A new point of view on Cantor's diagonalization arguments

In summary, the conversation is discussing a new perspective on Cantor's diagonalization arguments and thanking individuals for their contributions. The conversation also delves into the topic of alephs and the differences between conventional mathematics and the speaker's own system. The speaker's system claims to be more expansive than Cantor's transfinite universes, with different relationships between aleph0 and 2^aleph0. The conversation also touches on the concept of magnitude in relation to the binary tree representation.
  • #1
Organic
1,224
0
Hi,

In this pdf (+ its links)http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
you can find a new point of view on Cantor's diagonalization arguments.

I really want to send a BIG THANK YOU to Matt grime and Hurkyl for their hard time with me.

Yours,

Organic
 
Physics news on Phys.org
  • #2
One and exactly one post on this this time:

The alephs in that article are not the alephs of proper mathematics (conventional mathematics if that is what you prefer); they are not equivalence classes of sets modulo bijective correspondence; two sets that are bijective have different alephs associated to them in that article, at least that is the only way to read the sentence aleph-0 is not aleph-0+1 if they are both to be cardinals; whatever they are they do not obey the definitions that every one is used to; do not think that it is a commentary on the use and proof of Cantor's theorem, it is not, as it does not follow the same conventions; I don't know if he's still claiming this but an example would be organic's claim that the 'cardinality' of the reals was strictly greater than the Naturals yet both were enumerable (countable) despite his agreement there was no bijection between them.

Any issues that are raised are purely a function of refusing to follow the conventions,apparently under the impression that there is some higher pure definition of these things that we as mathematicians are ignoring by putting our dirty meanings on them.

My new motto will be don't feed the trolls.
 
Last edited:
  • #3
Matt,

Ok, prove by your system that my matrix does not have the complete 01 combinations.

...0101 and ...1010 are in the list, for example:

Let us take again our set:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]<--> 1
 ...,1,1,1,0 <--> 2
 ...,1,1,0,1 <--> 3 
 ...,1,1,0,0 <--> 4 
 ...,1,0,1,1 <--> 5 
 ...,1,0,1,0 <--> 6 
 ...,1,0,0,1 <--> 7 
 ...,1,0,0,0 <--> 8 
 ...,0,1,1,1 <--> 9 
 ...,0,1,1,0 <--> 10
 ...,0,1,0,1 <--> 11
 ...,0,1,0,0 <--> 12
 ...,0,0,1,1 <--> 13
 ...,0,0,1,0 <--> 14
 ...,0,0,0,1 <--> 15
 ...,0,0,0,0 <--> 16
 ...
Now let us make a little redundancy diet:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
...  [b]1[/b]-1-1-1 <--> 1
     \  \ \0 <--> 2
      \  0-1 <--> 3 
       \  \0 <--> 4 
       [b]0[/b]-[b]1[/b]-1 <--> 5 
        \ \[b]0[/b] <--> 6 
         0-1 <--> 7 
          \0 <--> 8 
 ... [b]0[/b]-[b]1[/b]-1-1 <--> 9 
     \  \ \0 <--> 10
      \  [b]0[/b]-[b]1[/b] <--> 11
       \  \0 <--> 12
       0-1-1 <--> 13
        \ \0 <--> 14
         0-1 <--> 15
          \0 <--> 16
 ...
and we get:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
          /1 <--> 1
         1 
        / \0 <--> 2
       1   
       /\ /1 <--> 3 
      /  0
     /    \0 <--> 4 
 ... [b]1[/b]    
     \    /1 <--> 5 
      \  [b]1[/b] 
       \/ \[b]0[/b] <--> 6
       [b]0[/b]  
        \ /1 <--> 7
         0
          \0 <--> 8
          
          /1 <--> 9 
         1
        / \0 <--> 10
       [b]1[/b]  
       /\ /[b]1[/b] <--> 11
      /  [b]0[/b] 
     /    \0 <--> 12
 ... [b]0[/b]    
     \    /1 <--> 13
      \  1
       \/ \0 <--> 14
       0  
        \ /1 <--> 15
         0
          \0 <--> 16
 ...
you have repeatedly said that proper mathematics is wrong and its error means that Cantor is wrong, and that there is only one kind of infinity.
My system is reacher then Cantor's transfinite universes bacause:

1) By my system aleph0+1 > aleph0 , 2^aleph0 < 3^aleph0

2) By Cantor's system aleph0+1 = aleph0 , 2^aleph0 = 3^aleph0


By the way, when we move from the 01 matrix representation to the Binary Tree representation, the meaning of the word magnitude become clearer, because several sequential 1 or 0 notations of each column in the matrix, are compressed to a single notation, which its magnitude equivalent to the quantity of the notations that it represents.
 
