A new point of view on Cantor's diagonalization arguments

[Organic]

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

[matt grime]

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.

[Organic]

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:

{...,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
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /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.

[suyver]

Originally posted by matt grime
My new motto will be don't feed the trolls.

Good motto! [:))]

[Hurkyl]

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.


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

...

What's the next digit?

[Organic]

Hurkyl,

You asked what is the next?


{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
/1 <--> 1
1
/ \0 <--> 2
1
/\ /1 <--> 3
/ 0
/ \0 <--> 4
?.. 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /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.

{..4,3,2,1,0}=Z*
2 2 2 2 2
^ ^ ^ ^ ^
| | | | |
v v v v v
/1
1
/ \0
1
/\ /1
/ 0
/ \0
1
|\ /1
| \ 1
| \/ \0
/ 0
| \ /1
| 0
| \0
..1
| /1
| 1
| / \0
\ 1
| /\ /1
| / 0
|/ \0
0
\ /1
\ 1
\/ \0
0
\ /1
0
\0

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

[Hurkyl]

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.

[Organic]

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:

{...,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

...0 101010 <--> 6 XOR ...1 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.

[Hurkyl]

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.

[Organic]

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 wich 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"?

[Hurkyl]

Can you choose any infinitely long binary sequence wich is not a trivial one like ...01010 or ...111010 and so on?

Such as the sequence <s_n> where:


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


Or


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


Or, given a list L,

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


Or what about this nifty sequence:

If n = p^m 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 s_n is one minus the p^m-th digit of the m-th row of the k-th list you have (or will have) chosen. Otherwise, s_n = 0.

[Hurkyl]

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.

[Organic]

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.

[Organic]

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.

[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. *shrug*

[Organic]

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?

[Hurkyl]

You said you had a list. I'm using it as "input" to create my sequence.

[Organic]

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?)

[Hurkyl]

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.

[Organic]

Hurkyl,

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


{..4,3,2,1,0}=Z*
2 2 2 2 2
^ ^ ^ ^ ^
| | | | |
v v v v v
/1
1
/ \0
1
/\ /1
/ 0
/ \0
1
|\ /1
| \ 1
| \/ \0
/ 0
| \ /1
| 0
| \0
..1
| /1
| 1
| / \0
\ 1
| /\ /1
| / 0
|/ \0
0
\ /1
\ 1
\/ \0
0
\ /1
0
\0

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

[Hurkyl]

How exactly are you using it for output?

[Organic]

Hurkyl,

What exactly do you want to check about this tree?

[Hurkyl]

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?

[Organic]

Hurkyl,

The property of my Binary tree is based on this invariant structure:


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

[phoenixthoth]

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.

[Organic]

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).

[phoenixthoth]

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?

[Organic]

Dont you see that we can always find any given 01 sequence and its opposite in the tree?


{..4,3,2,1,0}=Z*
2 2 2 2 2
^ ^ ^ ^ ^
| | | | |
v v v v v
/1
1
/ \0
1
/\ /1
/ 0
/ \0
1
|\ /1
| \ 1
| \/ \0
/ 0
| \ /1
| 0
| \0
..1
| /1
| 1
| / \0
\ 1
| /\ /1
| / 0
|/ \0
0
\ /1
\ 1
\/ \0
0
\ /1
0
\0

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

[phoenixthoth]

isn't "always" another word for "all?"

[Organic]

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.

[Hurkyl]

You can find any finite-length binary sequence in the tree. You miss most infinite-length sequences.

[Organic]

Prove it.

But first you have to prove that |Z*| < aleph0

[phoenixthoth]

in standard math, is there anything wrong with cantor's diagonal argument?

[Organic]

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:

{...,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
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...

[phoenixthoth]

can you turn that into a rigorous argument? most people eschew "proofs by picture."

[Organic]

Hurkyl wrote,

You can find any finite-length binary sequence in the tree. You miss most infinite-length sequences.


My answer is:

First you have to prove that |Z*| < aleph0

[phoenixthoth]

this is a dumb question, but isn't |Z*| defined to be alpeh0 so aleph0=|Z*|? i can prove that |Z*| has the smallest infinte cardinal number if you like...

[Hurkyl]

Well, seeing how you only tell us about a small portion of the tree, and keep changing it anytime one of us asks you about a specific sequence, I do have to admit I'm only presuming it does not contain every sequence.

So while I don't have a mathematical proof of my claim, it doesn't matter since we're not talking about a mathematically described object. [:)] In most situations, when one person keeps changing their idea every time an objection is raised, it is taken as pretty solid proof that the idea does not cover all objections.


Were you asking me to prove an infinite length sequence exists? I can do that mathematically.


You want me to prove |Z*| < aleph0? What do you mean by Z*? All the meanings I could imagine you mean satisfy |Z*| = aleph0, nor do I see how this statement relates to my assertions.

[Organic]

phoenixthoth,

You missed the point, because |Z*| = aleph0 and I use all Z* members to construct my list, than Hurkyl argument does not hold, see for youself:

...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
{...,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
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...

[quote]

[phoenixthoth]

yeah, but won't you be using standard math? [o)]

i think organic thinks that nothing is wrong with cantor in standard math. i think organic thinks that it is standard math that is wrong. if so, good luck proving that one. one way to do it is to develop another consistent system...

