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

[Hurkyl]

I still don't know what "x", "model()", nor "X" are.

The reason I presented a specific formal system for you to analyze was so that you could explicitly write down what "X" is, et cetera.

e.g. you say that X represents some constant. Well, show us the constant. Fill in the blank: "X = _"

 

[Organic]

For example:

X='INFINITY'


Please look at: http://www.geocities.com/complementarytheory/Theory.pdf

 

[Hurkyl]

How does "X='INFINITY'" relate to the system I presented?

 

[Organic]

Infinity has two basic sides:

Potential ----|
              |-- Actual   
Potential ----|

Please look at:

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

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

 

[matt grime]

Originally posted by Organic
Matt,Another typical example of your emotional response that does not give you the chance to understand what you read.

I did not ask anyone to show me how good mathematician he is, but asked for some legal brach in mathematics that researches our cognition's abilities to create math language.

Actually this is what you asked:

"Show some mathematical reseach that take your abilites to develop math language as a legal part of math."

It is a sentence that makes little sense, but my respsonse was one of the possible interpretations.

 

[matt grime]

Define 'actual infinity', what is it? Seriously?

As a general aside, if my emotional responses do not allow me to understand (mathematically) what I read, how come I've got several qualifications in mathematics?

 

[Hurkyl]

So is 'actual infinity' a P-thing or an L-thing? And what about 'potential infinity'? And there are two different potential infinities??


P.S. you asserted:

Axiom 1: There are no self interactions between L L or P P.

Which is incorrect. Here is a model that is a counterexample:


The only object in the model is M. M is an L-thing and a P-thing, and M interacts with M.

Formally:

P(M), L(M), I(M, M)

This model satisfies the three axioms of my formal system, yet there is an interaction I(P, Q) where P and Q are both L-things, and where P and Q are both P-things.

 

[Organic]

Your first axiom: "Interactions only occur between a P-thing and an L-thing."

(P(M) interacts with P(M)) is not (P(M) interacts with L(M))
(L(M) interacts with L(M)) is not (L(M) interacts with P(M))

(P(M) interacts with P(M)) or (L(M) interacts with L(M)) are not
allowed by your first axiom.

Therefore Axiom 1: There are no self interactions between L L or P P.

So is 'actual infinity' a P-thing or an L-thing? And what about 'potential infinity'? And there are two different potential infinities??

By your system I do not care about P-thing or L-thing because I am looking only for the invariant product of your axiomatic system, with is:

----|
    |-- 
----|

and the reason that P-thing or L-thing are not the invariant of your axiomatic system is because there can be:

Pa <-->|
       |-L
Pb <-->|

OR

La <-->|
       |-P
Lb <-->|

So, the invariant thing is the interaction sctucture itself, which is:

----|
    |-- 
----|

Now, let us look again at this:

Infinity has two basic sides:

Potential ----|
              |-- Actual   
Potential ----|

Actual infinity is {__} or {} contents (the word many is not allowed).

Potential infinity is {a,b,...} (the word many is allowed).

Again, if you want to understand my point of view, you have no choice but to understand my models here:

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

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

I'll say it again, if you choose not to read my papers, it is equivalent to:

"I asked you a question but I don't care about your answer".

 

[Hurkyl]

Your first axiom: "Interactions only occur between a P-thing and an L-thing."

You've made the unjustified assumption that something cannot be both a P-thing and an L-thing; A P-thing can interact with another P-thing if one of them is also an L-thing.

By your system I do not care about P-thing or L-thing because I am looking only for the invariant product of your axiomatic system, with is:

My system doesn't have an invariant product; that's something you created.

I'll say it again, if you choose not to read my papers, it is equivalent to:

"I asked you a question but I don't care about your answer".

The thing is, I'm asking you about horses and you're telling me about mermaids!

You are right, I don't care about your answers because you have not been answering my questions.


All of your attempts to explain the phrase "x = model(X)" have been little (if anything) more than a facy way of saying "It's a mathematical phrase with three parts; 'x', 'X', and 'model()'". You've tried giving examples, but all of your examples are your ideas that the rest of us have been telling you over and over that we don't understand.


My purpose over the last couple threads was to try and create a concrete, yet simple example for you to explain your ideas. In particular, I was hoping for precise definitions of the individual components of "x = model(X)". I purposefully tried to create an example that beared little resemblance to the other examples you have tried to make, in hopes that you would stick to the example instead of replacing it with the concepts which you should know are not sufficiently clear to the rest of us.