Last edited:
  • #4
Originally posted by matt grime
My new motto will be don't feed the trolls.

Good motto!
 
  • #5
Cardinality, as defined by mathematics, is useful because it tells us things about set functions.

|A| = |B| iff there is a bijection between A and B.
|A| <= |B| iff there is a surjection from B onto A.
|A| <= |B| iff there is an injection from A into B.
|A| < |B| iff |A| <= |B| and not |A| = |B|.


Yours does not do this, thus it cannot even serve as a substitute for cardinality.


Code:
{...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
          /1 <--> 1
         1 
        / \0 <--> 2
       1   
       /\ /1 <--> 3 
      /  0
     /    \0 <--> 4 
 ... [b]1[/b]    
     \    /1 <--> 5 
      \  [b]1[/b] 
       \/ \[b]0[/b] <--> 6
       [b]0[/b]  
        \ /1 <--> 7
         0
          \0 <--> 8
          
 ...

What's the next digit?
 
  • #6
Hurkyl,

You asked what is the next?

Code:
{...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
          /1 <--> 1
         1 
        / \0 <--> 2
       1   
       /\ /1 <--> 3 
      /  0
     /    \0 <--> 4 
 ?.. [b]1[/b]    
     \    /1 <--> 5 
      \  [b]1[/b] 
       \/ \[b]0[/b] <--> 6
       [b]0[/b]  
        \ /1 <--> 7
         0
          \0 <--> 8
Answer 1: Both cases and their opposites are already in the complete Binary Tree, therefore no sequence has do be added to the tree.
Code:
 {..4,3,2,1,0}=Z*
   2 2 2 2 2
   ^ ^ ^ ^ ^
   | | | | |
   v v v v v
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]1[/b]      
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
  
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]0[/b]     
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
 ...
Shortly speaking, this tree has the magnitude of 2^aleph0 enumerable unique combinations of infinitely wide (= aleph0 magnitude) 01 sequences.


--------------------------------------------------------------------------

Answer 2: Maybe this time you are going to understand the beauty of redundancy and uncertainty as inherent fundamental properties of Math language.

1) Please this time look and read carefully this pdf:

http://www.geocities.com/complementarytheory/Identity.pdf


2) Also please read this pdf about the symmetry proprty:

http://www.geocities.com/complementarytheory/LIM.pdf

--------------------------------------------------------------------------

Cardinality, as defined by mathematics, is useful because it tells us things about set functions.

|A| = |B| iff there is a bijection between A and B.
|A| <= |B| iff there is a surjection from B onto A.
|A| <= |B| iff there is an injection from A into B.
|A| < |B| iff |A| <= |B| and not |A| = |B|.


Yours does not do this, thus it cannot even serve as a substitute for cardinality.
Because I proved that there exists an enumerable list with 2^aleph0 magnitude, all what you wrote holds only between collections with finitely many objects.

Another alternative is to accept my dynamic point of view on collections of infinitely many elements saying that aleph0 is a general and flexible quantity, which its particular magnitude determinates by operations that are based on finite and/or infinite values, for example:

a=aleph0+1 > b=aleph0 means that there is always 1 element in a that cannot be covered by b.

Also 2^aleph0 < 3^aleph0, 2*aleph0 > aleph0, aleph0^aleph0 > 2^aleph0,
and so on.

Shortly speaking the elements are based on unknown or incomplete quantity.
 
Last edited:
  • #7
Answer 1: Both cases and their opposites are already in the complete Binary Tree, therefore no sequence has do be added to the tree.

Where?

Last time, you said that ...101010 was row #6. However, we now see that is not row #6.

You now seem to assert it is row #22. However, if you go out 2 more digits, you'll find that it is not row #22.

And it is not row #54. Nor is it row #118.


In fact, for every natural number M, I can tell you a specific digit in which the sequence in row #M does differs from ...10101010.

In other words, for any natural number M, ...10101010 is not row #M.



what you wrote holds only between collections with finitely many objects.

What I wrote are the actual definitions of the symbols involved.
 
  • #8
Hurkyl,

In other words, for any natural number M, ...10101010 is not row #M.
And this is exactly what happens when we try to find a mapping between infinitely long enumerable collections with different unique structural properties.

For example: aleph0 < 2^aleph0 < 3^aleph0 ... but each one of them can be represented by its own unique enumerable list.