[phoenixthoth]

this is known as an "ad nausum" fallacy. repeating the same argument doesn't make it correct.

[Organic]

yeah, but won't you be using standard math?

i think organic thinks that nothing is wrong with cantor in standard math. i think organic thinks that it is standard math that is wrong. if so, good luck proving that one. one way to do it is to develop another consistent system...

Did you read my paper here: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

[Hurkyl]

I find it hard to see how a matrix with 16 rows and 4 columns can contain every infinite binary sequence.

[Organic]

Hurkyl,

But you don't find it hard to see how Z* = {0,1,2,3,...} isn't it?

[Hurkyl]

Nope. But, you see, Z* comes with a definition. (or axioms, if you prefer)

[phoenixthoth]

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?

[Organic]

Hurkyl,

Because I use Z* members to construct my tree, it stands on Z* definitions.

see for yourself:
...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
{...,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
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /1 <--> 13
\ 1
\/ \0 <--> 14
0
\ /1 <--> 15
0
\0 <--> 16
...

[Hurkyl]

Because I use Z* members to construct my tree, it stands on Z* definitions.

Yes. You used exactly 16 Z* members to label rows, and exactly 4 Z* members to label columns, and some mysterious ellipses which don't come with definitions or axioms to describe them.

[Organic]

Hurkyl,

Please don't do that, you know exactly how my list is constructed so why are you playing this game?

[phoenixthoth]

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

[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
{...,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
...

Now let us make a little redundancy diet:

{...,3,2,1,0}=Z*
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
... 1-1-1-1 <--> 1
\ \ \0 <--> 2
\ 0-1 <--> 3
\ \0 <--> 4
0-1-1 <--> 5
\ \0 <--> 6
0-1 <--> 7
\0 <--> 8
... 0-1-1-1 <--> 9
\ \ \0 <--> 10
\ 0-1 <--> 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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /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
... 1
\ /1 <--> 5
\ 1
\/ \0 <--> 6
0
\ /1 <--> 7
0
\0 <--> 8

/1 <--> 9
1
/ \0 <--> 10
1
/\ /1 <--> 11
/ 0
/ \0 <--> 12
... 0
\ /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
{...,1,1,1}<--> 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
{...,1-1-1-1} <--> 1 Geometric magnitude(based on the
...,1,1,1,0 <--> 2 | thin notations)
...,1,1,0/ |
...,1,1/0, |
...,1,0, , |
...,1/0, , |
...,1|0, , |
...,1|0, , |
...,0/ , , |
...,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

[phoenixthoth]

ok.

[Hurkyl]

The problem, as I see it, is that you seem to be spending a lot of effort to assert that your tools talk about things that they don't.

For instance, what ever concept of "quantity" you have in your system, it is different from what a mathematician calls cardinality. You've done a great deal of asserting that your tools are proving that things mathematicians know about cardinality is wrong, when all you're really doing is discovering the differences between cardinality and your concept.

You seem almost fanatical in this pursuit, which is why many people get turned off by your theorizing. To be frank, because of your approach, I lost all interest in your ideas because you just couldn't seem to get past the "Look, I've proved mathematics wrong!" mentality.

The main reason I keep posting in your threads is because I think you're not a hopelessly lost cause like most so-called crackpots. (and that I'm a glutton for punishment!) You still seem to have the "This is so obvious, why can't they see it the way I do?" mentality about things, but you do sometimes seem to learn and adapt in the face of criticism.


Also, I hope you don't think that rigor is all mathematicians do. It is the ultimate standard, but intuition and heuristics have a lot to do with how we do things. For instance, on a great many problems I will start with "this is how my gut says to do this problem", and then if it looks promising, I begin to fill in the details. Filling in the details often gives me a good proof, and other times it illuminates a flaw I made in my intuitive reasoning. There is a lot of mathematics built up on things that are still conjecture; for instance, there are a lot of theorems of the form "If the Riemann hypothesis is true, then this other statement is true". Mathematically, one does not need to have proved the hypothesis with full rigor to reason about things; you just have to acknowledge that you haven't derived an "absolute" fact, but instead a "relative" fact.

[phoenixthoth]

hmm...

[matt grime]

Oh, well, time to feed my habit again.

It appears by 'complete' you mean a set is complete iff you can write down every single element of it on a bit of paper (possibly a very big bit of paper) without omitting anything. Is that a reasonable interpretation of your definition of a 'complete set'?

[matt grime]

Well, assuming you are using all those terms in the correct manner, the infinite binary tree has no leaves; there is not last level.

So perhas you ought to explain what all those things are. Given you didn't understand why a tree wasn't an uncountable totally disconnected set, it's very hard to even begin to say what you think any of these things are. That, and the fact you refuse to state what they are, of course.

By definition the root and leaf set of a tree is in the tree, when they exist: there is the infinite unrooted binary tree which has neither.

[Hurkyl]

Anyways, I would like to make my point about removing the leaves again:

One of these two diagrams is your "binary tree" with the lables removed.
The other of these two diagrams is your "binary tree" with the rightmost column chopped off, and the labels removed.
To draw each of them, I drew enough of the tree to represent the first 16 rows:


/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
...


/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
...

Can you tell which one is which? I can't.

[phoenixthoth]

seems like a fractal. at least it seems self-similar.

