What is the Collatz Problem and how can it be solved?

[moshek]

matt : Thank you for your long answer i feel now comftable with you.
I will study cerfully your answer and replay to it tommorow.

Best

Moshek

 

[matt grime]

IN particular Organic has never produced any evidence to back up his assertion that the countable set of rows in the diagram for N corresponds to a set of card 2^aleph-0. In the case for n columns the rows correspond to the power set of card 2^n, however this is not true in the infinite case, as has been clearly demonstrated. The orginal diagram had only rows with a finite number of non-zero entries, then there came a diagram interleaving with a second array where each row had only finitely many zero entries. The proofs of these facts are elementary and organic has continually refused to either accept these proofs or indicate where they are wrong. His only tactic has been to introduce yet more irrelavant things. If you wish to treat him as a mathematical thinker perhaps you, Moshek, could try and peresuade him to explain where these elementary proofs are wrong; they aren't as anyone can see, but Organic does not extend to us the courtesy he demands of others in reading and trying to understand what others write. We have read what he has written and do understand what he is attempting to do, and we can also see where he is going wrong.

 

[moshek]

Thank you Matt for the extension of your view and your really deep care for mathematics that teach me. I promise you to do all the best I can do for understanding of the situation were we are now !

sincerely
Moshek

 

[Organic]

Matt,

We have read what he has written and do understand what he is attempting to do...

No Matt you you do not understand what I am doing because my point of view is a paradigm change of the infinity concept used by Math language.

There are two kinds of it:

1) Actual infinity (the word "many" cannot be used) which is the unreachable limit of Math language.

2) Potential infinity (infinitely many objects) which is the only form
that can be used by mathematics language.

If you don't distinguish between these two forms of infinity concept, then you cannot say even a single word about my mathematical definitions.

Simple as that.


And for you Moshek, it looks that you have no opinion of your own and you are trying to be nice to everyone.

So, please let me tell you, this is not the way, you have to choose your side.

 

[moshek]

Well Organic you know already that I am on the side were you are since I work on this discovery by myself for the last 20 years. This is not the point here . I try to understand matt as much as I can to find some bridge among the gap now. you must understand that the fact that you don’t have many background in the tradition mathematics make it very difficult to understand you. and they may say by a mistake that it is a Garbage. but they are really care from there heart on mathematics.

Moshek

 

[Organic]

Dear Moshek,

In this case you have to show clearly what is your fundamental paradigm about the infinity concept and also clearly show what do you think we have to do if we want to show to the people of the old paradigm, how they can make Math under the new paradigm.

Can you do that?

 

[matt grime]

As I've said before you are prefectly entitled to whatever philosophical position you want. What I will not accept is you telling us that mathematics cannot cope with something that it makes no claims to explain, and is not mathematical. The only problems arise when we accept your ideas, ideas which are inconsistent and ill-defined. If you want to show mathematics is inconsistent you cannot work from outside, you must demonstrate it internally.

Noticably you don't defend yourself against the allegations that you do not read or understand other people's posts.

It is only your opinion that the rows you write down are in correspondence with a set of cad 2^aleph-0. We can prove it isn't as the property of being in bijection is an elementary equivalence relation, only in your theory is there a problem with this; it is only if one declares that the words 'many' and 'for all' are incompatible with logic that your problems with cantor arises, however that isn't mathematics.

It is now clear that you cannot refute the disproof that the rows are not in correspondence with the power set, which shows how poor your abstract thought is.

Do you want another thing to talk about?

You've demanded we provide a multiplication on N that is complementary to addition.

Let's do that.

1. What do you mean by complementary

2. What is a multiplication?Give the rules this notional operation must define.

Note yours

1. Is not a binary operation from NxN to N

2. Does not associate,

3. Either does not distribute over addition, or have 1 as the identity (if it had both then it would be commutative).

So what must something be to qualify as a 'multiplication' and be complementary to addition.

(And you claim we don't read and understand you)

 

[matt grime]

Originally posted by Organic
Hurkyl,

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.

It's very simple, Organic, and if you understood this your life would be so much better. Although there are 'aleph-0' entries, each entry is either in an odd or even position (at a finite point in the sequence), therefore it makes perfect sense to say the string

(x_i) i in N is 1 when i is even 0 when i is odd.

