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

[Organic]

x=model() or x=CONCEPT.

In both cases x cannot be a product of some useful system.

Therefore x has its meaning only by theory(=model() )and X(= some concept) relations.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

By the way x=model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

Er, I think you missed the joke again. you say concept_of_A_number, well which one is 1 the realization of the concept of. I was pointing out the imprecisoin of your terminology again.

Concept_of _a_number is the intuitive ability to count things without any supported theory.

 

[matt grime]

but surely 2=model(conceptof anumber) as well, and thus 1=2. Stop using equals when you don't mean equals. In fact, stop using extant mathematical terms for your own bizarre and ill-informed asinine concepts full stop.

 

[Organic]

The '=' in x=model(X) is not an equal sign, but like in a cumputer program where x gets as input the result of model(X) output.

 

[matt grime]

I wouldn't know whether to laugh or cry if I cared anymore

 

[Organic]

What is you problem know?

I used a computer program syntax to show that any useful product of some theoretical system cannot be but a model(CONCEPT) combination, as I explained in my last posts.

 

[matt grime]

I've been reading your crap for the last couple of months and telling you repeatedly that you are not using mathematical terms correctly (and last time I checked this was a maths forum, not a computer science one) and after all that time only now do you tell me that you're using the equals sign to not mean equal? You don't find that slightly amusing?

 

[Organic]

Most of the time I used the words 'input' and 'output' when a wrote x=model(X) expresion.

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

Instead of using titles, all you have is to open your mind and try to understand what infront of your eyes, and not in what is the title of the forum, because sometimes the difference between areas is not so clear.

 

[Hurkyl]

Warning: variable X is used without having been initialized.

 

[matt grime]

As with the vast majority of your output you fail to define what you use. You cannot expect people to second guess you. But this is worse than second guess you, becuase you are horribly misusing extant terms in your own private theory without explaining what the **** is going on. You cannot expect people to understand you if you do this, nor care to read what you write. If your material is difficult to read because it is poorly written you will get no interest. Your standard of presentation is poor enough as it is without deliberately obfuscating the issue in inappropriate notation.

 

[Organic]

No Dear Hurkyl,

This is the whole point. an actual concept does not have to be initialized.

Only x in the expression x=model(X) has to be initialized.

 

[Organic]

Matt,

Only in the expression x=model(X) I used '=' as an input sign, so again what is the problem?

Please show me another part of my work where '=' is unclear.

Thank you.

 

[Hurkyl]

Only x in the expression x=model(X) has to be initialized.

Try telling a C compiler that.


If you are using '=' as assignment (which, btw, is generally frowned upon in Computer Science; using a symbol like $\leftarrow$ is generally preferred), then any previous value of 'x' is irrelevant because it's being assigned the return value of model(X). However, since X has not been initialized, (IOW you haven't given us any indication whatsoever what X is supposed to be) model(X) is an entirely meaningless expression.

Incidentally,

error: no definition found for function model.


P.S. whoops, I didn't notice that this was put in the math forum! Back to TD you go

 

[Organic]

Well Hurkyl,

I see that you did not read the part where I write exactly What is X and why it is not have to be initialized.

The reason is very simple, X represents a constant in x <-- model(X) expression.


Also please can you answer to what is written below?

Thank you.

If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existence of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)

Any x in some theory cannot be but a model(X), therefore we always have to be aware to the combination of model() and X, where model() is the global state and X is some local state.

Model() is the container (global state).

X is the content. (some local state).

x is the product of model(X) (the combination or relations between global and local states).

Shortly speaking any theory is first of all model() or if you like an empty container (a global state) "waiting" to some X (some local state) to be its examined concept.

From this point of view, any theory must be aware to the relations between the global and the local, otherwise it cannot use its full potential.

If Peano Axioms is a theory then first of all is a model()(a container) that "needs" some X(a content) to deal with.

For example:

1 is in N (also can be understood as: 1 is a natural number)

x <-- model(X) where x cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Concept_of _a_number is the intuitive ability to count things without any supported theory.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

It means that if we want to understand x we have to "put on the table" its combination (container-content) property.

Shortly speaking, 1 cannot be but 1 in N, whether you say it or not.

