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

[matt grime]

And I thought I was the only one dumb/obstinate enough to try and explain stuff to Organic. I don't suppose by any chance you're a set theorist, are you Hurkyl? I'm one of those representation theory people and attempting to explain ZF and set theory is not something I'm particularly comfortable with - my statements here are reasonably informed opinions, but I'm absolutely convinced I've stated things that would have the genuine set theorist spitting blood. The use of 'model' throughout is not in the sense of model theory (or if it is it was a lucky guess).

 

[Hurkyl]

And I thought I was the only one dumb/obstinate enough to try and explain stuff to Organic.

I've spent probably a whole year trying to explain things to Organic. I'm on a break.

I don't suppose by any chance you're a set theorist, are you Hurkyl?

I don't really know enough about anything, yet, to have earned any sort of specialist title!

 

[Organic]

Hurkyl and Matt,

"There is a set, S, such that if n is in S, then n + 1 is in S".

Nothing in the Peano postulates say anything about sets. ZF does nothing but postulate the existence of a set with a given property. Induction is a mathematical technique for proof and thus cannot be one or more axioms. None of the three things mentioned in your sentence can be the same.

Before we can speak about a concept like set we have to define its existence.

For example:

ZF axiom of the empty set

There is a set, A, such that for any x, x not in A.



Also before we can use Peano postulates we have to define some "minimal input" to start with.

For example:

Peano's first postulate

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



In both axiomatic systems, there are at least two basic concepts, which are:

1. A container (in the above examples they are A, N)
2. A content (in the above examples they are x, 1)



Now let us examine ZF axiom of infinity and Peano's second postulate:

ZF: There is a set, S, such that if n is in S, then n + 1 is in S.

Peano: If n is in N, then its "successor" n' is in N.

As we can see, both axioms are forcing induction on some content, which means that they are the same axiom.

The result of this forced induction is infinitely many elements called “the natural numbers”.

These infinitely many elements cannot be defined without the forced induction axiom.

Without the axiom of infinity, I can prove that there does not exist a finite set that contains all of the natural numbers. Here's the sketch:

Suppose F is a set of all natural numbers. The function f(n) = n + 1 is a 1-1 mapping from F onto a proper subset of itself. Therefore F is infinite.

Therefore, no finite set can contain all natural numbers.

Isn't f(n) = n + 1 equivalent to the axiom of infinity?

Tell me how many sets you want proved exist and I'll do it for you without any form of induction. I'll do two sets for you, as a freebie.

{} exists by the axiom of the empty set.
{{}} exists by the axiom of the pair set.

{} is unequal to {{}} because {} is in {{}} but {} is not in {}.

Thus, I've proven two sets, {} and {{}} exist.

Thank you Hurkyl for correcting my mistake here.

The axiom of the empty set and the empty set are the same.

In this address (http://www.cs.bilkent.edu.tr/~akman/jour-papers/air/node5.html ) I have found this:

“The Null Set Axiom guarantees that there is a set with no elements, i.e., the empty set {}. This is the only set whose existence is explicitly stated".

Again, let me put it this way, the ZF axiom of the empty set uses x
where x is something and then it says that for any x(=something),
x(=something) not in some set X.

This is nothing but an indirect and complicated way of what we can say directly and simply: "there exist set X with no elements" notated as {}.

(In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it).

"The Pair Set Axiom states the existence of a set which has a member when the only existing set is {}. So the set {{}} can now be formed now and we have two objects {} and {{}}. The application of the axiom repetitively yields any finite number of sets, each with only one or two elements.

It is the Sum Set Axiom which states the existence of sets containing any finite number of elements by defining the union of already existing sets. Thus
{{},{{}}}U{{{},{{}}}} = {{},{{}},{{},{{}}}}

However it should be noted that all these sets will be finite because only finitely many sets can be formed by applying Pair Set and Sum Set finitely many times.

It is the Axiom of Infinity which states the existence of at least one infinite set, from which other infinite sets can be formed. The set which the axiom asserts to exist is {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}.

Thus, the ZF universe simply starts with the {} and extends to infinity. It can be noticed that cumulative hierarchy produces all finite sets and many infinite ones, but it does not produce all infinite sets.”


Well Pair Set Axiom and Sum Set Axiom are the basis of Von Neumann recursion.

If you don't see the invariant symmetry that exists in both Von Neumann recursion(micro level) and Collatz sequences(macro level) then you can't understand my proof.

The ZF axiom of infinity products (if n exists then n+1 exists implies infinitely many N members) are in the level of the macro tree, where Collatz sequences exists, therefore we cannot distinguish between them and the Collatz sequences because in the macro level they are the same elements.

Eech n in the induction macro level has its own internal unique structure produced by Von Neumann recursion.


When we examine the invariant Binary tree that stands in the basis of both Collatz sequences and Von Neumann recursion, then and only then we can see that Collatz sequences and ZF axiom of infinity are the same iteration.

General: any iteration that is based on root or exponent value 2, is equivalent to the ZF axiom of infinity iteration.