Aleph-0 is not a number, there is not an aleph-0'th entry in the sequence, every element in the sequence occurs at some n in N, which is either an even or an odd number.

 

[moshek]

Dear Matt !

I am afraid that i will not be able to answer you today i am writing a paper on 01 laws and i have to finish it until thursday.

But i will certanly do that in few days.

take care
Moshek

 

[moshek]

and also to you Dear Organic

Sorry

Moshek

 

[Organic]

Hurkyl,

For example, instead of saying:

...010101

You could say

"The sequence whose n-th term is 1 if n is even and 0 if n is odd"

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.

If we examine again the structure of the most right column of the matrix which is drawn below we find that the length of the list has at least aleph0 magnitude.

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V

Now I am going to show my new point of view about the potential infinity.

First let us look at this list:

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]... 1 1 1 1[b]}[/b]   geometric series 
 ... 1 1 1                    |
 ... 1 1   1                  |
 ... 1 1  /                   |
 ... 1   1 1                  |
 ... 1   1 /                  |
 ... 1   //1                  |
 ... 1  // /                  |
 ...   1 1|1                  |
 ...   1 1|                   |
 ...   1 ||1                  |
 ...   1 //                   |
 ...   /|1 1                  |
 ...   /|1                    |
 ...  / || 1                  |
 ... /  ||                    |
 ... 1  ||                    V

[b]the same can be done with '0' notations[/b]

Shotly speaking the main structure here is:

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]... 1 1 1 1[b]}[/b]   geometric series 
 ... 1 1 1                    |
 ... 1 1                      |
 ... 1 1                      |
 ... 1                        |
 ... 1                        |
 ... 1                        |
 ... 1                        |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          V

[b]again, the same can be done with '0' notations[/b]

Now we shall show thet this information is greather then aleph0 magnitude.

Step 1: we will show again our list in this way:

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,                   |
 ...,1,1,0,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          V

step 2: to make it clearer we shall show it now in this way:

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b] <--> 1 geometric series 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...                          |
 ...,0,0,0,0 <--> 2           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,1 <--> 3           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
...                           |
 ...,0,0,0,0 <--> 4           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,1 <--> 5           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
                              |
 ...                          V

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

Therefore we can conclude that the length magnitude of our list > aleph0 magnitude.

Please pay attention that in this case ...0101 most right column length has 2^aleph0 magnitude.

Therefore the width and length of my 01 matrix are both countable.

    3 2 1 0                   3 2 1 0
   2 2 2 2                   2 2 2 2
   ^ ^ ^ ^                   ^ ^ ^ ^
   | | | |                   | | | |
   v v v v                   v v v v
...0 0 0 0 <--> 1    or   ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2    or   ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3    or   ...1 1 0 1 <--> 3    
...0 0 1 1 <--> 4    or   ...1 1 0 0 <--> 4    
...0 1 0 0 <--> 5    or   ...1 0 1 1 <--> 5   
...0 1 0 1 <--> 6    or   ...1 0 1 0 <--> 6    
...0 1 1 0 <--> 7    or   ...1 0 0 1 <--> 7    
...0 1 1 1 <--> 8    or   ...1 0 0 0 <--> 8   
...1 0 0 0 <--> 9    or   ...0 1 1 1 <--> 9   
...1 0 0 1 <--> 10   or   ...0 1 1 0 <--> 10  
...1 0 1 0 <--> 11   or   ...0 1 0 1 <--> 11  
...1 0 1 1 <--> 12   or   ...0 1 0 0 <--> 12  
...1 1 0 0 <--> 13   or   ...0 0 1 1 <--> 13  
...1 1 0 1 <--> 14   or   ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15   or   ...0 0 0 1 <--> 15  
...1 1 1 1 <--> 16   or   ...0 0 0 0 <--> 16
...                  or   ...

or a mixing of them

 

[matt grime]

your view on infinity means that the rows are simultaneously countable, and that they are NOT in bijection with N. Thus your theory proves a statement that is simultaneous true and false. And you think our is the faulty system?

 

[Organic]

No Matt,

It is true and false iff countability is related to N (as it understood by standard Math).

My point of view clearly shows that this is not the case, and standard Math does not understand the infinity concept.