Shortly speaking, when we deal with collections with infinitely many elements, their unique structural properties can't be ignored.

So when we try to compare between two collections with infinitely many elements, first we have to compare between their unique invariant structural properties, and if they are not the same, there cannot be a bijection between these infinitely long collections.

Cantor did not pay attention to the invariant structural property that exists in any collection of infinitely many elements.

An example of 2^aleph0 and 3^aleph0:
Code:
 {...,3,2,1,0}=Z*           {...,3,2,1,0}=Z*
     2 2 2 2                    3 3 3 3
     ^ ^ ^ ^                    ^ ^ ^ ^
     | | | |                    | | | |
     v v v v                    v v v v
{...,1,1,1,1}<--> 1        {...,2,2,2,2}<--> 1
 ...,1,1,1,0 <--> 2         ...,2,2,2,1 <--> 2
 ...,1,1,0,1 <--> 3         ...,2,2,2,0 <--> 3
 ...,1,1,0,0 <--> 4         ...,2,2,1,2 <--> 4
 ...,1,0,1,1 <--> 5     /   ...,2,2,1,1 <--> 5
 ...,1,0,1,0 <--> 6    /    ...,2,2,1,0 <--> 6
 ...,1,0,0,1 <--> 7    \    ...,2,2,0,2 <--> 7
 ...,1,0,0,0 <--> 8     \   ...,2,2,0,1 <--> 8
 ...,0,1,1,1 <--> 9         ...,2,2,0,0 <--> 9
 ...,0,1,1,0 <--> 10        ...,2,1,2,2 <--> 10
 ...,0,1,0,1 <--> 11        ...,2,1,2,1 <--> 11
 ...,0,1,0,0 <--> 12        ...,2,1,2,0 <--> 12
 ...,0,0,1,1 <--> 13        ...,2,1,1,2 <--> 13
 ...,0,0,1,0 <--> 14        ...,2,1,1,1 <--> 14
 ...,0,0,0,1 <--> 15        ...,2,1,1,0 <--> 15
 ...,0,0,0,0 <--> 16        ...,2,1,0,2 <--> 16
 ...                        ...

Another very important conclusion:

From this point of view there is no fixed platonic realm waiting for us to discover it.

For example:

In base 2 there can be at least to different results to this mapping
Code:
...[b]0[/b] 101010 <--> 6    XOR    ...[b]1[/b] 101010 <--> 6
In this case we have to choose between more than one alternatives, therefore the "right" mapping depends on our decisions as living creatures.
 
Last edited:
  • #9
Would you agree that the following two statements are true:

For any binary sequence I choose, there exists a list of binary sequences that contains said sequence.

For any list of binary sequences I choose, there exists a binary sequence not on that list.
 
  • #10
Hurkyl,
For any binary sequence I choose, there exists a list of binary sequences that contains said sequence.
Can you choose any infinitely long binary sequence which is not a trivial one like ...01010 or ...111010 and so on?
For any list of binary sequences I choose, there exists a binary sequence not on that list.
What do you mean when you say "I choose"?
 
Last edited:
  • #11
Can you choose any infinitely long binary sequence which is not a trivial one like ...01010 or ...111010 and so on?

Such as the sequence [itex]<s_n>[/itex] where:

[tex]
s_n := \left\{
\begin{array}{ll}
0 \quad & \mbox{n is even} \\
1 \quad & \mbox{n is odd}
[/tex]

Or

[tex]
s_n := \left\{
\begin{array}{ll}
0 \quad & n = m^2 \mbox{(where m is some integer)} \\
1 \quad & \mbox{otherwose}
[/tex]

Or, given a list L,

[tex]s_n := 1 - \mbox{(the n-th digit of the n-th row of L)}[/tex]


Or what about this nifty sequence:

If [itex]n = p^m[/itex] for some prime p and some integer m, and p is the k-th prime, and you will have chosen at least k lists in your lifetime, then then [itex]s_n[/itex] is one minus the [itex]p^m[/itex]-th digit of the [itex]m[/itex]-th row of the k-th list you have (or will have) chosen. Otherwise, [itex]s_n = 0[/itex].
 
  • #12
What do you mean when you say "I choose"?

I mean that if, by any method, we happen to have a list in our consideration, one for which there is no "choice" to be made in constructing it (so it is really a list, and not just a method for generating lots of lists), then we can find a sequence not on that list.
 
Last edited:
  • #13
Hurkyl,

But what you show is the general structure that someone has to "break" and give a specific 01 sequence as a result.