[Organic]

Can you tell which one is which? I can't.

If each notation represents more then one value then there is a difference between those trees (finite of infinte).

Again you counting the number of the notations and not the magnitude that each one of them representing.

If you don't remember then my answer to you was:

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
{...,1,1,1}<--> 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
{...,1-1-1-1} <--> 1 Geometric magnitude(based on the
...,1,1,1,0 <--> 2 | thin notations)
...,1,1,0/ |
...,1,1/0, |
...,1,0, , |
...,1/0, , |
...,1|0, , |
...,1|0, , |
...,0/ , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
...,0, , , |
... V


After your operation we have:

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

Maybe this can help:


/1_1
1_2
/ \0_1
1_4
/\ /1_1
/ 0_2
/ \0_1
... 1_8
\ /1_1
\ 1_2
\/ \0_1
0_4
\ /1_1
0_2
\0_1

/1_1
1_2
/ \0_1
1_4
/\ /1_1
/ 0_2
/ \0_1
... 0_8
\ /1_1
\ 1_2
\/ \0_1
0_4
\ /1_1
0_2
\0_1
...


/1_2
1_4
/ \0_2
1_8
/\ /1_2
/ 0_4
/ \0_2
... 1_16
\ /1_2
\ 1_4
\/ \0_2
0_8
\ /1_2
0_4
\0_2

/1_2
1_4
/ \0_2
1_8
/\ /1_2
/ 0_4
/ \0_2
... 0_16
\ /1_2
\ 1_4
\/ \0_2
0_8
\ /1_2
0_4
\0_2
...

[moshek]

Dear Organic

What is the relation betwen your theoy to Frege fundaumanetal work fron 1878 that establish the foundation of mathematical Logic ?

Thank you
Moshe
[:))]

[Organic]

Hi Moshek,

Please look here:

http://www.physicsforums.com/showthread.php?s=&threadid=16502&perpage=15&pagenumber=1

[moshek]

Thank you Organic !!

There is allot of similarity to your concept of a a number to the way Prege develop first order logic but without the redundancy that you look on. So in some sense Prege work may be consider as a one example to your theory.

What about relativity thoery of Ablert Einstein, since Prege was before Him ?

Thank you
Moshek

[:))]

[Organic]

Moshek,

All I have now is Frege's basic information structures + uncertainty and redundancy as bulit-in properties of it.

Through my point of view these proprties have to be taken as basics properties of any modern theoretical research of any information system, like Math language for example.

[Organic]

After Albert Einstein there is no meaning to talk about space without time and vise versa.

Through my point of view there is no meaning to talk about quantity without structure and vise versa.



General conclusion:


The internal structure of any given quantity (finite or infinite) cannot be ignored.

[moshek]

Dear Organic :

The similarity and also the deferent
of your theory to Prege theory
is really amassing me !!!

Do you mean that Einstein develop new way
to look on the world but he still
was using Newton mathematics
and you suggest us and
alternativ mathematics ?

Moshek
[:))]


www.icm2006.org

[Organic]

Hi Moshek,

Newton mathematics was a real break through that gave us the ability to deal with momentum in the real world.

Now it is about time to deal with complexity in the real world, but in my opinion it cannot be done if structural property of any mathematical product is not examined together with its quantitative property.

Shortly speaking any mathematical product is at least structural/quantitative product.

Ferge started to develop the connection between information's structure and logic, but its colleagues did not understand its attitude, ignored the information structure and developed only the private case of information structure of no_redundancy_no_uncertainty form.

[moshek]

Organic:

I think that Your are talking about mathematics without any
modeling or equations !!

How all this is relate
to how Wittgenstein see mathematics?

Thank you
Moshek

[:))]

[Hurkyl]

I forgot I had aborted my last reply.


If you'll allow me to use your analogy:

Einstein's great idea was that we should stop pretending we know the answer to "What IS the universe?" and focus on the question "What do we know ABOUT the universe?"


Modern mathematics uses the same idea; we don't care what things are, we only care about what we can do with them.


From a purely logical perspective, this is a phenomenal trick; if all of our theorems are based only on "What can we do with these things?", and are completely independant of "What are these things?", then all of our theorems are valid even if we have the wrong answer to "What are these things?"


This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!

(And, for the record, I don't think your answer is even a valid one to this question)

[Organic]

This is why I think your approach is fundamentally flawed. You are focusing so hard on answering the question "What are these things?", but the question is irrelevant!

What is the connection between not ignoring the structural property
of the natural numbers and the question "What are these things?"

The inner structure of the natural numbers can be shown here:

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

[Organic]

Dear Hurkyl (this time please answer to this post),


My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.

Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.

Cantor's thing is a frozen one, mine is not.

I think that my approach is much more interesting and fruitful than Cantor's approach.

Please let me put these two different approaches "on the table" and I would like to examine them together with you.

I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.

Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.

So here it is:

1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.

2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].

( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )

Now, from my point of view I see these basic problems when RT AND LF are in C:

1) If both RT AND LF are in C, then C must be a finite set.

2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.

We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .

Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.

Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.

Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

[Hurkyl]

What is the connection between not ignoring the structural property
of the natural numbers and the question "What are these things?"

Exercise: try defining "structural property" without making any reference to what a natural number "is".