Therefore There is a container concept in Peano arithmetic.

Again, there can be a big problem for us to understand and develop deeper connections between so called different areas of research, if we don't take in account the global-local or container-content relations.

What I wrote here also can explain why ZF axiom of infinity and Peano first axiom are the same axiom.

And this axiom can be called "The forced-induction axiom".

My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".

All products of some theory are nothing but an x-model, an axiom is a product therefore an x-model.

I am talking about the hierarchy of dependency among these products.


The basic level is the axioms level, and on top of it there is the hierarchy of products, which can exist iff they do not contradict the axioms level.

But when we need an axiom that directly determines the existence of some element, it means that there is no hierarchy here but "the same lady with a different dress".

By the way x <-- model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

 

[matt grime]

Let us take some other theory: group theory. The axiom of identity does not 'create' the identity. groups exist whether or not we have a theory of groups. How did Galois manage to use them since the modern axiomatic form of group theory wasn't developed until around 70 years after his death.?

 

[Organic]

Dear Matt,


Maybe I am a living example of concepts without (yet) a rigorous framework as their theory.

This is exactly the meaning of x <-- model(X) expression.

By this expression we can understand that any useful x is at least a combination of some initial concept X(some constant) and model or a theory where we can develop it.

Galois case is a beautiful example that supports x <-- model(X) where Galois' work is the initial X and modern axiomatic form of group theory is model(X), and its useful product is x <-- model(X).

Be aware that I am not talking about the existence of X, but on the useful existence of x as a product of some model() and X combinations.

I made major corrections is this paper:

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

Pleae write your remarks on it.


Thank you.


Yours,


Organic

 

[matt grime]

At no point do you even define what these xs and Xs are. They could be goats, cars, or George Bush's DUI criminal record. What is 'actual infinity'? Oh, I know we're supposed to read Riemann's Balls pdf aren't we. Where you don't define it either. What's a set, Organic? What are any of these things?

The first two sentences of you reply above are just rubbish that makes no sense in the English language. The meaning of your postion on x<--model(X) is EXACTLY that you are a living ecample of concepts without a rigorous framework as their theory? Eh? You are a mathematical ill-defined concept?

What are you rambling on about with Galois? group theory is now a refinement of the ideas of galois and a generalization of the key points in his work. it is nothing to do with models or anything.

So go back and explain everything you write, explain the meaning of all the terms you misuse. I for one am no longer going to read your musings as evidently learning maths is not something you want to do.

 

[Organic]

The first two sentences of you reply above are just rubbish that makes no sense in the English language. The meaning of your postion on x<--model(X) is EXACTLY that you are a living ecample of concepts without a rigorous framework as their theory? Eh? You are a mathematical ill-defined concept?

You missed the point, concepts without a theory is only X.



Another example:

By x <-- model(Taniyama-Shimura) we mean that x is a product a more comprehensive theory which is based on the deep relations between modular forms and elliptic curves.

 

[Hurkyl]

X represents a constant

If X was a constant, you would be able to say exactly what it is, or specify a list of properties that characterize the constant symbol 'X'.

However, you later state:

some initial concept X(some constant)

...

Which seems to indicate that X isn't a fixed thing in your theories; it can be different things depending on the situation. We call that a variable.




Anyways, let's work through an example. Take this formal system. (I won't call it a theory becuase there is likely some disagreement between what mathematicians call a theory and what you call a theory)


In addition to ordinary logic, this formal system has the unary predicates P and L, and the binary predicates I and = (with = written in the usual infix notation) and the following axioms:

$$\begin{array}{l} \forall a, b: I(a, b) \implies (P(a) \wedge L(b)) \\ \forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c)) \\ \forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d) \end{array}$$


So what happens when you apply your ideas to this?

 

[Organic]

Hurkyl,

X in my theory is a geneal represetation for a constant, for example:

X is 'Taniyama-Shimura theorem'

x <-- model('Taniyama-Shimura theorem')

X is 'The concept of a number'

x <-- model('The concept of a number')

X is '...'

x <-- model('...')

 

[Hurkyl]

Hurkyl,

X in my theory is a geneal represetation for a constant, for example:

X is 'Taniyama-Shimura theorem'

x <-- model('Taniyama-Shimura theorem')

X is 'The concept of a number'

x <-- model('The concept of a number')

X is '...'

x <-- model('...')