 

[Hurkyl]

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

It looks to me like there's one block of length 8 for every natural number. (and nothing else)

Thus, I can make a list such that there's one row for each natural number (and nothing else) through the following mapping:

For any natural number n, write it as 8p + q for p and q integers and 1 < q <= 8. Then, I map n with the q-th row in the p-th block.

    3 2 1 0                   3 2 1 0
   2 2 2 2                   2 2 2 2
   ^ ^ ^ ^                   ^ ^ ^ ^
   | | | |                   | | | |
   v v v v                   v v v v
...0 0 0 0 <--> 1    or   ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2    or   ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3    or   ...1 1 0 1 <--> 3    
...0 0 1 1 <--> 4    or   ...1 1 0 0 <--> 4    
...0 1 0 0 <--> 5    or   ...1 0 1 1 <--> 5   
...0 1 0 1 <--> 6    or   ...1 0 1 0 <--> 6    
...0 1 1 0 <--> 7    or   ...1 0 0 1 <--> 7    
...0 1 1 1 <--> 8    or   ...1 0 0 0 <--> 8   
...1 0 0 0 <--> 9    or   ...0 1 1 1 <--> 9   
...1 0 0 1 <--> 10   or   ...0 1 1 0 <--> 10  
...1 0 1 0 <--> 11   or   ...0 1 0 1 <--> 11  
...1 0 1 1 <--> 12   or   ...0 1 0 0 <--> 12  
...1 1 0 0 <--> 13   or   ...0 0 1 1 <--> 13  
...1 1 0 1 <--> 14   or   ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15   or   ...0 0 0 1 <--> 15  
...1 1 1 1 <--> 16   or   ...0 0 0 0 <--> 16
...                  or   ...

or a mixing of them

Both of these are countable (you've even "counted" them!), and so is any "mix" of these.

 

[Hurkyl]

It is true and false iff countability is related to N.

And countability is related to N by definition.

 

[Organic]

Hurkyl,

It is true and false iff countability is related to N (as it understood by standard Math).

Standard math language is a "quntitative orianted system"

My point of view is first of all "structural orianted", therefore I can see standatd math universe But standard Math has no ability to my univerese.

Shortly speaking , in my univerese there is information That standard math ignore.

"STRUCTURAL ORIANTED" THEORY OF NUMBERS IS REACHER THAN "QUANTITATIVE ORIANTED" THEORY OF NUMBERS(= the current standard Math language).

 

[Organic]

Hurkyl,

You did not understand my example so, I'll write is and explain it again:

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,[b][i]1[/i][/b][b]}[/b] <--> [b][i]1[/i][/b] geometric series 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...                          |
 ...,0,0,0,[b][i]0[/i][/b] <--> [b][i]2[/i][/b]           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,[b][i]1[/i][/b] <--> [b][i]3[/i][/b]           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
...                           |
 ...,0,0,0,[b][i]0[/i][/b] <--> [b][i]4[/i][/b]           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,[b][i]1[/i][/b] <--> [b][i]5[/i][/b]           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
                              |
 ...                          V

Standard mapping looks at the block as if it is a one element, but each block has an extra information that gives to the 0 1 sequences the magnitude of 2^aleph0 in this list.

Shortly speaking, when we deal with infinitely many elements the structural information of our data cannot be ignored, and it has a tremendous impact on the results.

In this case, the same ...10101 sequence can have aleph0 magnitude or greather.

Shortly speaking through my more sensative "structural orianted" point of view we can write this:

aleph0 < 2^aleph0 < 3^aleph0 < 4^aleph0 < ...

aleph0 < aleph0+1

As you can see, my aleph0 is not your aleph0 because your aleph0 does not exist, and the reason is: Standard Math system does not distinguish between actual infinity and potential infinity, as I clearly and simply show here:

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

Again no collection of infinitely many elements can be completed.

Therefore the +1 in alpeh0+1 is always beyond the scope of aleph0, which means: there is no bijection between a collection with aleph0 magnitude and a collection with apeh0+1 magnitude.

Simple as that.

 

[matt grime]

Originally posted by Organic
No Matt,

It is true and false iff countability is related to N (as it understood by standard Math).

My point of view clearly shows that this is not the case, and standard Math does not understand the infinity concept.