If both RT AND LF are in C, then C must be a finite set.
if RT AND LF are in C AND C is a non-finite set, then the structural property of our information ... collapsed into itself ... and cannot be used as an input by Math language

We math people call that a contradiction. And it CAN be used in "Math language" through the Law of contradiction:

"If P implies a contradiction, then P is false"

Or, more formally:


P \rightarrow (Q \wedge \neg Q) \vdash \neg P


or equivalently


P \rightarrow Q, \neg Q \vdash \neg P



Or, informally, it goes like this:

Let's make an assumption (call it P).
We derive a contradiction.
We conclude our assumption was wrong.

Or, if I may try and translate into Organic-speak:

If P causes our information to collapse, then P cannot be true.



Therefore the expression 2^aleph0 cannot exist and we cannot construct the transfinite universes.

And how does 2^aleph0 relate to anything above this statement in your post?

[Organic]

Exercise: try defining "structural property" without making any reference to what a natural number "is".

I did more then thet, it can be found here:
http://www.geocities.com/complementarytheory/count.pdf

You ignore the inner information structure of the natural numbers,
I don't.

Therefore your natural numbers are private case of my natural numbers
as can be clearly shown here:
http://www.geocities.com/complementarytheory/ETtable.pdf

Or, if I may try and translate into Organic-speak:

If P causes our information to collapse, then P cannot be true.

You cannot translate my ideas by your mathematical tools, because you ignore the information structure of the Binary tree as irrelevant to you, and looking only on its quantitative "shadow" that falling on the "real line".

And how does 2^aleph0 relate to anything above this statement in your post?

So you did not understand me then:

1) On what basis you translate me?

2) After you read all what I wrote can you answer to this?:

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

[Hurkyl]

You cannot translate my ideas by your mathematical tools

The reason for this is because you do not convey your ideas effectively, and you are self-contradictory.

(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")



2) After you read all what I wrote can you answer to this?:

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Can you translate it into math-speak?

[moshek]

Hurkyl :

I there is in mathematics
a definition to "definition" ?

Thank you

Moshek
[:))]

[Hurkyl]

I there is in mathematics
a definition to "definition" ?


I'm sure there are mathematical theories which have a class of objects called definitions, but in general the term "definition" is meta-mathematical, not mathematical, so to answer your question literally, the answer (in general) is no.

[moshek]

Dear Hurkl!

I was glad for your kind answer to me !

Since you are a matematition
but you can't defind "definition"
Way you ask it from Organic ?

Thank you
Moshek
[:))]

[Hurkyl]

I'm not asking him to define definition, I'm asking him to provide one. [:)]


I'm probably somewhat more abstract than "mainstream" mathematics, but I would define "definition" metamathematically as simply one or more (precise) mathematical statements initially taken to be true.


The point, then, is that we can carry out logical deductions from the definitions to derive theorems.

[moshek]

No Hurkl !

The point is only a point !

Let me ask you my question in anothr way:

Do you belive that what happend to physice at 1905
chould Hapend also to mathematics in one day?

Thank you
Moshek

[:))]

[Hurkyl]

It happened to mathematics first. [:)]

[moshek]

when ?

[Organic]

It happened to mathematics first.


If you speak about non-Euclidian geometry then you right but what about QM and Bohr's complementary attitude?


(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")

No, N and R are enumerable by my definitions because my aleph0 is not your aleph0, no more no less.

Also I see that you don't understand my post about the difference between Cantor's aleph0 and my aleph0.

So, here it is again, but please this time stop on each part of it and please ask me about it, if you think that you don't understand it, thank you:

Dear Hurkyl (this time please answer to this post),


My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.

Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.

Cantor's thing is a frozen one, mine is not.

I think that my approach is much more interesting and fruitful than Cantor's approach.

Please let me put these two different approaches "on the table" and I would like to examine them together with you.

I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.

Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.

So here it is:

1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.

2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].

( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )

Now, from my point of view I see these basic problems when RT AND LF are in C:

1) If both RT AND LF are in C, then C must be a finite set.

2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.

We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .

Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.

Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.

Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.

[Hurkyl]

I'm not talking about non-Euclidean geometry.

I'm talking about the idea of eliminating unnecessary assumptions.

Remember what SR was all about; it was proven that the speed of light is measured to be the same in all reference frames. Physicists of the time were trying to add new and mysterious things to physics to "save" their ideas of how the universe should work. Einstein said to heck with it and said that we might as well study what our measurements say.

My memory of timelines sucks, but I think mathematical formalism has several decades on Einstein. If not, it's in full swing now. *shrug* For instance, look at category theory; it cares nothing about the objects themselves, just the fact that they are objects, and that there are functions between objects.

[Organic]

Einstein said to heck with it and said that we might as well study what our measurements say.

What measurements?

Again, what about QM and Bohr's complementary attitude?

[Hurkyl]

What complementary attitude, and what about it?

[Organic]

http://plato.stanford.edu/entries/qm-copenhagen/#4

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

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

More details can be found here:

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

[moshek]

Hurkyl :


I am sorry, but what happend
to physic at 1905 by Einstein
Did not happend yet
to mathematics!

Moshek
[:))]

[matt grime]

Originally posted by moshek
Hurkyl :


I am sorry, but what happend
to physic at 1905 by Einstein
Did not happend yet
to mathematics!

Moshek
[:))]

I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?

