A new point of view on Cantor's diagonalization arguments

[Hurkyl]

I know how I think the list is constructed, and that construction doesn't have any of the magical properties you ascribe to your list.

 

[Organic]

Hurkyl,

Ok, please read in your rigorous way how my list is constructed.

 

[phoenixthoth]

as an aside, this reminds me of the battle between darth maul, obi-wan and qui-gon in star wars episode i. i love this!

organic, I'm not saying you're darth maul or bad or anything and i admire your spirit. will you reply to my posts about three dots not being a proof nor a definiton?

 

[Organic]

phoenixthoth,

three dots don't work for definitons either. can you define real numbers using three dots?

Please write the full represetation of pi in base 2.

 

[Hurkyl]

Ok, please read in your rigorous way how my list is constructed.

It is an array whose rows and columns are labelled by the natural numbers.

The s_{i,j} entry (where the first index is the row and the second index is the column) is a 1 if and only if \lfloor i / 2^j \rfloor is an even number.

 

[phoenixthoth]

how does that address what real numbers are? is your definiton of "real number" "pi written in base 2?" examples prove nothing. you can give 10,000 examples, each with three dots, and not have a definiton. so, tell me, what is the definiton of real numbers, or anything, using three dots?

for example, you say that Z*:={0,1,2,...}.

well, how do i know what's hidden in those three dots? is 3.14 in the list or not? that's something a definiton can decide for you. with just three dots, you have to just shrug your shoulders and say, "well, don't you know what i mean?" well, to play devil's advocate, i don't know what you mean. only a rigorous definiton will elucidate what you mean in a way acceptable to my mathematical standards; anything less will not be tolerated. it is a closed minded approach but view it this way. suppose you're in karate class. you punch in a way not in accordance with teaching and your sensei scolds you (or hits you with a stick!) and says, "NO! YOU SHOULD PUNCH THIS WAY!" you're thinking, what the heck? my punch would hurt someone, so why can't i punch this way? if you were courageous enough to ask, the sensei would reply with this: "we know the best way already. this way of punching is the most effective."

if you want to prove the sensei wrong you need to go to another dojo because most senseis are too caught up in their own ego to listen to you and your what they call ignorance. only once in a blue moon is a new martial art created and not everyone is cut out to make their own style. it's even harder to get people to follow your lead, if anyone even listens to you.

do you understand the metaphor?

so it is closed minded in a way to do what we are asking but it's also the most effective way. how do i know this is more effective? well, because your proof would mean that all infinite sets have the same size, which detracts from the richness of the infinities of sets; that is aestecally unappealling and can be disproven by couterexample (in standard math, of course). it says that there are as many real numbers as there are natural numbers, right? well, can't you see by my picture that the two are *differnent* sizes:
...
___

?

 

[Organic]

Hurkyl,

It is an array whose rows and columns are labelled by the natural numbers.

Now please write what is the magnitude of the length and what is the magnitude of the width of this matrix? , but first pay attention that the length of the matrix depends on all Z* members that are used as power_values of this matrix and determine its length, as we can see here:

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

Let us take again our set:

 {...,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:

 {...,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:

 {...,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
 ...

 

[Hurkyl]

The rows and the columns are indexed by the natural numbers. Thus, they are both of cardinality aleph0.

 

[Organic]

Hurkyl,

Ok, use Cantor's diagonal method on my list and prove that its opposite not in the list.

 

[phoenixthoth]

but we've already told you what's wrong with your proof; therefore, there is no need to do that.

 

[Hurkyl]

If

The s_{i,j} entry (where the first index is the row and the second index is the column) is a 1 if and only if \lfloor i / 2^j \rfloor is an even number.

is accurate, then it doesn't contain the all zeroes sequence. (Which, incidentally, is the sequence produced by the basic diagonal argument)

 

[Organic]

phoenixthoth,

Prove that your three dots example also holds in my case.

 

[Organic]

Hurkyl,

is accurate, then it doesn't contain the all zeroes sequence. (Which, incidentally, is the sequence produced by the basic diagonal argument)

One index is power_value index the other is not, so how you can find a bijection between their results?

 

[phoenixthoth]

Originally posted by Organic
phoenixthoth,

Prove that your three dots example also holds in my case.

i don't need to. I've proven that three dot arguments don't always work. therefore, i don't trust *your* three dots argument. why should i?

i've proven that your tools are flawed. end of story.

 

[Organic]

phoenixthoth ,

And I don't trust the results of your bijection map between Z* members used as arithmetic index and Z* members used as geometric (power_values) index.