Your tools cannot do that, because you cannot describe a result witch is not a trivial 01 repetitions.
 
  • #14
Hurkyl,
I mean that if, by any method, we happen to have a list in our consideration, one for which there is no "choice" to be made in constructing it (so it is really a list, and not just a method for generating lots of lists), then we can find a sequence not on that list.
By the way I used to construct my 01 list, we can find any 01 unique sequence and its opposite in the list.

But again you have no mathod to define a non-trivial sequence.
 
Last edited:
  • #15
If you've constructed the list, then there should be a method to compute the n-th digit of the n-th row of the list. I can then use this method to construct the sequence whose n-th digit is 1 - the n-th digit of the n-th row of your list. *shrug*
 
  • #16
Hurkyl,

If you've constructed the list, then there should be a method to compute the n-th digit of the n-th row of the list. I can then use this method to construct the sequence whose n-th digit is 1 - the n-th digit of the n-th row of your list.
But first you have to define some input, can you do that?
 
  • #17
You said you had a list. I'm using it as "input" to create my sequence.
 
  • #18
Hurkyl,

Can you use a matrix of aleph0 x 2^aleph0 as an input?

All you can do is first choose your unique 01 path until some finite place, and then it is easy to find this finite 01 sequence and its opposite in infinitely many places in the above matrix.

(By the way why did you move my thread to theory development?)
 
  • #19
You said it was a list. (which, by definition, has only aleph0 rows)


And yes, if you output this list, I don't see why I cannot use it as an input.


I moved it here because you're not doing mathematics. You may be intent on studying the topics that mathematics likes to study, but you're not doing it in a mathematical fashion. I don't remember the circumstances, but you seemed to prefer theory development to philosophy, so I move your posts here once I think it's clear that you don't want to do things in a mathematical fashion.
 
  • #20
Hurkyl,

Please look again on this tree and tell me exactly how to you want to use it as an input.

Code:
 {..4,3,2,1,0}=Z*
   2 2 2 2 2
   ^ ^ ^ ^ ^
   | | | | |
   v v v v v
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]1[/b]      
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
  
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]0[/b]     
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
 ...
 
  • #21
How exactly are you using it for output?
 
  • #22
Hurkyl,

What exactly do you want to check about this tree?
 
  • #23
A list of properties that uniquely specify it would be nice.


Just to be entirely clear, let me ask this question:
Can I label each leaf with a unique natural number?
 
  • #24
Hurkyl,

The property of my Binary tree is based on this invariant structure:
Code:
            1 = child
           /
          /
Father = ?
          \ 
           \ 
            0 = child
The number of the Childs depends on any existing Z* member = {0,1,2,3,...} used as the power_value of each level in the tree.

Because |{0,1,2,3,...}| = aleph0, and these members are used as power_values for each level in the tree, the result can't be but a tree width with aleph0 magnitude and a tree length with 2^aleph0 magnitude.

Each child is the beginning of infinitely long sequence of 01 unique combinations.

We can label each child with a unique natural number but this is only an illusion of a bijection that can clearly shown here:

http://www.geocities.com/complementarytheory/Countable.pdf
 
Last edited:
  • #25
here it is organic, plain and simple.

these guys just aren't going to buy into your theories until you can show them how cantor's arguments fail using their language. you have to start with the axioms of set theory. you have to define functions and such. you have to define onto functions. you have to look at powersets. you have to use their language or else they won't believe you. and they're not going to necessarily try to learn your language, which ain't math (no offense intended), so you have to come to their level and do the following: write out cantor's argument as he wrote it and tell them exactly which line or axiom or whatever you think is wrong. and you may not convince them until you give a countexample they can believe. it's just not credible to draw a tree with dots on it and call that a proof. mathematicians eschew such "proofs." they worry a heck of a lot about what hidden assumptions you might be making when you write three little dots.

my problem with your three little dots is that each "dot" represents an infinite enumerable set.

i on the other hand have written them something very similar to what you are intending. in my article, i show the following:
1. there is a set x such that there is a function f that maps x onto the powerset of x.
2. if there is a function that maps x onto its powerset then its powerset contains at least one "fuzzy set".
3. if a set's powerset contains no fuzzy sets then there is no function that maps x onto its powerset. (this and 2 are logically equivalent)

i spell out all my assumptions and all that good stuff. i also claim that what i do fits with set theory rather than being a replacement that no one should bother looking at. (i could, for example, have a set theory where there is only one axiom, the universal set axiom, but that wouldn't be too interesting.) i contend that my tuzfc is an interesting set theory and it has some cool implications.