[suyver]

Originally posted by matt grime
I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?

1905 is Einsteins "magic year": it is the year in which he published special relativity, the resolution of the photoelectric effect (for which he would get the Nobel prize later on), and the theoretical description of Brownian motion (that almost proved beyond doubt the existence of atoms). These three theoretical discoveries gave a profound paradigm shift in physics.

I have no idea if such coincedences have also been seen in the field of mathematics (I am not an expert), but I guess they probably have (Euler, Gauss, other giants?)...

[moshek]

Dear Matt

Thank you for your nice question to me !!

I will share with you today here very clear evidents that mathematics is standing today in front of changing paradigms like Einstein did in 1905 to Physics.

Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml


Best
Moshek[:))]

[matt grime]

Arguably these are partly mathematical discoveries, at least I was taught about two of the three of them in my mathematics degree.

I suppose Russell would have to qualify, as would the category theory 'paradigm shift' (notice that, organic, shift, not change, shift) and then there was the bourbaki school, and the current shift away from that line of thinking. The mathematics of today is vastly different from that in Gauss's day - a lot of the proofs of that period don't stand up to scrutiny now. Then science was seen as a branch of philosophy, natural philosophy, and had according standards of proof. The shift in mathematics to today's view was gradual, and didn't have this alleged sudden epiphany, but it is arguable that saying physics fundamentally changed in 1905 is missing the years of research that went on up to that point that allowed these new discoveries to be accepted.

And we haven't even mentioned Cantor, Godel, Zermelo-Frankel, Nash, chaos theory, Mandelbrot, and various others to a lesser degree (Brylinski's loop space, Tsygan's simultaneous and independent discovery of Cyclic homology with Connes, Keller and Rickard's independent and simultaneous work on derived equivalences, Brauer's ground breaking work on representation theory...). What standards do you want to use to compare them?

[matt grime]

Originally posted by moshek Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml


Best
Moshek[:))]


I doubt it, Organic has not shown the lsightest interest in learning anything about mathematics to tihs point, why should he start now.

[moshek]

Matt :

Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?


Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.

That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.

Take care
Moshek

[:))]

[Organic]

The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

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

More details can be found here:

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

Man is no longer an observer but a participator, which its influence must be included in any explored system.

[matt grime]

Originally posted by moshek
Matt :

Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?


Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.

That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.

Take care
Moshek

[:))]


That'd be the Hilbert who inspired Organic's username? The one who also said we need never stray from the paradise Cantor created? Wonder why that hasn't sunk in?

You posted saying organic might be interested in some mathematics conference. What evidence do you have that he would even understand the simplest talk there, or care? I remember offering him a link to some papers of Keller explaining operads and A infinity algebras which give structures on trees that he said mathematics cannot describe. He dismissed them as worthless in less than 20 minutes.

[moshek]

Matt:

I am sorry to hear that Organic dismissed significant mathematical paper. Maybe because he have not background to read it.
Well at list you can see that he was influence from Hilbert in the end of his famous lecture since he is trying to develop Organic type of a number with the duality of Quantity and structure.

As far as I can understand big change always started from to most simple thing.

Do you familiar with Wittgenstein attitude to mathematics?


Best
Moshek

[:))]

[Organic]

Matt,

Let me refresh your memory:

I wrote:

Matt, can you give me some address where i can read about A-infinity algebras?

Thank you.

Organic

Last edited by Organic on 01-22-2004 at 01:50 PM

Your response was:

And,no *I* won't tell you anything about A-infinity algebras. I don't beleve that will accomplish anything as you've yet to understand the basics of common mathematics, and these are very advanced concepts.

Then I have found myself these articles, and then I wrote to you:

Matt,

If you wrote this:

Your multiplication is not well defined then, because your 'numbers' are not well defined, that is there is no single element that represents it uniquely. For instance you give me several different answers for 2*3. To deduce anything from this you must acknowledge that '3' is not a number in your system and to call them numbers is misleading.


Then I have to say that you are much more closed system then I thought, that has 0 ability to understand new fundamental ideas about what numbers are.

You are in your closed system cannot understand how multiplication and addition are complement operations that do not changing the quantity of the number, but only its internal scructure, by breaking and unbreaking its internal self symmetry.

Now, I see you jump and saying: "here comes more undefined terms".

All the rigoruos terms are infront of your eyes but for you terms is written text, not for me.

Also I looked at:

http://arxiv.org/PS_cache/math/pdf/9910/9910179.pdf

http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf


They do not based on my fundamental ideas, therefore cannot express my mathematical system.

[matt grime]

My apologies for my poor recollection. But you did dismiss out of hand these objects without thinking about them for anywhere near long enough to even begin to understand them.


I am familiar with Wittgenstein, and Gowers, who is currently the most vocal proponent of this position on mathematics.

This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.


It is an operation on (certain) rooted trees, some subsets of which you label with elements of N. If you cannot even begin to see why you ought to learn about operads given this then it is you who is closed off to 'new' ideas. You've almost never accepted anyone else might have a valid point of view on this.

[moshek]

Dear Matt !

Please sent me more information about Gowers work.

I am really glad that you know already Wittgenstein attitude to the possibility of creation new mathematic and the place of Klein bottle in this creation by adding one more dimension to mathematics as it appear in today paradigm which base on Logic in it center !