So you want to claim this is a contradictin in ordinary maths? Well, then you must prove that the enumerated rows are in bijection with a set of cardinality 2^aleph-0 using our mathematics. Since in our mathematics bijectivity is an equivlaence relation, this cannot be true, and therefore something in your logic has gone wrong. This is easily spotted at the point where you claim a geometric series has 2^aleph-0 elements. That is not true, we can prove it is not true. It is only your opinion that it is true. If it is not just an opinion prove it. You can't as you'd have to demonstrate that the notion of bijection is wrong in our mathematics,.

 

[Organic]

Matt,

Please read the last two posts that I wrote to Hurkyl.

Thank you.

 

[matt grime]

I did read your last posts to Hurkyl. They are unclear and suffer from not having enough words explaining them. However they are also self-contradictory as by enumerating them by N you are providing a bijection with N. You are then claiming a bijectio with 2^aleph-0 if we can abuse notation like that. As I can prove that there is no bijection from N to P(N) then you've made a mistake somewhere. The proof is valid in mathematics even if you don't like it. We can use the words 'for all' and 'many'.

Why when you generate a contradiction do you assume the entireity of mathematics is wrong and not even consider the option that perhaps you, who has no formal mathematical training and appears to have never read a mathematics textbook (beyond high-school) could have misunderstood something?


There is nothing in your posts which explains why the cardinality of a geometric sequence is 2^aleph-0. I imagine this will provoke the response that I have a closed mind unwilling to be imaginative. However, as you are attempting to make this claim be true in my interpretation of mathematics you have to write it mathematically, not in your personal language.

So, provide some compelling evidence such as a proof that the set of rows in your diagram has cardinality 2^aleph-0.

 

[Organic]

Dear Matt,

I did read your last posts to Hurkyl. They are unclear and suffer from not having enough words explaining them. However they are also self-contradictory as by enumerating them by N you are providing a bijection with N. You are then claiming a bijectio with 2^aleph-0 if we can abuse notation like that. As I can prove that there is no bijection from N to P(N) then you've made a mistake somewhere. The proof is valid in mathematics even if you don't like it. We can use the words 'for all' and 'many'.

Please read this again:

In this case, the same ...10101 sequence can have aleph0 magnitude or greather.

Shortly speaking through my more sensative "structural orianted" point of view we can write this:

aleph0 < 2^aleph0 < 3^aleph0 < 4^aleph0 < ...

aleph0 < aleph0+1

As you can see, my aleph0 is not your aleph0 because your aleph0 does not exist, and the reason is: Standard Math system does not distinguish between actual infinity and potential infinity, as I clearly and simply show here:

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

Again no collection of infinitely many elements can be completed.

Therefore the +1 in alpeh0+1 is always beyond the scope of aleph0, which means: there is no bijection between a collection with aleph0 magnitude and a collection with apeh0+1 magnitude.

Simple as that.

 

[Organic]

Hurkyl,

It looks to me like there's one block of length 8 for every natural number. (and nothing else)

No it has infinitley many finite collections of '0' or '1' notations for each natural number, that are ignored by Standard Math mapping.

 

[matt grime]

So in order to create a problem with mathematics you see fit to redefine its terms to mean something different.

That does not prove anything. You cannot prove something is incorrect if you are not using it properly.

It is only possible to say that there is a mistake if ou stick within the system.

So try and prove the error without redefining terms to suit your needs. Using only the correct mathematics prove mathematics is not consistent. That is your challenge, that is what you've been claiming is going on.

 

[Organic]

Dear Matt,

That does not prove anything. You cannot prove something is incorrect if you are not using it properly.

By Standard Math aleph0+1 = aleph0 because Standard Math is first of all a "quantitative oriented" system.

My Math is first of all a "structural oriented" system.

Here is some simple example:

By my Math system there is a structural difference between a solid line and a collection of segments that have the same length of this solid line.

Shortly speaking no collection of segments can be a solid line.

They can have the same length, but this is only the quantitative point of view of this comparison.

My point of view is looking for both structural and/or quantitative properties before it air her view about some conclusion.

The structure of a solid line is what I call “an actual infinity”.

This structure cannot be reached by any collection of segments, for example: no collection of glass’s broken pieces can construct a smooth “one piece” glass.