my problem is that no one reads it, for whatever reason, and gives me feedback. so for all i know it's complete trash. I've stared at it so many times i don't know heads from tails. it looks fine to me but what do i know?

so these three results and my article i think are what you want to accomplish: an ammendment to the cantor argument. a revision. i fully agree that cantor needs revision but you're not going to convince anyone the way you're trying to do it. but that shouldn't be the point. you do it because you enjoy discovering mathematics as do i. i don't honestly really care if my theory is right or publishable because it was so fun to create. if it was a waste of time, then c'est la vie. not the first time I've had a set (lol) back.

i urge all of you, organic and hurkyl especially, to really give my paper a chance and read it. all feedback is welcome.

hurkyl, i hope I've given organic enough feedback so that i can throw that plug in for my thread; hope this won't be considered trolling his thread.
 
  • #26
Dear phoenixthoth,

One of the big problems of highly advanced systems is that some fundamental property was forgotten behind.

For example, let us say that you finished building a house and then you discover that some fundamental calculations about the strength of the first floor are wrong, and it means that you can't let people to live in this house.

You have no choice but to rebuild the house.

The way I constructed the binary tree give it length of 2^aleph0 magnitude on width of aleph0 magnitude.

If Standard Math using the word "all" when defines set Z* then aleph0=2^aleph0 because there is a bijection between N and P(N).
 
Last edited:
  • #27
If Standard Math using the word "all" when it define set Z* then aleph0=2^aleph0 because there is a bijection between N and P(N).

i simply don't understand why that should be or is the case. can you give me more details in your reasoning without overloading me? can you sketch your proof of that?
 
  • #28
Dont you see that we can always find any given 01 sequence and its opposite in the tree?

Code:
 {..4,3,2,1,0}=Z*
   2 2 2 2 2
   ^ ^ ^ ^ ^
   | | | | |
   v v v v v
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]1[/b]      
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
  
          /1  
         1 
        / \0  
       1   
       /\ /1   
      /  0
     /    \0   
     [b]1[/b]    
    |\    /1   
    | \  [b]1[/b] 
    |  \/ \[b]0[/b]  
   /   [b]0[/b]  
   |    \ /1  
   |     0
   |      \0  
 ..[b]0[/b]     
   |      /1   
   |     1
   |    / \0  
   \   [b]1[/b]  
    |  /\ /[b]1[/b]  
    | /  [b]0[/b] 
    |/    \0  
     [b]0[/b]    
     \    /1  
      \  1
       \/ \0  
       0  
        \ /1  
         0
          \0  
 ...
 
  • #29
isn't "always" another word for "all?"
 
  • #30
Yes they are the same, but now i am talking about Standard Math, so when we are using "always" or "all" then Cantor's diagonal method does not hold because we can always find any given 01 unique sequence and its opposite in the tree.
 
  • #31
You can find any finite-length binary sequence in the tree. You miss most infinite-length sequences.
 
  • #32
Prove it.

But first you have to prove that |Z*| < aleph0
 
Last edited:
  • #33
in standard math, is there anything wrong with cantor's diagonal argument?
 
  • #34
In standard Math The opposite of any given diagonal has to be added to the list, therefore no list of magnitude aleph0 can be in a bijection with R members which means that |R| is uncountable.

But look at this:

...0101 and ...1010 are in the list, for example:

Let us take again our set:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]<--> 1
 ...,1,1,1,0 <--> 2
 ...,1,1,0,1 <--> 3 
 ...,1,1,0,0 <--> 4 
 ...,1,0,1,1 <--> 5 
 ...[b],1,0,1,0[/b] <--> 6 
 ...,1,0,0,1 <--> 7 
 ...,1,0,0,0 <--> 8 
 ...,0,1,1,1 <--> 9 
 ...,0,1,1,0 <--> 10
 ...,[b]0,1,0,1[/b] <--> 11
 ...,0,1,0,0 <--> 12
 ...,0,0,1,1 <--> 13
 ...,0,0,1,0 <--> 14
 ...,0,0,0,1 <--> 15
 ...,0,0,0,0 <--> 16
 ...
Now let us make a little redundancy diet:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
...  [b]1[/b]-1-1-1 <--> 1
     \  \ \0 <--> 2
      \  0-1 <--> 3 
       \  \0 <--> 4 
       [b]0[/b]-[b]1[/b]-1 <--> 5 
        \ \[b]0[/b] <--> 6 
         0-1 <--> 7 
          \0 <--> 8 
 ... [b]0[/b]-[b]1[/b]-1-1 <--> 9 
     \  \ \0 <--> 10
      \  [b]0[/b]-[b]1[/b] <--> 11
       \  \0 <--> 12
       0-1-1 <--> 13
        \ \0 <--> 14
         0-1 <--> 15
          \0 <--> 16
 ...
and we get:
Code:
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
          /1 <--> 1
         1 
        / \0 <--> 2
       1   
       /\ /1 <--> 3 
      /  0
     /    \0 <--> 4 
 ... [b]1[/b]    
     \    /1 <--> 5 
      \  [b]1[/b] 
       \/ \[b]0[/b] <--> 6
       [b]0[/b]  
        \ /1 <--> 7
         0
          \0 <--> 8
          