Have you notice already on the similarities of Organic work in mathematics to Prague work on creating first order logic?

Thank you
Moshek
[:))]

[matt grime]

He's got a fields medal, I think you can locate him withtout too much trouble.

It is hard to notice any similarities with what organic writes and extant maths because he misuses terms all the time. My favourite at the moment being the insistence that the infinite rooted binary tree is a Cantor Set.
His abuse of the term enumerable or countable is now just annoying, as is his refusal to learn about any of the words he uses.

His use of trees is reminiscent of operads, and the operations are those of semi-groupoids or monads. He often asks us to offer current mathematical objects that does what his do, we have, with these, yet I doubt he's ever read about these to see. Nor does he state what his things 'do' (cf my repeated requests to find out how one decides which trees exhibit 'uncertainty') which as a Wittgenstein-follower you should admit is the most important thing.

[Organic]

Matt,

This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.

You can't understand it do you?

Any product in my number system is a structural/quantitative product.

Therefore multiplication and addition must be examined through this point of view, and from this point of view multiplication and addition are complementary operations.

This is a new approach, and partial treatment about it maybe can be found in A-oo algebras here:
http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf
in what is called "multiplication inside the inner-product-diagram" (page 26).

[moshek]

Matt:

Wittgenstaein said : "..Mathematics have no foundation in Set theory or in any other theory. Mathematics is depend only on our living.." .

maybe this is way Organic ask us what is the real meening of the 3 points in :

{1,2,3,... }

Or about the very similarity of Rashel paradox and Cantor diagolazation metod to prove that R in uncontable.

I found something about Gowers , thank you !
I understand that his new idea about mathematics deal with the connection between mathematics and Physics.

Well i defiantly agree with that direction.

Do you know the Michel Atiya ( Also a well known fields medal) said in his lecture at the conference about the unity of mathematics at Harvard university ( I was there)
that we are waiting to a new Newton that will broke the Enigma between mathematics and Physics

So do you think that maybe Gowers
is our new Newton ?

Thank you
Moshek

[:))]

[matt grime]

I do get it, Organic, and because I get it I see that you're being imprecise:

you are calling it a multiplication on N, aren't you?

take two elements in N, say [2] and [3], you have shown what [2]*[3] isn't another natural number, [n].

[2] and [3] in your model are a sets of trees satifying certain properties, their product give you some of the trees in the set [6], doesn't it?

I don't think you can then be justified in calling it a multiplication on N.

I am a pure mathematician, and am happy with abstraction and objects that are apparently meaningless, but you do not provide enough information to say what's going on! Repeatedly I've said you're more than welcome to develop all this machinery, it's just that no one else understands what you're doing because you do not offer enough explanation or define anything in terms that allow other people to reconstruct your objects. If you forgot all the mumbo-jumbo about maths needing to remember the people participating in it, you'de get a lot further. Of course you can't do that because to you it's vitally important, though no one else can see why.

[moshek]

Hurkel !

I ask you about the futher
so way did you answer me
about the past ?

Moshek
[:))]

[matt grime]

Originally posted by moshek
Matt:

Wittgenstaein said : "..Mathematics have no foundation in Set theory or in any other theory. Mathematics is depend only on our living.." .

maybe this is way Organic ask us what is the real meening of the 3 points in :

{1,2,3,... }

Or about the very similarity of Rashel paradox and Cantor diagolazation metod to prove that R in uncontable.

I found something about Gowers , thank you !
I understand that his new idea about mathematics deal with the connection between mathematics and Physics.

Well i defiantly agree with that direction.

Do you know the Michel Atiya ( Also a well known fields medal) said in his lecture at the conference about the unity of mathematics at Harvard university ( I was there)
that we are waiting to a new Newton that will broke the Enigma between mathematics and Physics

So do you think that maybe Gowers
is our new Newton ?

Thank you
Moshek

[:))]


No.

Gowers prefers to deal with elementarily stated problems, that are therefore hard to solve.

If you want a mathematical physicist then Baez, Dolan, Atiyah (still. Mathematically I am his partial grandson as I was supervised in my phD for a year by one of his students), Witten, Grothendieck (if he's stoppped doing biology), Orlov or Lusztig seem more likely.

Perhaps Hurkyl doesn't desal with the futher because there's no such word as futher. Do you mean future or father?


Mathematics is precisely what we make it, I agree, which seems to conflict with Organic's apparently previously held belief that the natural numbers do not exist independently of set theory.

[matt grime]

Originally posted by Organic
Matt,

[2]*[3]=[6] but because my nanural number is a structural/quantitative product, there is more than one solution to [6], which are dependend in the inner structure of [2] and [3], for example:

[2]=(1,1) , ((1),1)
[3]= (1,1,1) , ((1,1),1) , (((1),1),1)

Therefore:

[2]*3=((1,1),(1,1),(1,1)) or (((1),1),((1),1),((1),1))
[3]*2=((1,1,1),(1,1,1)) or (((1,1),1),((1,1),1)) or ((((1),1),1),(((1),1),1))


which justifies my assertion that the is no unique element that represents [n] and the multiplication is therefore not well defined - something you riduculed in that reposted message.