My Math word products are the complementary association between these different structural states.

If we use some analogy then my Math system is the associations between the collection of glass’s broken pieces AND the smooth “one piece” glass.

Therefore my natural number is first of all examined by its structural properties, where quantity is the invariant state which is being kept when structural property is changed (by this way we have more information, which is unreachable by quntitative point of view).

The result of this point of view define the natural numbers not only by their quantitative property but also by their internal structural changes when some quantity is given.

If you understand this fundamental point of view on Math language, then please look again at this example:

[b]
A set is only a framework to explore our ideas.

The concept of an oredered set does not depend on the quantity concept as shown here:

By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication 
and addition are complementary opreations that can be ordered 
by different symmetry degrees where quantity remains unchanged 
for example:

A Number is anything that exists in ({},{__})

Or in more formal definition:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and 
addition ((((+1)+1)+1)+1) ,and vice versa. 

This fading can be represented as:
 

(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)


============>>>

                Uncertainty
  <-Redundancy->^
    3  3  3  3  |          3  3             3  3
    2  2  2  2  |          2  2             2  2
    1  1  1  1  |    1  1  1  1             1  1       1  1  1  1
   {0, 0, 0, 0} V   {0, 0, 0, 0}     {0, 1, 0, 0}     {0, 0, 0, 0}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    (1*4)            ((1*2)+1*2)      (((+1)+1)+1*2)   ((1*2)+(1*2))
 
 4 =                                  2  2  2
          1  1                        1  1  1          1  1
   {0, 1, 0, 0}     {0, 1, 0, 1}     {0, 0, 0, 3}     {0, 0, 2, 3}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1))  ((1*3)+1)        (((1*2)+1)+1)

   {0, 1, 2, 3}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this [i]no-redundancy_no-uncertainty_symmetry[/i])
    |_____|  |
    |        |
    |________|
    |    
    ((((+1)+1)+1)+1)
 

Multiplication can be operated only among objects with structural identity, 
where addition can be operated among identical and non-identical 
(by structure) objects.

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) , ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) , ( ((1,1),1),((1,1),1) ) , ( (((1),1),1),(((1),1),1) )
[/b]

 

[matt grime]

So does this mean that you now accept that there is no problem in Cantor's Diagonal argument in the extant mathematical world where aleph-0 and 2^aleph-0 are clearly and demonstrably not the same, and that the constructions you claim must be true are not in fact constructions that hold in mathematics as is held to be true today? In short do you accept that there is no contradiction in the current mathematics which uses 'infinite' and 'many' and 'for all'?

If not then explain why it is inconsistent within itself. Do not mention you're structural view point, do not mention your definition of any objects, unless they are things that current mathematics holds to be valid. Further, reread the posts where I ask you to explain in what sense your 'multiplication is a 'multiplication' - it is not even a binary operation from NxN to N, is it?

 

[Organic]

Matt,

First let us see what the meaning of a non-empty set is.

A non-empty set of natural numbers does not depend on the quantity of the numbers which included in it.

All we have to know is that each number which included in it fits to the definition of the natural numbers.

The definition is important and not the set that contain its products.

Now, there is one definition and infinitely many products which we call natural numbers.

We define some set, called N, that contains any product that can be defined by the natural numbers definition.

Now we want to know how many products included in N or in other words, we want to know the cardinality of N.

The value of a cardinal is the answer to the question "How many?".

So, what is the meaning of the cardinal concept when we have infinitely many elements?

It cannot be a quantity because we cannot ask "how many?" when we deal with infinitely many elements.

So there are two solutions to this condition.

Solution 1) We can change the basic meaning of cardinality, by saying that the cardinality of infinitely many elements does not measured by quantity but by the magnitude (or size) of the collection.

By choosing this way we define two different meanings to cardinality in Math language that cannot agree with each other.

More than that, by using concepts like magnitude or size we actually saying that there is a way summarize infinitely many elements.

But there is a fundamental conceptual problem here which is: The very nature of infinitely many elements is that they cannot be summarized, because no collection of infinitely many elements is a complete collection.

So there are to big problems two solution 1, which are:

a) A main concept like cardinality has two different meanings that do not agree with each other.

b) By summarizing infinitely many elements we are going against the very nature of infinitely many elements which is: they cannot be summarized, because no collection of infinitely many elements is a complete collection by definition.