Please look again at: http://www.geocities.com/complementarytheory/3n1proof.pdf

Thank you,

Organic

 

[matt grime]

You seem to not be able to distinguish between different objects.

A function is not an axiom, a set and an axiom, etc.

You also appear to have invented another new term: ZF axiom of infinity products.

I don't really have the inclination to even think about what you mean in your new view of it.

 

[matt grime]

quote:
In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it

I missed that one. that's probably one of the more silly things you've said, and boy is that a hotly contested fight.

so many counter-examples and corrections spring to minf, but what's the point? You keep insisting you are correct and have this proof of something, and you keep on having to correct it and each time you come back just as vehement that we are wrong and you are right... hell some times you don't even correct it, yet still claim it is correct despite the evidence to the contrary (I wonder if you are still claiming a set is equal to an inequality...?)

What is it you are claiming anyway - that the Collatz conjecture is undecidable? Do you take this from the assertion that COnway proved some Collatz like conjectures are undecidable? did you even look at his proof? do you even know what undecidable actually means? Seeing as number theory uses ZF, and you claim that he Collatz conjecture is EQUIVALENT to the axiom of infinity, then you've probed Collatz is true (in ZF).

You haven't done that though.

Tell you what, you produce a nice picture that, say, proves the 4-subspace problem is wild, and maybe we'll take you a little more seriously. Until then I'll just keep pointing out the gross abuses of the English language and mathematics you are disguising your lack of ability in.

 

[Hurkyl]

Before we can speak about a concept like set we have to define its existence.

Why would you think that?

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)

In both axiomatic systems, there are at least two basic concepts, which are:

1. A container (in the above examples they are A, N)

There is no container concept in peano arithmetic. (There are also quite a few other theories that have no container concept)

The peano axioms give a list of properties for three undefined terms: the unary predicate "is a natural number", the constant symbol "1", and the unary function "successor".

"Natural number" is no more a container than "human" is; in laymen's terms, "Natural number" is just a name we give to a 'type' of thing; it is a 'category' (I mean this in the lay sense, not the technical sense).


An aside on notation: sometimes, we like to use notation that looks like set notation, even when there are no sets involved. To use a different example, we might choose to write the logical predicate "Z is an ordinal number" in the form Z \in \mathrm{OR}, even though there is not (and in ZF, cannot be) a set OR that contains all ordinal numbers.

ZF: There is a set, S, such that if n is in S, then n + 1 is in S.

Peano: If n is in N, then its "successor" n' is in N.

As we can see, both axioms are forcing induction on some content, which means that they are the same axiom.

First off, we have to assume that you intend the phrases "n + 1" and n \cup \{ n \} to be synonymous.

Secondly, how can they be the same axiom?

The axiom of infinity guarantees the existence of a set, but I can't use the Peano axiom "If n is a natural number, then the successor of n is a natural number" to prove the existence of any set.


And, incidentally, the set guaranteed by the axiom of infinity cannot be proven to be an inductive set under the "n + 1" operation.

Isn't f(n) = n + 1 equivalent to the axiom of infinity?

As per a later statement you make, I would like to point out that the existence of the function I used here is given by the axiom of the power set and the axiom of subsets.

The axiom of power sets guarantees the existence of the set P(P(F)). (the power set of the power set of F)

P(P(F)) contains, among other things, all ordered pairs (a, b) where a and b are elements of F.

The axiom of subsets guarantees the existence of a set, call it f, which is the subset of P(P(F)) that contains only ordered pairs of the form (x, x + 1). This is the function we would more commonally write f(n) = n + 1.

More precisely:

<br /> f := \{ x | x \in \mathcal{P}(\mathcal{P}(F)) \wedge(\exists a \in F : x = (a, a+1) \}<br />

(In my opinion any element whose existence is explicitly stated, is equivalent to the axiom that defines it).

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



Finally, I would like to make a comment on models.

I mentioned earlier that:

The peano axioms give a list of properties for three undefined terms: the unary predicate "is a natural number", the constant symbol "0", and the unary function "successor".

(I changed "1" to "0" because I prefer formulations of the natural numbers that include zero)

Allow me to emphasize that these are undefined terms of the theory of peano arithmetic. They are NOT "n is in N", "{}", and "n U {n}".


When we make these identifications, we are making what mathematicians call "a model of the natural numbers in ZF". A model, more or less, consists of taking the undefined terms of one theory and defining them in another theory. There are additional details, such as you have to verify that all of the axioms of the one theory are satisfied when you define the terms in the other theory.

 

[Organic]

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).

It means that if we want to understand x(=1 in this case) 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".

 

[matt grime]

Originally posted by Organic
x=model(X) where x=1 cannot be but a combination of model()(= theory of numbers) and X(= the concept of a number).

Therefore any product in any theory cannot be but a combination of container and content.

The first thing in there translates as

1= theoryofnumbers(conceptofanumber)

which is beyond words.

 

[Organic]

No it is simple.

x=thory_of_some_concept(THE_CONCEPT).

thory_of_some_concept() is the container.

THE_CONCEPT is the content.

x is the container-content relations.

 

[matt grime]

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.

 

[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:

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


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.