[moshek]

Matt :

Do you know that Alein Connes the great mathematician who invent Non -commutate Geometry ( also a fields medal) end with his interesting lecture the conference "100 to Hilbert" at August 2000 at U.C.L.A when he said that we need today a new understanding in mathematics that is base on Geometry and not on logic ?

Organic:

Thank you for sharing with us
your very interesting noncomutative arithmetic.


Moshek
[:))]


Who was killed by a solders and said:

Bring me one point so i can change the whole world .

[moshek]

Matt :

Good to know that !!!

I am Sorry for my English mistake in my question to Hurkel
yes you are right i mean future. ( of mathematics)

Well your Grandfather gave the best and maybe the only lecture that was relate really to the question of the unity of mathematics ( as was posed by Hilbert at the end of his lecture).

Please ask him if i quote him correctly about a change in the paradigm of mathematics !!!.

And if he said also :

"We are waiting today to a new Newton ?

Maybe you know another possibility if it is not Gowers ?

Thank you
Moshek
[:))]

[Organic]

Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

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


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


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

The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

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

More details can be found here:

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

Man is no longer an observer but a participator, which its influence must be included in any explored system.

It mean that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

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

[moshek]

Dear Organic:

We all know already that you are not so good in what is called mathematics...


But :

Your gentle treatment of number with there inherent duality is seeing the world in the eyes of a child.

Like Einstein did to relativity theory it is the only way to create a unmoors change with the paradigm.

So you share with us by this
a positive intepatatino to Gödel theorem.

My Best wishes
Moshek


[:))]

[matt grime]

The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?

[moshek]

Matt:

well It's all about modeling of the world, like Newton did !
Einstein put us in the world but still use Newton mathematics.

Did you heard or read Wolfram Book "new kind of science"


Please Ask Michel Atiya about redefinition of the 6 problem of Hilbert ! maybe Organic is trying to share with us his solution to this very difficult problem.

as you know the 6,8,16 problems
are still opme in the list.

Thank you
Moshek


[:))]


p.s I study now his very nice book on commutative algebra of Atiya and Mecdonalds.

[Organic]

Matt,

Instead of reading the details of my previous post you take some name and play with it.

Please show me some influence on basic thing like the natural number, by one of these theories.

[moshek]

Dear Matt:

We can say that the origin of the possibility to a paradigm change in mathematic was at the end of Hilbert famous lecture. Unfortanly that math world relate mostly to the famous 23 list of open problem which Hilbert himself said that the are only example.
This was 5 years before Einstein appears to the world. Wittgenstaein pose already about this possibility also when he talk about develop anew dimension to mathematics in the geometry of the Klein bottle. In the Epilog of the nice book of Ian Stewat “Nature Numbers” he also talk about he need to invent new –mathematic which will be flexible and rigid together. Alein Connes talk about a new understanding that may come to mathematic beside the regular logic. And Michel Atiya lecture about the need to broke the enigma of the connection between mathematics and physics

“We are looking for a new Newton today..” he said at the conference about the unity of mathematics in Honored of Israel Gelfand 90 birthday.

Yes Hurkal is right !

A similar change happened already but it was ~2,500 years ago when Hipasus of metapontum discover that irrational number exist. Which Contradict Pythagoras “Every thing is Number”.

I ma sure you know the legend what his teacher did to him after that.

Organic share with us here his discovery about Number when time is Vanish.
So you must hold the quantity and the structure always together.



Take care
Moshek

















[:))]

[matt grime]

Organic would need to demonstrate the the natural numbers are not sufficient for the job they are used for for that parallel to hold here.

The greeks thought that all numbers on their number line were rational originally. This is not true, therefore their assumption is wrong.

Where do we go wrong with the definition that the natural numbers are the basic counting blocks of finite sets? What has gone wrong?

Sure we can add all the extra structure to it you want, but it doesn't alter the basic facts.

You could argue the 'inventing' the negative integers, then the rationals, then the reals and then the complexes are all paradigm shifts, but they don't mean the naturals are incomplete within themselves.

We dont' assume those are all the arithmetical objects that exist.

As I keep telling organic he's just playing around with basic operads and monads, if he wants to add extra structure to them that's fine, but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers. That is not the objection to what he's doing. The objection is that he's making wild unsubstantiated and plainly false claims about things he doesn't understand.

[moshek]

Matt:

You Ignore by your question to Organic
all my background that i share with you here
as i promise you few hour ago!

Moshek


I will let Organic to answer your question about his numbers.
[:))]

[Organic]

Matt,

First, you have to show me that you understand what I am doing.

For example, it is the second time that you write general things about my work instead of reading and and reply detailed comments on this:

Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

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


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


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

The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

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

More details can be found here:

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

Man is no longer an observer but a participator, which its influence must be included in any explored system.

It mean that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

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


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

The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?

Please show me some influence on basic thing like the natural number, by one of these theories.

[Organic]

but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers.

So, you dont understand that the stantard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.

Do you get it?

A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:

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

[matt grime]

Originally posted by Organic Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

This would only hold true if I had ever said that the Natural Numbers are the only thing in mathematics. We have N, simply, N, the set of natural numbers. Then there is (N,+) which is a semi-ring. Then the is (N,<) the ordinals. And who can forget (N,+,<) the ordered semi-ring. On top of that one might do (N[-1],+), ie Z under adding formal inverses. Not forgettiing the ordering too. What about localizing and getting a field, Q, and completing to get R, and closing to get C? Then there's the division rings and Wedderburns' structure theorem over that.