Solution 2) We do not change the meaning of cardinality, but the clarity of its result, when we deal with infinitely many elements, which means: when we have a collection of infinitely elements quantity is unknown, then our only well known information is that any element that included in this collection, fits
to the definition of the natural numbers.

Shortly speaking, by using ", ..." notation we clearly saying that we accept the vogue nature if cardinality, when we deal with a collection of infinitely many elements like the set of Natural numbers, which is notated in this way:
{1, 2, 3, ...}.

By this attitude we do not force any artificial summa on infinitely many elements and accept their nature of not being completed.

Cantor chose solution 1, and the result is two separated worlds (finite and infinite) in today's Math.

Also by the transfinite word aleph0+1 = aleph0, 2^aleph0 = 3^aleph0 (and so on) which means that Cantor's attitude has no ability to distinguish between (for example) aleph0+1 and aleph0, and by this result we can understand that this mathematical system looses information (that can be important) on each step of it.

On the contrary solution 2 can distinguish between aleph0+1 and aleph0 because through its point of view aleph0+1 > aleph0 and also 3^aleph0 > 2^aleph0 (and so on) and can use this differences as a valuable information to make Math.

My Math is based on solution 2 that can be used to develop a much more sensitive and interesting systems then solution 1.

 

[matt grime]

Firstly, the cardinals do not need to come with any innate ring structure such as addition - we choose to define one which has certain unintuitive (to some) properties, all of which summarize the simple idea that if S is a set that is not finite, then then S can be put in bijection with plenty of other sets such as Su{1}, Su{1,2},..,SuS etc.

There are symbols (that can be) associated to sets that aren't finite called ordinals, w is the first infinite ordinal and has the property that w is not equal to w+1, because of the nature of order preservation required.

All the things you allude to have
already been defined properly.

A perfect example of these is realized in the (well ordered) sets

{k-1/(n+1) | n,k in N}

with the element r-1/(s+1) corresponding to the ordinal rw+s (there is a small error here in the exact labelling, I hope you will allow that this is just supposed to be an indicator of this area)

There are therefore infinitely many well ordered, distinct sets with distinct ordinals that have the same cardinality.

I'm sorry that you cannot use the word complete for a set that is not finite, no one else has a problem with N being the complete set of Natural numbers, and it not being finite. Perhaps you ought to think that your interpretation of the word 'complete' is at fault?

Notice that I've only used the word infinite once in that, and that was as part of a noun, and at no point did I have to even use the word infinity.

When a mathematician speaks of cardinals measuring 'quantity' or size of sets, it is a generalization of the finite case, such generalizations do not share all of the properties of the original case, but they are useful, and provide us with lots of tools, such as the ability to prove that transcendental numbers exist without having to construct them.

Cardinality does not have two different meanings, it has one.

 

[Organic]

Matt,

Cardinality does not have two different meanings, it has one.

So please tell me exactly how many n's there are in N.

Also ordinals has to be distinguished from each other before we can use them, so redundancy and uncertainty are not allowed.

 

[matt grime]

Asking me to state 'exactly how many ns there are in N' is an ill-founded question. Where have I said there are X many natural numbers (or elements of an infinite set) in anything other vein than an attempt to communicate in your language or illuminate a point?

I have attempted whenever I am trying to be rigorous to switch to more correct language and use the phrase 'the set of rows has card aleph-0' rather than use your abuse of notation and say 'there are aleph-0 many rows' though I have often for expediency abused the notation in the same way. The word 'many' in such instances is not being used in a rigorous way.

The natural numbers have cardinality aleph-0. Aleph-0 tells us what the isomorphism type of the set is. Two sets have the same cardinality iff they are isomorphic. This would be lost in your definition. Cardinality is about the set-maps between sets, that's all.

www.dpmms.cam.ac.uk/~wtg10/ordinals.html

see what a fields medallist thinks.

 

[Organic]

Matt,

Two sets have the same cardinality iff they are isomorphic. This would be lost in your definition

By my definition I can distinguish between aleph0+1 and aleph0.

By your definition you cannot distinguish between them.

So, where is the advantage of your system, that we have to keep?