 

[Hurkyl]

One index is power_value index the other is not, so how you can find a bijection between their results?

In my version of the construction, both the rows and the columns were labelled by natural numbers.


There is, of course, a bijection between the set of natural numbers and the set of powers of 2. Or more trivially, a surjection from the natural numbers onto the set of powers of 2. (Or the set of powers of numbers)

 

[phoenixthoth]

Originally posted by Organic
phoenixthoth ,

And I don't trust the results of your bijection map between Z* members used as arithmetic index and Z* members used as geometric (power_values) index.

thank you for being candid.

we're near the end of how all debates that end in draws end: we'll have to agree to disagree. i respect your postion, otherwise i wouldn't be bothering, but i disagree. you reject my proof and i reject yours. we've both shot our cannons and we both dodged each other's shots by basically saying "i reject your argument". that's it. the is getting into a debate on what constitutes proof (or in your case, what doesn't) and i don't feel like debating that.

but let me get one last shot in before i go:
i have proved that there is a set such that it can be mapped onto its powerset!

but the set of natural numbers ain't that set! the universal set is the only example i know of.

 

[Organic]

Hurkyl,

Some analogy:

Let as say that you want to compare between red photons and blue photons do you think that you can ignore their energy and look only for their quantity?

 

[phoenixthoth]

here, i'll even prove to you that I'm right.

here's my proof:


.............
.............
.............
.............
.............
.............
.............
.............
.............
.............
.............

if:
1. you accept dots as proof and
2. you don't want to admit that dots don't prove squat
then
3. you will contradict yourself.

 

[Hurkyl]

How about another analogy:

Let's say I want to count how many photons you have. What use is knowing their energy?

 

[Organic]

phoenixthoth,

You know what, please write in your way how my list is constructed,
without using ... .

Can you do this for me?

Thank you.

Organic

 

[Organic]

Hurkyl,

Let's say I want to count how many photons you have. What use is knowing their energy?

Because we are talking about the magnitude of this quantity.

 

[phoenixthoth]

Originally posted by Organic
phoenixthoth,

You know what, please write in your way how my list is constructed,
without using ... .

Can you do this for me?

Thank you.

Organic

i will seriously consider trying but this is really your job as a math researcher.

show me what you get to a rigorous statement and i'll look it over for you. we are here to try to help you, you know.

 

[Organic]

Hurkyl,

You comparing between notations , I comparing between the results behind these notations.

 

[Organic]

phoenixthoth ,

How we can represent this idea in standard notations?

 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
{...,1,1,1,1}<--> 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

 

[phoenixthoth]

i've done this already in my paper. please read it. I've shown exactly where cantor's argument would fail if the rules are expanded. and yes, expanded, and not contracted. *if the rules stay the same, there is nothing wrong with cantor's argument.* so what you're asking me to do is impossible.

 

[Organic]

Dear phoenixthoth,

You asked me what are ",..." trough my point of view.

My answer is: No collection of infinitely many elements is a complete collection, for example:

 {...,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
 ...

To this tree there cannot be a one common father, because in this case
the tree has a finite size.

By using ... notations we say that we cannot reach the state of the one common father.

Shortly speaking, any collection of infinitely many elements is an open collection of "never ending" story.

My aleph0 is an open collection that its magnitude is unknown.

The open collection can appear in infinitely many magnitudes of different open collections, for example:

When we write a=aleph0+1 > b=aleph0 we mean that a is always bigger then b by one more element, end this ratio between a and b does not change unless we choose to change it.

Shortly speaking there is a relative ratio between open collocations that determinate by us, where aleph0 stands for an open collection.

Through this point of view any arithmetic operation keeps its unique influence on the results for example:

aleph0*2 < 2^aleph0 < 3^aleph0 > aleph0^3 > (aleph0^3)/2 ...

By this attitude our information is richer then the transfinite point of view of aleph0, and it can be used in more interesting ways then the way it is used by the transfinite approach.

 

[phoenixthoth]

you're still not writing in the language of math. until you do, mathematicians will ignore you.

having said that, i fully agree with you. it's quite cool, isn't it? but heed the advice in the opening lines of this post.

i believe that i already have formalized what you mean in my tuzfc. so anyone who's half-way interested in what you are writing, including you, should seek to understand my arguments there.

