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

[Organic]

Hurkyl and Matt,

Please look again at this model:

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

Now, please show me a map between infinitely many intersections
representing R set, and the element notated by oo.

If there is no such a map then you have a simple proof shows that infinity or infinite concept in your system is not well defined.

Shortly speaking, you don't know what are you talking about when you use concepts like infinite or infinity in your system.

Cantor, Dedekind, and each one of you as professional mathematician who continue to use their conceptual mistake about the infinite or infinity (by forcing infinitely many elements or intersections of R set on oo) have no reasonable model to talk about.

By forcing infinitely many elements or intersections of R set on oo all you get is a circular and closed system that running after its own tail, therefore prove meaningless proofs when researching infinite or infinity concepts.

And the reason is very simple:

You are not aware to the limits of your system.

And Matt stop telling me about the difference between infinity and infinite, because in both cases a mapping between infinitely many elements is used by standard Math, see for your self:

http://mathworld.wolfram.com/Infinite.html

http://mathworld.wolfram.com/Infinity.html

Don't try to tell me that what is written in Wolfram is wrong, because
I'll send you immediately to the philosophy forum.


Another "great" example of infinity by standard Math can be found here:

http://mathworld.wolfram.com/PointatInfinity.html

 

[matt grime]

Originally posted by Organic Now, please show me a map between infinitely many intersections
representing R set, and the element notated by oo.

why? what do you mean by the infinitely many intersectiosn representing R set? why is it important to have this map - there is a trivial one sending everything in the domain to the symbol infinity. it's not clear what you want here, or why.

If there is no such a map then you have a simple proof shows that infinity or infinite concept in your system is not well defined.

but it is. something is infinite if it is not finite. i agree that infinity is not well defined, it is contextual, in the same way as there are different kinds of multiplication operations on different groups

Shortly speaking, you don't know what are you talking about when you use concepts like infinite or infinity in your system.

i don't think you understand what we mean by anything in mathematics

Cantor, Dedekind, and each one of you as professional mathematician who continue to use their conceptual mistake about the infinite or infinity (by forcing infinitely many elements or intersections of R set on oo) have no reasonable model to talk about.

what does that sentence mean? force infinitely many elements onto something? intersections of R set on oo? they don't make sense.

By forcing infinitely many elements or intersections of R set on oo all you get is a circular and closed system that running after its own tail, therefore prove meaningless proofs when researching infinite or infinity concepts.

And the reason is very simple:

You are not aware to the limits of your system.

And Matt stop telling me about the difference between infinity and infinite, because in both cases a mapping between infinitely many elements is used by standard Math, see for your self:

http://mathworld.wolfram.com/Infinite.html

http://mathworld.wolfram.com/Infinity.html

Don't try to tell me that what is written in Wolfram is wrong, because
I'll send you immediately to the philosophy forum.


Another "great" example of infinity by standard Math can be found here:

http://mathworld.wolfram.com/PointatInfinity.html [/B]

why must wolfram be correct? infinity as they have it is a useful notion that encapsulates the idea of being 'not finite' and perhaps it isn't they who are wrong but you who does not understand what is writte there?

you are the one misusing (mathematical) language and saying infinity is a set of some kind or is {__}.

think for a second and define multiplication. see? probably not.

the symbol infinity is used in a variety of ways, the point at infininty of the Riemann sphere, the sum from 1 to infinity and so on. they all have the common thread of denoting 'not finite', or 'at no finite point'. Why do you insist that there is this ACTUAL INFINITY out there? what is it? please, define it clearly. if you are going to use {__} again try and define that becuase you have not produced a defintion that anyone has accepted or understood.

look on the websites you list. show me where

"mapping between infinitely many elements is used"

is written, or anything approaching it. are you trying to use the idea that a set is infinite iff it is in bijection with a proper subset of itselt? but that doesn't tell you what infinity is does it? people abuse language by saying 'there are an infinity' of real numbers, but the key here is that it is a phrase 'infinity of', and it means that there are an infinite number of, it doesn't mean infinity is a set in the way you think it is.

 

[Hurkyl]

Suppose the real numbers is countable. Choose any enumeration of them.

Create a countable collection of intervals such that the i-th interval contains the i-th real number, and has length 1/2ⁱ.

This collection of intervals contains every real number, however, the total length of all of the intervals is 1.