 

[matt grime]

But what is your definition of aleph-0 and aleph-0 + 1?

Two sets with an isomorphism between them no longer have the same cardinality

Now suppose S and T are sets and there is an injection from S to T. Does your system imply |S|<|T|?

 

[Organic]

Matt,

Because by my point of view the cardinality of infinitely many elements is unknown (no collection of infinitely many elements is completed) I use aleph0 as the notation of this open (non completed) state.

By doing this I can use any operation that we use between finite collections for example aleph0/2 < aleph0.

For example: the number of odd numbers in N is aleph0/2.

 

[matt grime]

But then what is the cardinality of an infinite set that isn't a set of natural numbers, or a subset of them?

For instance, what is the cardinality of the set of finite groups, what is the cardinality of the set of algebraic integers, what is the cardinality of the set of functions from projective n space to projective m space? What is the cardinality of the underlying field of the rank one free module of D_2n over an algebraically closed field of characteristic 2?

There is more to life than just the natural numbers. Unless you about to invent a different cardinal for every set you ain't getting much.

And I can differentiate between the set of integers and the set of even integers because they are different (but isomorphic) sets.

 

[Organic]

Matt,

Aleph0 is only the basis of a collection of infinitely many elements.

For example 2^aleph0 = |R|

For example (that you already know):

<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V

Also please pay attention that 3^aleph0 > 2^aleph0

 

[matt grime]

But why is the card of R 2^aleph-0?

Every real number has an integer part, and there are aleph-0 of those, for each of these surely there are 10^aleph-0 possible decimal expansions of the non-integer part? so |R| is aleph-0*10^aleph-0. But wait what if I chose base 2,3,4 or 7 expansions, surely then |R| is aleph-0*3^aleph-0, so they must be the same, cancelling aleph-0 surely 2^aleph-0=3^aleph-0...

so, what's your justification for defining |R|=2^aleph-0

you said any cosntruction I could do with finite sets I could do with infinite ones, so I just did.So, what's the cardinality of the set of all finite groups?

 

[matt grime]

I've got an even better one.

What's the cardinality of the rationals?

 

[Hurkyl]

By my definition I can distinguish between aleph0+1 and aleph0.

By your definition you cannot distinguish between them.

So, where is the advantage of your system, that we have to keep?

There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

 

[Organic]

Matt,

so, what's your justification for defining |R|=2^aleph-0

My mistake, I mean |R|=base_value>1^aleph0.

surely 2^aleph-0=3^aleph-0...

surely 2^aleph-0<3^aleph-0...

because in 3 notations we have much more combinations in both width and length of the 0,1,2 matrix, then in the width and length of the 0,1 matrix.

Shortly speaking, my system does not ignore these differences between 2^aleph0 and 3^aleph0 matrixes and can use them to make math.

Your system cannot do that.

What's the cardinality of the rationals?

Please look at page 5 in this paper:

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

 

[Organic]

Hurkyl,

There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

But I can be much more sensative then you and find an interesting information that exists between your 0 1 statistic results, for example:

2^aleph0 < 3^aleph0 because base value of 3 creates more information (that can be explored) then base 2.

The same is about aleph0 < 2^aleph0 or aleph0+1 > aleph0.

The finite part of each operation here can be used to give us statistical results, which are much more interesting then any general statistics, which is reduced to and based on 0 1 results.

Your definitions work, but what a price you pay (by ignoring information that can be very important)?

Shortly speaking, your word is a synthetic 0 1 digital world.

My world is an analogical world that can use details and also can give statistical results, which are much more accurate then your 0 1 digital statistical world.

For example, from my point of view the CH problem is an artificial problem that was forced on infinitely many elements that can be understood only by 0 1 digital point of view.

Shortly speaking, from my analogical point of view Cantor's world is nothing but a collection of shortcuts that do not distinguish between the simple and the trivial.

Let me give you some example:

We have this inifintely long periodic patterns list:

0
0
1
0
0
1
0
0
1
.
.
.

Let us say that we want to know the ratio between notations '1' and '0' in this infinitely long list.

By your system we can find a bijection between '1' and '0' notations:

0 <--> 1
0 <--> 1
0 <--> 1
...