In other words, X is a variable.

 

[Organic]

Let us say that X is like a dummy variable.

Shortley speaking, in any use of the general form of x <-- model(X),
X place is taken by some constant.

X is only a place holder.

 

[Hurkyl]

Let us say that X is like a dummy variable.

Shortley speaking, in any use of the general form of x <-- model(X),
X place is taken by some constant.

X is only a place holder.

In other words, X is a variable. (as the term is used in logic)


Anyways, I presented a formal system. How do your ideas apply to it?

 

[Organic]

If you represent a formal system please show me where are you in this system?

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

For example look at this paper:
http://www.geocities.com/complementarytheory/count.pdf

 

[Hurkyl]

I didn't say I represent a formal system, I said I presented a formal system.

 

[Organic]

Hurkyl,

Please see my answer to some old post of you in this tread.

If I have a list of properties that something called a "set" obeys, I don't need to know anything about the existence of a "set" in order to reason about those properties.

(This is a familiar idea even in "everyday" logic; we often call it a "hypothetical scenario" in such a context)

Any x in some theory cannot be but a model(X), therefore we always have to be aware to the combination of model() and X, where model() is the global state and X is some local state.

Model() is the container (global state).

X is the content. (some local state).

x is the product of model(X) (the combination or relations between global and local states).

Shortly speaking any theory is first of all model() or if you like an empty container (a global state) "waiting" to some X (some local state) to be its examined concept.

From this point of view, any theory must be aware to the relations between the global and the local, otherwise it cannot use its full potential.

If Peano Axioms is a theory then first of all is a model()(a container) that "needs" some X(a content) to deal with.

For example:

1 is in N (also can be understood as: 1 is a natural number)

x <-- model(X) where x cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Concept_of _a_number is the intuitive ability to count things without any supported theory.

Concept_of_a_number is not a Theory_of_numbers, and theory_of_numbers without Concept_of_a_number is an empty model(=model() ).

It means that if we want to understand x we have to "put on the table" its combination (container-content) property.

Shortly speaking, 1 cannot be but 1 in N, whether you say it or not.

Therefore There is a container concept in Peano arithmetic.

Again, there can be a big problem for us to understand and develop deeper connections between so called different areas of research, if we don't take in account the global-local or container-content relations.

What I wrote here also can explain why ZF axiom of infinity and Peano first axiom are the same axiom.

And this axiom can be called "The forced-induction axiom".

My limited comprehension of your ideas may be at fault here, but this sounds awfully like confusing "x-content" with an "x-model".

All products of some theory are nothing but an x-model, an axiom is a product therefore an x-model.

I am talking about the hierarchy of dependency among these products.


The basic level is the axioms level, and on top of it there is the hierarchy of products, which can exist iff they do not contradict the axioms level.

But when we need an axiom that directly determines the existence of some element, it means that there is no hierarchy here but "the same lady with a different dress".

By the way x <-- model(X) is not under Russell's Paradox, because the global level(=container) transcendent any local level(=content).

 

[Hurkyl]

How do your ideas apply to this formal system?


In addition to ordinary logic, this it has the unary predicates P and L, and the binary predicates I and = (with = written in the usual infix notation) and the following axioms:

$$\begin{array}{l} \forall a, b: I(a, b) \implies (P(a) \wedge L(b)) \\ \forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c)) \\ \forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d) \end{array}$$

 

[Organic]

Please translate it to plain English.

 

[Hurkyl]

Let's say P(a) means "a is a P-thing", L(b) means "b is an L-thing", and I(a, b) mean "a and b interact".

$$\forall a, b: I(a, b) \implies (P(a) \wedge L(b))$$

Interactions only occur between a P-thing and an L-thing.

$$\forall a, b: (P(a) \wedge P(b) \wedge \neg(a = b)) \implies (\exists c: L(c) \wedge I(a, c) \wedge I(b, c))$$

For any two different P-things, there is an L-thing that interacts with both of them.


$$\forall a, b, c, d: (I(a, c) \wedge I(a, d) \wedge I(b, c) \wedge I(b, d)) \implies (a = b \vee c = d)$$

If each of two L-things interact with each of two P-things, then the L-things are the same or the P-things are the same.

