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

[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:

<br /> <br /> \begin{array}{l}<br /> <br /> \forall a, b: I(a, b) \implies (P(a) \wedge L(b)) \\<br /> <br /> <br /> \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)) \\<br /> <br /> <br /> \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)<br /> <br /> \end{array}<br /> <br />

 

[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".

<br /> \forall a, b: I(a, b) \implies (P(a) \wedge L(b))<br />

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

<br /> \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))<br />

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


<br /> \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)<br />

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

 

[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)