          /1 <--> 9 
         1
        / \0 <--> 10
       [b]1[/b]  
       /\ /[b]1[/b] <--> 11
      /  [b]0[/b] 
     /    \0 <--> 12
 ... [b]0[/b]    
     \    /1 <--> 13
      \  1
       \/ \0 <--> 14
       0  
        \ /1 <--> 15
         0
          \0 <--> 16
 ...
 
Last edited:
  • #35
can you turn that into a rigorous argument? most people eschew "proofs by picture."
 
<h2>1. What is Cantor's diagonalization argument?</h2><p>Cantor's diagonalization argument is a mathematical proof used to demonstrate the existence of uncountable sets. It was developed by Georg Cantor in the late 19th century and is based on the idea that the cardinality of the real numbers is greater than the cardinality of the natural numbers.</p><h2>2. How does Cantor's diagonalization argument work?</h2><p>Cantor's diagonalization argument works by assuming that a set is countable and then constructing a new element that is not in the set, thus proving that the set is uncountable. This is achieved by creating a diagonal sequence of elements that are different from each element in the original set.</p><h2>3. What is the significance of Cantor's diagonalization argument?</h2><p>Cantor's diagonalization argument is significant because it revolutionized our understanding of infinity and led to the development of set theory. It also has important implications in other areas of mathematics, such as the study of real numbers and the concept of infinity.</p><h2>4. Can Cantor's diagonalization argument be applied to other sets?</h2><p>Yes, Cantor's diagonalization argument can be applied to any set that is suspected to be uncountable. It is a general proof technique that can be used to show the existence of uncountable sets in various mathematical contexts.</p><h2>5. Are there any criticisms of Cantor's diagonalization argument?</h2><p>Yes, there have been some criticisms of Cantor's diagonalization argument, particularly in regards to its philosophical implications and its use of the concept of infinity. Some argue that it relies on assumptions that are not necessarily true, while others question the validity of using a proof by contradiction in mathematics.</p>

1. What is Cantor's diagonalization argument?

Cantor's diagonalization argument is a mathematical proof used to demonstrate the existence of uncountable sets. It was developed by Georg Cantor in the late 19th century and is based on the idea that the cardinality of the real numbers is greater than the cardinality of the natural numbers.

2. How does Cantor's diagonalization argument work?

Cantor's diagonalization argument works by assuming that a set is countable and then constructing a new element that is not in the set, thus proving that the set is uncountable. This is achieved by creating a diagonal sequence of elements that are different from each element in the original set.

3. What is the significance of Cantor's diagonalization argument?

Cantor's diagonalization argument is significant because it revolutionized our understanding of infinity and led to the development of set theory. It also has important implications in other areas of mathematics, such as the study of real numbers and the concept of infinity.

4. Can Cantor's diagonalization argument be applied to other sets?

Yes, Cantor's diagonalization argument can be applied to any set that is suspected to be uncountable. It is a general proof technique that can be used to show the existence of uncountable sets in various mathematical contexts.

5. Are there any criticisms of Cantor's diagonalization argument?

Yes, there have been some criticisms of Cantor's diagonalization argument, particularly in regards to its philosophical implications and its use of the concept of infinity. Some argue that it relies on assumptions that are not necessarily true, while others question the validity of using a proof by contradiction in mathematics.

Similar threads

  • Set Theory, Logic, Probability, Statistics
2
Replies
55
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
17
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
797
  • Atomic and Condensed Matter
Replies
0
Views
282
  • Biology and Medical
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
3
Replies
93
Views
16K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
3K
  • Other Physics Topics
Replies
4
Views
4K
Replies
8
Views
1K
Back
Top