 

[Organic]

phoenixthoth


Your wrote:

three dots are not a complete list of R={0,1,...}. as i said earlier, each dot has too much information in it for that to be more than an *infinitesimal* partial list.

three dots don't work in proofs. (they only help you see. so while we may see what you see, or not, that is *not* a proof.)

proof:
let x equal 1+(-1)+1+(-1)+...
1+(-1)+1+(-1)+...=
(1+(-1))+(1+(-1))+...=
0+0+...=0.
therefore, x=0.

also, x=
1+(-1)+1+(-1)+...=
1+((-1)+1)+((-1)+1)+...=
1+0+0+...=1.
therefore, x=1.

therefore, 0=1.

therfore, if we allow three dots to be a proof then we will have to sacrifice the law of identity. do you see this? do you believe 0=1?

The way you use () in (1+(-1))+(1+(-1))+... case, is different from the way you use () in 1+((-1)+1)+((-1)+1)+... case.

Therefore you get different results, and I don't see any connection between these results and ...

 

[phoenixthoth]

i was wondering if you were going to spot that. you are truly an exceptional student of mathematics.

try this on for size:

x=1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+... can be truncated into EITHER this:
x=1+(-1)+1+(-1)+...
OR but NOT XOR
this:
x=1+(-1)+1+(-1)+1+...

ok?

so assume x=x and get a contradiction. copy my original proof from this point forward and you get 0=1.

 

[Organic]

Dear phoenixthoth,

x=1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+1+(-1)+... can be truncated into EITHER this:
x=1+(-1)+1+(-1)+...
OR but NOT XOR
this:
x=1+(-1)+1+(-1)+1+...

What do you mean by truncated?

Nothing is truncated without SASs interference and this is exactly the meaning of ..., which is: x result can be 0 XOR 1.

Shortly speaking, we have to choose the value of x, which means: x value determinate by SASs.

There is no x result out there without SASs determination, when we deal with infinity.

( By the way, please read this paper: http://www.geocities.com/complementarytheory/CQ.pdf )

 

[phoenixthoth]

you cannot use conjecture as the basis of a proof. not yet.

 

[Organic]

No phoenixthoth,

This invariant information structure

                             ^
            0 XOR 1 = child  |
           /                 |
          /                  |
father = ?               redundancy    
          \                  |
           \                 |
            0 XOR 1 = child  |
                             v   
        <--uncertainty-->

is not a conjecture but a "rigorous" proof which some of its legal properties are uncertainty and redundancy.

If you don't clearly show this in your theory then you miss the whole point.

Also please read this paper:
http://www.geocities.com/complementarytheory/CuRe.pdf

Thank you,

Organic

 

[phoenixthoth]

in no way, shape, or form, is that a rigorous proof.

it is, however, a "RIGOROUS" proof, PErHAps. :P

 

[Organic]

Dear phoenixthoth

If you still keep in your theory the stuffed form of infinity and the stuffed proofs of Euclidian mathematics which are based on objective platonic realm, then we have nothing to talk about Math, because from my point of view Math is meaningless without us as its SASs.

For example please read this:

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

 

[Hurkyl]

I can't believe I never picked up on this before:

          /1
         1 
        / \0
       1   
       /\ /1
      /  0
     /    \0
 ... 1    
     \    /1
      \  1
       \/ \0
       0  
        \ /1
         0
          \0

          /1
         1 
        / \0
       1   
       /\ /1
      /  0
     /    \0
 ... 0    
     \    /1
      \  1
       \/ \0
       0  
        \ /1
         0
          \0

You assert that this is a tree with depth aleph0 and 2^aleph0 leaves, right?

What if we remove all of the leaves? We should be left with a tree with depth aleph0-1 and 2^(aleph0-1) leaves, by your reckoning, right?

Removing the leaves gives

         1 
        / 
       1   
       /\ 
      /  0
     /    
 ... 1    
     \    
      \  1
       \/ 
       0  
        \ 
         0
          
         1 
        / 
       1   
       /\ 
      /  0
     /    
 ... 0    
     \    
      \  1
       \/ 
       0  
        \ 
         0
...

But guess what? This is the exact same tree we started with! (If you don't see it, fill in the next level)

So we must have aleph0-1 = aleph0!

 

[phoenixthoth]

organic, i think "yoda" just had an ah-ha moment! :P

guess he was right about it not seeming to have any magic properties.

ready for a little breaking out of the box?

how about when someone maybe doesn't rigourously prove but rigourously tries to prove then that should be considered "circumstantial mathematical evidence" when something is correct.

 

[Organic]

Hurkyl,

Your calculation is not right because:

1) aleph0 in my point of view stands for a general notation for any collection of infinitely many elements, and its value is flexible.

2) for example: By saying that a=(aleph0-aleph0) < b=aleph0 we mean that there are aleph0 elements in b that are not covered by a.