So the ratio value does not exist when we deal with infinitely many elements (and it cannot be used to make Math)


By my system the ratio r1=aleph0/3 or aleph0*(1/3), the ratio of r0=aleph0*(2/3) and r1+r0=aleph0.

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

 

[matt grime]

|R| = base_value>1^aleph-0?

what does that mean? base value of what? why isn't 1^aleph-0 1 as it ought to be.
Did you understand why I wrote that 2^alpeh-0=3^alpeh=0 etc based upon the construction of R?Our stastice is about sets, yours is only about sets of natural numbers

if we have 001001001001... I can tell you the ratio as it's the limit of n/3n as n tends to infinity, ie 1/3.Why, if I can see that there is a copy of Q inside N, and a copy of N inside Q, can I not conclude they have the same number of elements? Why must the cardinality of a set depend on how it's written down. For instance, the evne integers are 'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

 

[Organic]

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

 

[matt grime]

But you've not correct th problemt that 'capturer tool does not belong to the element it captures'

your aleph0 is still not a natural number.Don't you in newdigaonlpdf state clearly the rationals are countable, that is |Q|=|N|?

but the even numbers are countable too, so |evens|=|N|, implies aleph0=aleph0/2?

Or would you like to clarify what countable means.

first you state that |R| is 2^aleph-0, then you change that to something that doesn't make sense (just put real numbers in there to see why), and now when asked to clarify that, you come up with something about uniqueness.What do you mean points of R are connected to each other?
The interval [x,x] contains exactly one point in it.
Q also has these properties, or at least the best guess I can make from what you actually write.

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

 

[Organic]

Matt,

'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?

 

[moshek]

I see that it is very exiting here ,
I will join you in few days.

Have fun with mathematics!

Moshek

 

[Organic]

Matt,

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

As I clearly show here:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

No infinitely many unique elements can construct (can be or can use a model of) a solid line.

So you have to deside between infinitely many unique elements XOR solid line.

If you choose infnitely many elements, then their cardinality is the unknown value of infinitely many elements that cannot be completed by definition ("infinite" means no end therefore no completeness).

If you choose solid line then you have no iput that can be used to make math or shortly speaking |R|=?^aleph0.

 

[Organic]

Matt,

(just put real numbers in there to see why),

base_value is any n>1


aleph0 is the cardinal of any arithmetic series of infinitely many elements.

n>1^aleph0 is the cardinal of any geometirc series of infinitely many elements.

 

[matt grime]

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?


The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

Take N, take NU{-1}, and NU{-2}

They both have card, in your world, of aleph-0+1, but they are different sets.

The card of {1,2...,n} is the same as {2,3,...,n+1} yet they are different sets. What makes you think cardinality is in anyway a measure of what the elements of the set are?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

 

[Hurkyl]

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

That is not all I have; I have other things like well-order types, topologies, algebras, and measures.

But it is frequent that cardinality is what I want. In fact, the idea of there being a map from the natural numbers onto a set is so important that it was given a special name.

 

[matt grime]

To echo Hurkyl, two sets having the same cardinaltiy is by definition saying there is a bijection between them, that is all it says. You seem to think that different sets cannot ave the same 'cardinality', at least that is how i interpret you assertion that I can't distinguish between aleph-0 and aleph-0+1, well, how can you distinguish between them? what is aleph-0? You assert |Q| and |N| are the same but surely from the way you construct them |Q| = aleph0*log(aleph0) approx.

we have a very good way of distinguishing between N and Q - they are not equal. They are in 1-1 correspondence, though.

 

[Organic]

Matt,

Please answer to each part of this post.

part 1:

'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?


part 2:

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?

By n>1 I mean any n value greater than 1.

The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

It is not well defined construction because it uses simultanuasly two different models that contradict each other (existing on the same level), which are:

1) A model of inifintly many elements.
2) A model of solid line with on gaps.

What makes you think cardinality is in anyway a measure of what the elements of the set are?

dependency Matt,

Can your body exist without the atoms of it?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

Again, please read page 5 in:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

((aleph0/1)*(1/aleph0))*aleph0=1*aleph0=aleph0


Another important thing:

When I use ", ..." notation in {1, 2, 3, ...} I mean that a set with infinitely many elements cannot be completed, because the meaning of the word "infinite" is "has no end".