 

[Organic]

Hurkyl,

This is exactly what R is, a fractal where each part of it is the same in any scale that we choose.

Cantor himself used this invariant self similarty upon scales to define R, or what is called sometimes "Cantor set":

http://mathworld.wolfram.com/CantorSet.html

Cantor set is nothing but a Binary-Tree.

Please look here:

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

As you can see Cantor set exists in the open interval ({},{__}),
Therefore R cannot use the model of a line.

Shortly sparking the "real line" (a collection of infinitely many objects that construct a one solid element) is a conceptual mistake of modern mathematics, and any result or research that is based on it is nothing but a waste of time.

You can use any collection of nice symbols that you want, but there is nothing but nonsense behind them.

Please read my paper about the CH problem:

http://www.geocities.com/complementarytheory/CL-CH.pdf

 

[Organic]

Matt,

The Math you use (when dealing with the non-finite) is valuable like a point at infinity.

you are the one misusing (mathematical) language and saying infinity is a set of some kind or is {__}.

think for a second and define multiplication. see? probably not.

1) There is no an objective thing like Mathematical language which is disconnected form the people who create it, so there is no use to repeat again on this false thing.

2) For Multiplication please read this:

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

 

[matt grime]

Originally posted by Organic This is exactly what R is, a fractal where each part of it is the same in any scale that we choose.

Cantor himself used this invariant self similarty upon scales to define R, or what is called sometimes "Cantor set":

that is not the definition of the cantor set; R is not a cantor set. A cantor set is a pefect hausdorf compact totally dsiconnected etc subset of the real line and is unique up to homeomorphism

Cantor set is nothing but a Binary-Tree.

no it isn't. a binary tree does not a priori come with a topology, but giving it one won't work because it is clearly never going to be totally disconnected and perfect etc

Shortly sparking the "real line" (a collection of infinitely many objects that construct a one solid element) is a conceptual mistake of modern mathematics, and any result or research that is based on it is nothing but a waste of time.

You can use any collection of nice symbols that you want, but there is nothing but nonsense behind them.

irony isn't dead!

 

[Organic]

Cantor set has (by standard Math) the power of the continuum.

Therefore |R|=2^aleph0.

It is easy to show that 2^aleph0 is Cantor set where Cantor set is a Binary Tree:

                ?
__________________________________

      1                    0
_____________        _____________

  1       0            1       0
_____   _____        _____   _____

1  0    1  0         1  0    1  0  
__ __   __ __        __ __   __ __

You know Matt, it is amazing to see how the educational system took your
Independent way of thinking and shaped it to its faceless uniformed shape
which is full of second hand bombastic names that sometimes there is nothing
behind them.

This system killed any flexibility and curiosity that has to be natural parts
of a good researcher, and it did it so good until you can't see simple things
that are standing in front of your eyes.

My heart with you because I think this is a real tragedy.

 

[matt grime]

the thing you draw isn't even the binary tree in you own paper, it isn't a tree - which are the leaves, the vetices, nodes, whatever? or at least it isn't a tree in anything other than a trivial way.

2^{aleph-0} is a cardinality, it isn't a set, why do you say things are the same when they aren't?

do you know what any of the words compact hausdorff disconnected mean?

thanks for your sympathy. the educational system has completely killed my research abilities, which is why I've done a phd (in maths); yes, your logic is faultless. what it has inculcated in me is a dislike of undefined and therefore unprovable assertions.

 

[Organic]

the thing you draw isn't even the binary tree in you own paper

As I said my heart is with you.

Please look at this:

http://www.mathacademy.com/pr/prime/articles/cantset/

 

[Organic]

                ?
__________________________________

      1                    0
_____________        _____________

  1       0            1       0
_____   _____        _____   _____

1  0    1  0         1  0    1  0  
__ __   __ __        __ __   __ __

Is exactly this:
http://www.geocities.com/complementarytheory/PTree.pdf

and this:
http://www.geocities.com/complementarytheory/3n1proof.pdf

 

[matt grime]

Originally posted by Organic
As I said my heart is with you.

Please look at this:

http://www.mathacademy.com/pr/prime/articles/cantset/

thank you for yet another pointless post, i know perfectly well what a cantor set is, i also know about graph theory. I'm sorry that you don't bother to look up any thing you use until too late, but the tree you draw in your own paper is the infinite bifurcating diagram (infinite in the sense of the number of leaves)and isn't a cantor set - it is connected for instance. as it must be, a tree has the property that any two nodes are connected by a unique path. or didn't you know that? oh look once more your ignorance leads to a problem in the mathematics.