Also by a=(aleph0-2^aleph0) < b=aleph0 we mean that there are 2^aleph0 elements in b that are not covered by a.

There are no absolute magnitudes when we deal with collections of infinitely many elements, and no arithmetical operation (finite or infinite) can change their property of being infinitely many elements.

Shortly speaking, no arithmetical operation (finite or infinite) can change the “–“ or “+” sign in a collocation of infinitely many elements.

3) in my Binary tree the aleph0 width magnitude and the 2^aleph0 length magnitude, depends on each other, therefore their relative proportion (notated as width=aleph0 < length=2^aleph0) was not changed by your operation.


You still ignore the inner structure of infinitely many elements, because after your operation we have this list:

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

So, as you see aleph0-1 < aleph0

By the way, the result of your oparation is ((aleph0)-1) < ((2^aleph0)-aleph0)and the reason that it is jusut -aleph0 and not -2^aleph0, can be clearly shown here:

 <---Arithmetic magnitude 

 {...,3,2,1,0} = Z*
     2 2 2 2  
     ^ ^ ^ ^   
     | | | |   
     v v v v  
{...,[b]1-1-1-1[/b]} <--> 1  Geometric magnitude(based on the 
 ...,1,1,1,[b]0[/b]  <--> 2          |          thin notations)          
 ...,1,1,[b]0[/b]/                   |
 ...,1,1/0,                   |
 ...,1,[b]0[/b], ,                   |
 ...,1/0, ,                   |
 ...,1|0, ,                   |
 ...,1|0, ,                   |
 ...,[b]0[/b]/ , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          V

After your operation we have:

( ((aleph0)-1) < ((2^aleph0)-aleph0) ) < ( aleph0 < 2^aleph0 )

 

[phoenixthoth]

this tree is very much like the tree of knowledge. but i agree with hurkyl, it is not magical.

 

[Organic]

phoenixthoth,

What do you mean by "magical"?

 

[phoenixthoth]

come on organic, don't play this game. you know exactly what "magical" means. magic means that you can prove something by a picture only without a rigorous proof.

btw, organic, my paper was not about SASs.

 

[Organic]

phoenixthoth ,

What is proof for you?

 

[phoenixthoth]

self-evidence. what's it to you?

there are two of me around here. but who is the master and who is the apprentice?

 

[Organic]

Then why a self-evidence thing has to be proved?

 

[phoenixthoth]

Originally posted by Organic
Then why a self-evidence thing has to be proved?

touche.

why don't you ask yourself the same question, or, rather, why you have to go around proving stuff all the time?

 

[Organic]

What I understand don't has to be proved to me unless someone (including myself) show me that there is a deeper way to understand this thing.

 

[phoenixthoth]

agreement is the seed of salvation, organic. we are in total agreement.

 

[Organic]

Dear phoenixthoth,

Please read the front page of my website:

http://www.geocities.com/complementarytheory/CATpage.html

 

[phoenixthoth]

it looks to me, although I'm just an amateur mathematician, that you have innovation and spirit. yet you lack the ability to articulate yourself within the confines of mathematical rigor.

i would suggest going to school and learning how to do this. this requires a lot of boring homework. i recommend you do your homework.

and, organic, help has arrived.

 

[Organic]

phoenixthoth,

For me my pictures are the best tools to understand my ideas.

I don't want to seat in a boring classroom and learn how to use other person's tools of understanding.

I enjoy every moment in my own independent way to give simple shapes to my ideas, and I don't care if the academic persons don't understand it. and specially pure "stuffed" mathematicians that are doing their best to escape from real life influence on their rigorous methods.

I can find persons that can understand my ideas, and these persons do not afraid to open themselves and their methods to the complexity of the real life, for example:

-----Original Message-----
From: Dr. A.M.Selvam [mailto:amselvam@eth.net]
Sent: Monday, March 01, 2004 10:04 AM
To: Shadmy Doron
Subject: A new approach for the definition of a NUMBER

1 March 2004

Dear Doron  Shadmi 

 I am indebted to you for your email dated 
16 February giving references of your valuable 
research work.
 
 I find your original research work very valuable 
for developing a simple unified theory with ramifications
in the numerical modeling of nonlinear dynamical 
systems/processes. 

 Your research work would benefit many of the scientists 
particularly those who are working in the area of 
numerical modeling.

                                   with best regards
                                    yours sincerely
                              Dr. (Mrs.) A. Mary Selvam

Papers of Dr. (Mrs.) A. Mary Selvam can be found here:

http://www.geocities.com/CapeCanaveral/Lab/5833/pub11.html