Apparently I didn't do a very good job; just about the only thing you've said about my system is an unjustifiable claim that there are no self-interactions between L's or P's. Apart from that, you've managed to rewrite everything else so you can start talking about invariant products and potential vs actual infinity again.


So yes, I don't care about your answer, because it's not an answer to my question.

 

[matt grime]

when someone asks you a simple direct question they are entitled to a simple direct answer addressing the question. It's as if someone asks you what what the annual level of rainfall in kinshasha is and you say, ah here is a link to the website of the encyclopedia britannica!

 

[Organic]

Hi Hurkyl,



First, thank you very much for your efforts to give me some bridge to your world.

OK, I see, I looked at P-thing as P AND its content (or at L-thing as L AND its content) , where you looked only on the contents of of each of them, isn't it?.

Any way, let us take inf <-- model('INFINITY') where 'INFINITY' stands for actual infinity that cannot be used as it is by Math.

But the model of it (notated as inf) which i call it a potential infinity, can be used by Math.

Actual infinity is too string or too weak to be used as information in Math language.

The srong limit is marked by {__} content, and the weak limit is marked by {} content (the word many cannot be used).

Potential infinity is marked by {a,b,...} (the word many is used).

Let us start from this model, that shows the difference between actual infinity that is marked by {___}, and potential infinity that is marked by {a,b,c,...}:

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

To understand better {___} and {} contents please look at:

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

The infinitely long base of the green trianlges is {___} content.

The infinitely long base of the empty trianlges is {} content.


Please look at both models and write your detailed remarks.



Thank you,

Orgainc

 

[matt grime]

when you write {a,b,...} you are implying that the 'model of infinity' is an infinite countable set of elements. Is that really what you want to do. Why is infinity a set like this? It is infinite, but that is not the same thing.

 

[Organic]

Matt,

Please show me how you notate an infinitley many elements which are not countable.

By my point of view there is no such a thing "uncountalbe infinitely many elements".

I already proved it here:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

 

[matt grime]

you proved no such thing there because of your repeated misuse of 'the axiom of infinity of induction'.in that article you claim certain things have certain properties which have been absolutely refuted beyond any doubt by my arguments in various threads on this issue. that you do not understand the refutations is because you do not understand mathematical arguments. At various times, you claim that an infinite list has a 'last' element, that something is a priori countable for no reason that is true in mathematics, you claim results about infinite sets from finite ones despite plentiful counter examples to this principle, you want me to go on?

As in this thread, you claim to prove something (incorrectly) and then immediately claim that in fact the opposite is true. in that you 'prove' by an incorrect method that something is uncountable, then state that it isn't. an educated monkey can see it isn't countable (picking a metaphor not at random). in this thread you claim something is deducible (is equivalent to!) some axiom in ZF and then state it is undecidable in ZF! proving only that you don't know what undecidable means.simply put I cannot understand anyone who cannot understand there is no bijection between N and its powerset and that therefore there are uncountable sets; it's just a defintion.

that wasn't the point i was making anyway - you say infinty is a set that is implicty enumerable. in what sense is that infinity. it is infinite, that is a different thing entirely. in all of this you have not provided a definition of 'infinity' just some claims about infinite things. that is not the same.you want a 'notation' for an uncountable infinite set? I'm not sure what that means, but how about R, the real numbers that is easily proven to be uncoantable?

 

[Hurkyl]

OK, I see, I looked at P-thing as P AND its content (or at L-thing as L AND its content) , where you looked only on the contents of of each of them, isn't it?.

I don't know whether to say "yes" or "no", because I don't know what you mean by "the contents of each of them".

The only things I know about P-things and L-things are the three axioms I've listed.

I -have- been specifically avoiding trying to ascribe any meaning to P-things and L-things; I've been considering them merely as things that obey the three axioms I gave.

My best guess is that I have done the exact opposite of "looking only on the contents of each of them".



Do P-things fit into your "x <- model(X)" equation? Do they get substituted for 'x' or do they get substituted for 'X'? What goes in the other spot?

Please show me how you notate an infinitley many elements which are not countable.

We don't try notate them by enumerating their elements. Examples of uncountable sets are $\mathbb{R}$, $\mathbb{N}^\mathbb{N}$, and {x | x is in the interior of triange ABC}.

(where the latter, of course, is in the context of Euclidean geometry, and I have already specified the noncollinear points A, B, and C)

 

[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