 

[Organic]

No, the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation.

For eample:

    ?
    |
   / \
  /   \
 /     \
 |     |
 1     0

And also Cantor set:

                ?
__________________________________

      1                    0
_____________        _____________

  1       0            1       0
_____   _____        _____   _____

1  0    1  0         1  0    1  0  
__ __   __ __        __ __   __ __

 

[ahrkron]

Originally posted by Organic
No, the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation.

Organic, you are blurring and confusing concepts. It seems that you have a lot of conceptual problems with basic math.

 

[Organic]

ahrkron,

Please give me an example.

 

[ahrkron]

I just did. Integration and differentiation have nothing to do with binary trees.

You can probably use both graphs and calculus to represent aspects of some specific problem, but the two concepts are independent of each other, and it is just false that

"the minimal building-block of a Binary tree is simultaneously in two complementary states, which are integration and differentiation."

Also, this statement shows that you are doing an incorrect use of math terminology. Integration and differentiation are operations, not states.

 

[Organic]

When integration and differentiation complement each other they become states of a structure, which I call the building-block of the Binary-Tree.

Because I used the word "simultaneously" their opposite operational property can be described also as states.

 

[ahrkron]

Originally posted by Organic
When integration and differentiation complement each other they become states of a structure

No, they don't. The fact that this statement is wrong may pass unnoticed in an informal conversation, but it definitely cannot be used as the basis for the definition of anything in math.

, which I call the building-block of the Binary-Tree.

You cannot "call" things as you please, because you cannot make sure that everybody understands that you are not talking the same language. Your use of words already used in math to designate other concepts can confuse people trying to learn math.

Because I used the word "simultaneously" their opposite operational property can be described also as states.

Again, this is a very informal way to express your ideas. You need to pay much more attention to the accuracy of your statements if you insist in working on math problems.

Just to make it clear: the problem is NOT your command of English, but the lack of precision of your assertions.

 

[matt grime]

Well done for proving you don't know what a tree is. I asked you about that repeatedly and i thought we established the tree in your article was a genuine tree - the infinite bileaved tree you draw. now we find out you don't know what's going on again. why do you insist on knowing more about maths than the rest of us when you can't even define a tree correctly?

 

[Organic]

Matt,

This is theory development forum, where I can define a tree in my way.

 

[Organic]

ahrkron,

No, they don't. The fact that this statement is wrong may pass unnoticed in an informal conversation, but it definitely cannot be used as the basis for the definition of anything in math.

Please look at my paper:
http://www.geocities.com/complementarytheory/ET.pdf

 

[matt grime]

Originally posted by Organic
Matt,

This is theory development forum, where I can define a tree in my way.

but you didn't define tree though. i asked you about it repeatedly but you never acutally described it properly. undoubtedly it made sense in your head, but you didn't explain it to anyone else. in fact anyone who actually looked in your article would see that you drew a tree as is understood in graph theory, albeit with an infinite vertex set. now you claim the 'tree' is the cantor set. yet the tree is the natural numbers, therefore you're talking crap again! as even allowing for your inconsistent notation you've said it is countable and uncountable, an impossible dichotomy.

 

[Organic]

Matt,

You don't remember the last version of my work on Cantor's diagonal methed, so please look at it again:

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

And this time please read all of it, if you want to understand it.

 

[matt grime]

ah lovely, that heap of garbage again. still using the axiom of infinity of induction despite there being no such thing, still claiming the number of rows is 2^aleph-0 because of the finite case. still wrong despite the number of revisions you've undertaken.

there is not justification for claiming there are 2^aleph-0 rows. there aren't. you are wrong and there really is no simpler way of saying this. so why are there 2^aleph-0 rows? go on pleae state here and now in mathematical terms why there are 2^aleph-0 rows which are enumerable.here is the counter proof to your assertion

the list you produce is enumerable and is alleged to be the power set of N. Let z be in the power set of N. it is in the list at some point, n(z). by construction though the element at n(z) has only finitely many non-zero entries, therefore as z was arbitrary we have a contradiction.

you've still not managed to refute that counter example to your unfounded assertion.

 

[Hurkyl]

Allow me to reemphasize my conclusion:

If we assume the real numbers are countable, we can find a set whose total length is 1, yet this set contains every point of the entire real line!


Are you actually comfortable with the implication that the entire real line a length no greater than 1?

 

[Organic]

Matt,

by construction though the element at n(z) has only finitely many non-zero entries

By what construction?

Please give a detailed example of this construction.

 

[Organic]

Hurkyl,

That’s exactly the Idea, only a solid line (which means no points in it) has length 1.

No collection of infinitely many points can use the model of a solid line.

Fullness = Solid line = {__} content = Mathematics language strong limit.

For better understanding please look once more at(please pay attention to the Continuity that stands in the basis of empty or full(green) triangles):

http://www.geocities.com/complementarytheory/4BPM.pdf


Emptiness = {} content = Mathematics language weak limit.


Mathematics language is already aware to {} content.


It is the time to fulfill the symmetry by being aware to {__} content.

 

[matt grime]

Originally posted by Organic
Matt,By what construction?

Please give a detailed example of this construction.

]]

by your construction. column 1 goes 010101010...

column two goes 001100110011...column 3 goes 000111000111...

you cycles based on 2^n remember.therefore the n'th column starts with n zeros. Te resulting infinite matrix you write down is thus strictly lower tringular - just look at the first few diagrams you've drawn in that newdiagonal.pdf

for any row, row r say, reading right to left, all the entries become 0 after the r'th place (if not sooner), thus there are only finitely many non-zero terms in the r'th row (at most r of them).

I've told you this on at least 5 occasions and you've never managed to disprove it. you can't because it's clearly true.

In fact the thing you constructed precisely enumerates the 'finite' power set - the set of finite subsets of N which is countable.

 

[Hurkyl]

That’s exactly the Idea, only a solid line (which means no points in it) has length 1.

But, by definition of length, [0, 2] has length 2. And [0, 2] is part of the real line, so the real line has to have length no less than 2.


And 1 isn't special; allow me to modify my proof a little:

Suppose the real numbers is countable. Choose any enumeration of them.

Create a countable collection of intervals such that the i-th interval contains the i-th real number, and has length 1/2^(i+1).

This collection of intervals contains every real number, however, the total length of all of the intervals is 1/2.

So now I've proven the length of the real line is no greater than 1/2.


(In fact, I can prove the length of the real line is equal to zero, with an addition to this argument)

 

[Organic]

Matt,

for any row, row r say, reading right to left, all the entries become 0 after the r'th place (if not sooner), thus there are only finitely many non-zero terms in the r'th row (at most r of them).

There is no r'th place where after it you know exactly what is the next notation (depends on the base value for example: in base 2 the notation can be 0 XOR 1, in base 3 the notation can be 0 XOR 1 XOR 2, in base 4 the notation can be 0 XOR 1 XOR 2 XOR 3, and so on).

Which means that when we dealing with fractalic(=a^b) subsets where b is non-finite, probability enters to the picture and can't be ignored, as i clearly show here:

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

This is one of the fundamental mistakes that Cantor did when he researched infinity, and he made this mistake because in his time information theory and fundamental concepts like redundancy and uncertainty were not “must have” concepts of infinity research.

 

[matt grime]

Your argument doesn't hold water - you fixed base 2. that is how you constructed this object, this 'list' of strings of 0s and 1s. To say that you might have 2s or 3s in the strings of 0s and 1s is frankly misleading, if not a down right attempt to change the subject away from something where you are wrong. You're off again on an unrelated topic.

Jus look at the construction YOU gave, the t'th column starts with t zeroes! You can cleary see that on the r'th row, every entry after the r'th column must be zero. Look at you'ure own diagram where you can see the pattern that all the numbers above the diagonal are 0 - it is trivial to show that this pattern continues in the 'list' as I've just proven.

Can we make it a bit clearer? the t'th entry in row r is the entry from column t, if t>r (and r is fixed remember) then as the t'th column starts with t zeroes and r<t it must be that the r'th entry in that column is 0 becuase all the entries from 1,2,...r,..,t are zero. (This is your construction, yet you do not even understand this simple observation.) So after the r'th place in row r all the entries are zero. thus the r'th row has only a finite number of non-zero entries, (at most r). Thus the corresponding element in the power set is a finite set.