Of course there is the general theory of semi-rings etc that 'has N as a private case of a larger language', in you words.

There is a whol rich tapestry of objects that have N, Z, Q or R as the simplest version. This apparent paradigm shift is already there.

If you incidentally knew about the quantized structures above and the rigorous framework behind quantum mechanics, then perhaps you'd understand the analogies.

[Organic]

Matt,
Two critical things you ignore when you define N members:

1) A research of your ability to count:
http://www.geocities.com/complementarytheory/count.pdf

2) that natural numbers are first of all information forms, therefore
Their minimal existence must start form here:

(1*4) ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's
(((+1)+1)+1*2) ={{{1},1},1,1} clarity-degree
((1*2)+(1*2)) ={{1,1},{1,1}} (no uniqueness)
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's
clarity-degree
(uniqueness)



After these two "MUST HAVE" steps, we can continue to develop the next numbers in the number system.

[matt grime]

Why are these must have steps? We can define the natural numbers without them, so they obviously aren't must have are they? Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?

[Organic]

Matt,

The answer is very simple.

Any number is first of all an information form, therefore any aspect of information form MUST be researched by us, where our cognition’s abilities to research information MUST be included too.

Form this point of view, redundancy AND uncertainty cannot be ignored, and through this approach(which is not an extra approach but the MINIMAL approach to understand the natural number concept) we can clearly show that the standard natural numbers are only a one and only one private case of verity of information forms, which are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.

Man is no longer an observer but a participator, which its influence must be included in any explored system.

The above is the QM paradigm shift that is not understood yet by the current community of pure mathematicians.

For example: Be aware that what you call a function is first of all a reflection of your memory.



Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?

A set is only a framework that helps us to explore our ideas, no less no more.

When there is a paradigm shift this framework is chaneged too.

[matt grime]

It would be nice if you told us what the uncertainty and redundancy of a number is. So that we know what we MUST be aware of.

[Organic]

The answer is here:

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

[matt grime]

Originally posted by Organic
The answer is here:

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

Ah, of course! The answer to my question is in another pdf! This one doesn't even mention the words uncertainty and redundancy. In what way is that an answer?

Do you mean to imply that numbers are uncertain because if we have set of identical objects we cannot distinguish between them? I don't see why that makes numbers uncertain. I see why it makes identifying identical objects impossible, but that has nothing to do with quantity. Suppose I just had one bead. I blink, I see an identical bead in the same place. is it the same bead? I do that for more than 1 bead.. what has quantity got to do with it?

[Organic]

From your response it looks that you did not read the pdf file.

Please reply if you have some technical problems to open the pdf.

Thank you,

Organic

[matt grime]

I opened the pdf link you gave and didn't see the word uncertainty mentioned once, please tell me which line it is on. Or, as the pdf is only a page long, try stating in plain simple English what you mean the the uncertainty of a number, or its redundancy. It's another simple request. Actually it's the same simple request isn't it? A simple paragraph starting:

The redundancy of a number is...

or perhaps it ought to start

A number is redundant if...


or even,

given a number n, and some object related to n, then the object is redundant if...

[Organic]

Dear Matt,

You are going too far.

This pdf is not a technical paper, but a simple test that answer to the question:

what are the minimal conditions that give us the ability to count?

Please read it again from this point of view, and don't search for any definitions there, just try to understand this simple test.

Thank you.

Organic

[cantormath]

Cantor's Diagonalization Argument

Theorem-The interval [0,1] is not countably infinite.
Proof:-The proof by contradiction proceeds as follows:

Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in order). In the case of numbers with two decimal expansions, like 0.499 ... = 0.500 ..., we chose the one ending in nines. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows:

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...
We shall now construct a real number x in [0,1] by considering the kth digit after the decimal point of the decimal expansion of rk.

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...

The digits we will consider are indicated in bold and illustrate why this is called the diagonal proof. From these digits we define the digits of x as follows.
if the kth digit of rk is 5 then the kth digit of x is 4
if the kth digit of rk is not 5 then the kth digit of x is 5
For the example above this will result in the following decimal expansion.

x = 0 . 4 5 5 5 5 5 4 ...

The number x is a real number (we know that all decimal expansions represent real numbers) in [0,1] (clearly). Hence we must have rn = x for some n, since we have assumped that ( r1, r2, r3, ... ) enumerates all real numbers in [0, 1]. However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3, ... ). This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction.

Hence the assumption that the interval [0,1] is countably infinite must be false.

Q.E.D.

It is a direct corollary of this result that the set R of all real numbers is uncountable. If R were countable, we could enumerate all of the real numbers in a sequence, and then get a sequence enumerating [0, 1] by removing all of the real numbers outside this interval. But we have just shown that this latter list cannot exist. Alternatively, we could show that [0, 1] and R are the same size by constructing a bijection between them.

[Tom Mattson]

Cantormath,

Welcome to Physics Forums! And you're right, there is no argument.

What you've stumbled upon here is a piece of crackpottery from the old days when we adopted an "anything goes" attitude in the Theory Development Forum. We've since tightened things up so that we only allow things that make sense. :smile:

So feel free to stick around and enjoy the Forums. Don't worry about this thread, because the orginal poster isn't even here anymore.