listing it this way, I can now show that I can draw a diagonal slash from upper left corner to the lower right corner and this slash contains a infinite number of "0"..

 

X= 000000000000....

 

 

Therefore it goes to show and follows from the above that by changing "any" and "every" nth digit of X I am able to make it "fit" any of the horizontal sequence of elements!!!

 

 

In general, for since I am able to order it in such a way that I can show that the progression of the natural numbers is "slow" enough that it never has a chance to catch up to the "diagonal", then effective at each and every sequence level "X" can be manipulated to be exactly that sequence!

 

Therefore Cantor's Diagonal Slash has been disproved.

 

 

 

Happy Halloween

 

update:

 

for further clarification look at this new graph I posted..

 

 

Essentially for / at every nth digit level of X, (say if X is only 2 digits) there is NO possible combination of X that

anyone could possibly come up with that isn't contained in the sequences demarcated by the red rectangle,

same is true for when x is at three digits, four digits, for any number of digits of X (indeed for any value, discrete or 'real') of

X, there is already EVERY possible combination of that X enumerated one by one according to the natural numbers already belonging in that list! As you can see, it is not possible to come up with any sequence of numbers no matter how "picky" you are that isn't already in the list!

 

Therefore again Cantor's Diagonal Slash is disproved.

[Ben]

Therefore Cantor's Diagonal Slash has been disproved.

Sorry, but what has been disproved is that you know a circular argument when you see it

 

You start with a countable list of natural numbers, and then proceed to claim, using Cantor, that any diagonal set obtained from this list is countable.

 

This is no surprise, in fact it is circular.

[CraigD]

Hi 137 – welcome to hypography. :eek2: Always good to see a set/number theory enthusiast grace our forums.

… However I have been able to visually disprove Cantor's Diagonal argument simply by reordering the sequence .. like such below...

I think you’ve misunderstood Cantor’s diagonal argument, 137. What you’ve sketched doesn’t look much like the usual sketch of a diagonalization proof.

 

Usually, the proof begins with a countable list of infinite sequences where each element is a 0 or 1 (any pair of elements could be used, but 0 and 1, or two different letters, are common). For example:

 

s₁={0,0,0,1,0,0,0,1,1,...}

s₂={0,0,1,0,1,0,0,1,0,...}

s₃={0,0,1,1,1,1,1,1,1,...}

s₄={0,1,1,0,0,0,1,0,1,...}

s₅={0,1,1,0,0,0,1,1,0,...}

s₆={1,0,0,0,1,0,1,0,1,...}

s₇={1,1,0,0,0,1,0,0,1,...}

s₈={1,1,0,0,1,1,0,1,1,...}

s₉={1,1,1,0,0,1,0,0,0,...}

 

“Countable” simply means that every sequence in the list has a natural number assigned to it – the numbers 1 through 9 in the above sketch. Note that the sequences don’t have to be generated in any special way, nor must each be unique. In the above example, I just generated 9 term of 9 sequences randomly, and sorted them. I needn’t have sorted them, or had the number of elements displayed in each sequence be the same as the number of sequences displayed.

 

In general, we describe one of the sequences in the list as:

s_n={s_(n,1),s_(n,2),s_(n,3),...}

 

Now, we construct a new sequence:

u={1-s_(1,1),1-s_(2,2),1-s_(3,3),...}

 

For the above example,

u ={1,1,0,1,1,1,1,0,1,1,...}

 

This making the sequence number and position within the sequence the same is the diagonalization for which such proof are named.

 

Because of the definition used to construct u, we’re guaranteed that it’s different from every s_n in at least one element, and thus, with a bit of thinking, we can see that there’s no way to assign (map) a natural number to every possible infinite sequence, because for any such mapping, this simple diagonal construction method can always produce an infinite sequence that’s not included in it.

 

Cantor's diagonal argument is, I think, a fun one, because a common reaction upon learning is “That’s obvious. How could nobody have thought of that before the late 1800s?!” It and similar kinds of thinking open up doors to the mind-bending realm of transfinite math. :hihi: I wish it were better taught in school, as practically anyone able to understand simple math can understand it, and it’s IMHO way cool.

Therefore it goes to show and follows from the above that by changing "any" and "every" nth digit of X I am able to make it "fit" any of the horizontal sequence of elements!!!

Again, this isn’t how a diagonalization proof works. Although you’re free to generate the infinite sequences any way you like, including changing all of their diagonals elements, the essence of such proofs is generating an new infinite sequence from the countable set of sequences.

 

Though it’s traditional to use only 2 element values in Cantor’s diagonal argument, it’s not an absolute requirement. We could actually use any finite collection of values for which a “make it different” operation is defined. For 137’s example, which uses {0,1,2,3,4,5,6,7,8,9}, we could define:

u={s_(1,1)+1 mod 10,s_(2,2)+1 mod 10,s_(3,3)+1 mod 10,...}

 

“mod” here means the modulo operation, where 1+1 mod 10 = 2, 9+1 mod 10 = 0.

 

In either of the sketches 137 gave, u={2,1,1,1,1,1,1,1,1,…}.