 

[Organic]

Please tell me if my conclusions are right.

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


Axiom 2 : There must exist this one to two structure

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

Axiom 3 : the first exists by axiom 2

Pa <--> La |(= Axiom 2)
Pb <--> La |

La <--> Pa
Lb <--> Pa

 

[Hurkyl]

So how does all this x = model(X) stuff fit in?


I'm not sure what your drawing for the third axiom is supposed to mean.

 

[Organic]

If each of two L-things interact with each of two P-things, then the L-things are the same or the P-things are the same.

Axiom 3 : the first structure already exists by axiom 2

Pa <--> La
Pb <--> La

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

La <--> Pa
Lb <--> Pa

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

Any way, the least structure of these axioms (as much as I see) cannot
be but one to two structure, which is the building-block of the Binary-Tree, or even more general the interactions between integration (sum) and differentiation (parts).

x is always a combination of model() and X, for example:

model() <-->|
            |-x
   X    <-->|

 

[Hurkyl]

I still don't know what the whole thing means. What are the structures? What can you do with them? Why are they useful? How do they relate to P-things, L-things, and interactions?

(I admit I'm pretty sure about a partial answer to the last of these questions)


And this is the most confusing of all:

model() <-->|
            |-x
   X    <-->|

How can this make any sense? Are you saying that, for instance, "model()" is a P-thing or an L-thing? How can this be if "model()" is supposed to be some sort of function? And if this is the meaning of "x = model(X)", how can it possibly apply to any other theory (such as set theory)?


(P.S. does it ever make sense, to you anyways, to say "X = model(x)", "y = model(Y)" or "j = model(C)"?)

 

[matt grime]

you want your detractors to demonstrate they can do maths (I know, but it's a like a habit that's bad for you, reading your latest murdering of mathmatics, and I'm finding it hard to go cold turkey) ok, here are some things I've proven that no one else has published as far as we are aware

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

How about this:

Let M be the module category of some finite dimensional group algebra over a (countable) algebraically closed field. Then the modules induced from a subgroup do not necessarily form a definable subcategory (in the sense of Krause). In particular, there are certain groups with normal subgroups of index p (=char of field) with the direct limit of induced modules from the subgroup not induced from any module.

that do you?

If you want I can give you some sufficient conditions on relative stable categories that ensures they are compactly generated. Interested?

 

[Organic]

Matt,

you want your detractors to demonstrate they can do maths

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.

Also I gave an example for this kind of a research:

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

When you read it then you will understand what I mean.

 

[Organic]

Dear Hurkyl,


My number one law is: think simple (which is not think trivial).

When we think simple then we get the chance to (sometimes) see the deeper interactions between so called different things.

The 3 axioms that you gave me define the building-block of one-to-many interactions structure.

1) side 1 of this building block is different from side 2 in this building-block by at least 2 properties.

Property 1) P is not L

Property 2) If P is one then L is at least two, and vise versa.

Therefore we can get:

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

OR

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

But the deep invariant thing is the one_to_many structure.

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

So (as I see it) our systems are the same in this level.

I take this level, mark my basic elements on it and get:

model() <-->|
            |-x
   X    <-->|

1) x cannot be but a model of X, where X cannot be anything but a thing that can be translated to a model of itself.

Shortly speaking, X-itself can be an x-model of X-itself.

2) x can exists iff there are at least two things: a theory of X(=model() ), X.

3) x is the interaction of model() and X, notated as x <-- model(X).

4) model() is the container, X is the content, therefore x is container-content interactions.

5) Also model() is the global, X is the local, therefore x is global-local interactions, which means that any x can be understood only by its global-local interactions or container-content interactions.

x needs at least two parents to exist, Mama model() and Papa X.

how can it possibly apply to any other theory (such as set theory)?

x cannot be but a container-content interaction where content can be at least nothing XOR something.

Know please read these short papers by this order, and see by yourself some examples that are based on this way of thinking:

1) http://www.geocities.com/complementarytheory/ET.pdf

2) http://www.geocities.com/complementarytheory/AHA.pdf

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

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

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

6) http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf