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

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The discussion centers on the Collatz problem and the implications of fixing the variable k within its mathematical framework. Participants debate whether k should be considered fixed or variable, with arguments suggesting that treating k as fixed leads to contradictions in the proof structure. The concept of decidability is also scrutinized, with claims that the Collatz problem is undecidable due to its reliance on the axioms of infinity and the inherent symmetry of the Binary Tree. The conversation highlights the complexity of proving the Collatz conjecture and the necessity of clarifying terms like "out of range" and "fixed" in mathematical discourse. Ultimately, the participants emphasize the need for rigorous definitions and logical consistency in mathematical proofs related to the Collatz problem.
  • #271
Evidently you don't understand the maths.

The Cantor set is not a binary tree. You have your own odd definition for binary tree, don't you. Care to explain what it is?

The cantor Set is the ***limit*** of that process.

So how does being able to draw a rooted binary tree on top of some other diagram provide you with a bijection with the Cantor set?
 
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  • #272
Matt,

Cantor set is a hybrid of base 2 and base 3 ratios.

Therefore Cantor set can be a base 2 fractal and base 3 fractal.

Your Cantor set is disjoint because you don't look at it from the complex level.
 
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  • #273
Full credit to him, it takes some doing to talk about binary trees (uniquely path connected), even drawing them so they appear to be what you I might think, and then after much pressing claim it is actually a Cantor Set (totally disconnected), and further that as this is TD he can do whatever he damn well pleases. (Despite the fact it was your decision to move it here, Hurkyl, and that he wanted to post it in the maths forum.) That's some cheek

You're right. He's dropped the attitude he had for a while that caused me to start moving his posts over here in the first place.



Anyways, Organic, the trick to the "binary tree" construction of the Cantor set is that it's not enough; it only generates the points in the Cantor set with terminating decimal expansions. In order to get the whole Cantor set, you have to take the (topological) closure of this set of points.


Given a set S, the closure of S is the set of all points that are limits of sequences of elements of S. For example, the closure of (0, 1) is [0, 1]. (the sequence 0.1, 0.01, 0.001, 0.0001, ... is entirely in (0, 1), and the limit is 0, so 0 is in the closures of (0, 1))
 
  • #274
Hurkyl,

You can't contradict it.

{10... ,1100... ,11110000... ,...} set is bijective to N.

The n-th notation of each member is:

Code: 
n-th 1 2 3 . .  1 2 3 . . . .  1 2 3 . . . . . . . .  1 2 3
    {1 0 . . . ,1 1 0 0 . . . ,1 1 1 1 0 0 0 0 . . . ,. . .}

By taking the n-th notation from each member we can build a list of unique sequences which its magnitude=2^aleph0:

Code: 
      3 2 1 0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]<--> n-th 1
 ...,1,1,1,0 <--> n-th 2
 ...,1,1,0,1 <--> n-th 3 
 ...,1,1,0,0 <--> n-th 4 
 ...,1,0,1,1 <--> n-th 5 
 ...,1,0,1,0 <--> n-th 6 
 ...,1,0,0,1 <--> n-th 7 
 ...,1,0,0,0 <--> n-th 8 
 ...,0,1,1,1 <--> n-th 9 
 ...,0,1,1,0 <--> n-th 10
 ...,0,1,0,1 <--> n-th 11
 ...,0,1,0,0 <--> n-th 12
 ...,0,0,1,1 <--> n-th 13
 ...,0,0,1,0 <--> n-th 14
 ...,0,0,0,1 <--> n-th 15
 ...,0,0,0,0 <--> n-th 16
 ...

If by closure you mean:

The collection of all points such that every neighborhood of these points intersects the original set in a nonempty set.

The result of this definition cannot be but at least Binary Tree where each point of it is a connaction (an intersection) with at least three neighbors (one above it and two below it).
 
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  • #275
The problem is that each of the
10101010...
11001100...
11110000...
...

are only aleph0 long.
 
  • #276
Hurkyl,

It is 2^aleph0 long because it "grows" in geometric series and not arithmetic series.

Code: 
<---arithmetic series
      3 2 1 0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]<--> n-th 1      geometric series 
 ...,1,1,1,0 <--> n-th 2      |
 ...,1,1,0,1 <--> n-th 3      |
 ...,1,1,0,0 <--> n-th 4      |
 ...,1,0,1,1 <--> n-th 5      |
 ...,1,0,1,0 <--> n-th 6      |
 ...,1,0,0,1 <--> n-th 7      |
 ...,1,0,0,0 <--> n-th 8      |
 ...,0,1,1,1 <--> n-th 9      |
 ...,0,1,1,0 <--> n-th 10     |
 ...,0,1,0,1 <--> n-th 11     |
 ...,0,1,0,0 <--> n-th 12     |
 ...,0,0,1,1 <--> n-th 13     |
 ...,0,0,1,0 <--> n-th 14     |
 ...,0,0,0,1 <--> n-th 15     |
 ...,0,0,0,0 <--> n-th 16     |
 ...                          V
 
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  • #277
The problem is that the geometric growth only exists in the finite realm. You have to do some other operation (such as take a limit, or a nested union) to transfer to the infinite case, and in this case, that operation doesnt' catapault you to 2^aleph0.
 
  • #278
Hurkyl,

Forget about the word "growth".

I am talking about infintley long arithmetic series = aleph0
and infinitely long geometric series = 2^aleph0.

More than that both serieses are depended.
 
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  • #279
There is no &omega;-th term in a geometric sequence, nor in an arithmetic sequence. An infinitely long geometric (or arithmetic) sequence has aleph0 terms, one for each finite ordinal.
 
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  • #280
Hurkyl,

Thank you, it is just something that I used to explain Matt my point of view in some particular case.

So here it is with no -th things:
Code: 
<---arithmetic series
      3 2 1 0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V
 
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  • #281
There is no infinite term in a geometric series. Each of the aleph0 terms are finite.
 
  • #282
Hurkyl,

Now you have it in front of your eyes, two depended serieses.


The first one (arithmetic) cannot be but with aleph0 cardinality,
therefore the other one (geometric) cannot be but with 2^aleph0 cardinality.

Code: 
<---arithmetic series
      3 2 1 0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V
 
  • #283
First off, the rows and columns are not in an arithmetic nor a geometric sequence; they're not even numbers!

It is the width and the height that are in those sequences... however, geometric and arithmetic sequences only have finite terms. The width and height of the infinite matrix cannot be in said sequence!


Let me put it this way. Except for the first, every term in an arithmetic or geometric sequence has a term immediately before it. So if the width and height of your infinite matrix are part of these sequences, then what are the terms that come immediately before them?
 
  • #284
Hurkyl,
then what are the terms that come immediately before them?
Please give some simple example in plain English.
 
  • #285
You're "sequences" arise from looking at pieces of the whole array. You look at a 2x1 piece, then a 4x2, then an 8x3, 16x4, ...

The thing is, each term in an arithmetic (geometric) sequence has a finite index. These sequences can only describe finite pieces of your whole array.


One of the many forms of the fundamental problem with progressing to the infinite case is this:

Each term (but the first) of an arithmetic sequence is defined to be a fixed constant plus the term before it.

You assert aleph0 is in the arithmetic sequence of widths. Well, what is the term before it?

The same complaint applies to the geometric sequence.



This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a &omega;-th step of the process (if one even exists)
 
  • #286
Hurkyl,
First off, the rows and columns are not in an arithmetic nor a geometric sequence; they're not even numbers!
If I get a list of 2^aleph0 unique elements, then their name is not important, first: they exists, second: give them a name.
This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a ù-th step of the process (if one even exists)
Hurkyl, there is no process here but an immediate existence based on ZF axiom of infinity iterations of the power_value of the matrix:
Code: 
<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V
 
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  • #288
How many times do we have to explain induction to you? There is a set of statements labelled by n in N, if the r'th implies the r+1st and if the 1'st is true then all the statements are ture by induction.

That is the statement is true for all n in N. It does not state the statement (ugly turn of phrase, sorry) labelled by N is true; that statement might not even make sense. You cannot write down something labelled by N and say it is true because it is true for each n in N, that is not induction, it is plain wrong.

COUNTER EXAMPLES, again.

the set of rows in the finite diagram is finite, therefore the set of rows for N should be finite as the first set is finite, and if the n'th set is finite the n+1st set only has twice as many and is hence finite. You can't just pick and choose the properties that you think pass through
Do you ever actually think about the counter arguments to your claims or do you just blithely turn the handle and crank out another piece of 'complementary theory'?The infinite case must be taken as a limit of something in the proper sense. You do not do this, and I am certain you do not even know how to do so. Listen to the people who do.

The 'limit' of a nested union of sets of cardinality 2^n is not 2^aleph-0. There is no substantiation for this claim other than your misuse of the axiom of infinity, which just states that the elements of N form an inductive set. |N| is not in N so you can't claim anything using it and induction.

And to make it absolutely clear, no one is saying that the power set of N does not have card 2^aleph-0, but that the list of elements you construct does not in anyway correspond to any set of that cardinality. The only reason you have for saying it is is that it follows from the finite cases - but it doesn't, otherwise it follows from the finite cases that the set of rows is finite - induction does not allow you to claim the things you are claiming *must* be true.
 
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  • #289
Matt,

aleph0 magnitude exists in ({},{__}) open interval where {} content is the weak limit and {__} is the strong limit.

To understand this please look at:

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

Again aleph0 cannot be something which is beyond "infinitely many" because no potential infinity can be an actual infinity and the reason is very simple.

Actual infinity is the limit of any information system including Math language.

Therefore only potential infinity which defined as "infinitely many objects" can be used as an input to Math language.

Therefore the "transfinite" universes that trying to "eat the cake" (to construct a solid line by using the model of points or segments) and also to keep it untouched (to insist that this "pointed/segmented" line is beyond its points/segments) is nothing but a conceptual mistake, when the logic is Boolean logic.

In Boolean Logic x and the negation of x is false, x=solid_line AND x=infinitely many points or segments, does not hold.
 
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  • #290
Hmm, looks like garbage. Seeing as you are attempting to prove things in mathematics are not sufficient your personal issues are not important - we can prove mathematics is perfectly consistent in this issue, and it is your ignorance of it that is causing you to misundertand it. Did I say aleph-0 is 'beyond n in N?' No I didn't. It isn't a natural number. Lots of things aren't natural numbers. Now, get back to the point in hand which is that you are now trying to prove that the rows of your diagram are in bijection with a Cantor set. Where is the proof of this? Their construction bijects them with N, but at no point is there a bijection witha Cantor Set. Your only indication as to why this is true is the patently false assertion that it must follow from the finite case because the rows there biject with a set of cardinality 2^n for n columns. This is not true.
 
  • #291
How can you use the word Garbage to somone who creat interest of more than 24 pages on this web-site ?

Don't you feel that Organic is trying to sow you somting that you still can't recognize as new interpetation to almost everyting in mathematics?

Moshek
 
  • #292
Moshek,

It is great to see that There is onther life in Math world which is not limited to Matt's understanding abilities.


Matt,

aleph0 magnitude exists in ({},{__}) open interval where {} content is the weak limit and {__} is the strong limit.

To understand this please look at:

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

Again aleph0 cannot be something which is beyond "infinitely many" because no potential infinity can be an actual infinity and the reason is very simple.

Actual infinity is the limit of any information system including Math language.

Therefore only potential infinity which defined as "infinitely many objects" can be used as an input to Math language.

Therefore the "transfinite" universes that trying to "eat the cake" (to construct a solid line by using the model of points or segments) and also to keep it untouched ( to insist that this "pointed/segmented" line is beyond its points/segments (it means a solid_line) ) is nothing but a conceptual mistake, when the logic is Boolean logic.

In Boolean Logic x and the negation of x is false, x=solid_line AND x=infinitely many points or segments, does not hold.
 
Last edited:
  • #293
Organic:

I did not convins yet that collatz problem 3n+1 is undisideabl by the self similarity fractal type to the infinit axiom in set theory but your arguments are really beutifulls:

Please look on:

www.as.huji.ac.il/midrasha04.htm[/URL]

Yours

Moshek
 
Last edited by a moderator:
  • #294
If I get a list of 2^aleph0 unique elements, then their name is not important, first: they exists, second: give them a name.

There's no good reason to believe the list has 2^aleph0 unique elements.
 
  • #295
Hi Moshek,

If the result of "to be eqivanet to an axiom of some system" is true then 3n+1 is true in ZF.

This is what I write at the end of:

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

"An axiom of some Mathematical system cannot be proved by definition.

Therefore Collatz sequences are true but cannot be proved within ZF axiomatic system."
 
  • #296
Hi Hurkyl,

This fundamental problem rears its ugly head for any iterative process where each step is built from the step immediately before it. Such a process only works for terms that can be generated in a finite number of steps. There are, of course, aleph0 of these terms, but none of these terms correspond to a ù-th step of the process (if one even exists)
Hurkyl, there is no process here but an immediate existence based on ZF axiom of infinity iterations of the power_value of the matrix:
Code: 
<---arithmetic series
      3 2 1 0 [b]<---The power_value of the matrix[/b] = aleph0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric series 
 ...,1,1,1,0                  |
 ...,1,1,0,1                  |
 ...,1,1,0,0                  |
 ...,1,0,1,1                  |
 ...,1,0,1,0                  |
 ...,1,0,0,1                  |
 ...,1,0,0,0                  |
 ...,0,1,1,1                  |
 ...,0,1,1,0                  |
 ...,0,1,0,1                  |
 ...,0,1,0,0                  |
 ...,0,0,1,1                  |
 ...,0,0,1,0                  |
 ...,0,0,0,1                  |
 ...,0,0,0,0                  |
 ...                          V
 
  • #297
Dear Moshek Hurkyl and Matt,

I did not sleep fore more then 36 hours, so please if you want we will continue after I take some good sleep.

Bye.
 
  • #298
Organic:

I really don't know yet if you are rigth, maybe.

But if you rally can show by this that the 3n+1 problem is undisidable problem in number theory, as a simple and natural Godel theorem. then it will consider as the greatest discovery in the history of the euclidian mathematics !

Good luck !
Moshek
 
  • #299
Hrm, &omega; is supposed to be the omega (\omega), the symbol typically used for the ordinal that is the well-order type of the natural numbers (which is equal to the set of natural numbers in the Von Neumann model). Is it not showing up on your computer?



Ok, let's try a different tact. Let's suppose I don't understand how to make the entire array based on the sample entries you've given. Let's also suppose that I still won't get it if you give me more sample entries.

So the question is, can you explain what is your array without simply writing sample entries and presuming the reader can fill in the details?

For example, instead of saying:

...010101

You could say

"The sequence whose n-th term is 1 if n is even and 0 if n is odd"

Instead of

...10000001000010011

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if n = p^2 for some integer p
f(n) := 0 otherwise"

Instead of

...1000100100101

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if n = g(p) for some integer p
f(n) := 0 otherwise
where g is recursively defined as
g(1) := 1
g(i) = g(i-1) + i for all integers i > 1"

Instead of

...000011110000111100001111

you could say

"The sequence whose n-th term is f(n) where:
f(n) := 1 if floor(n / 4)1 is even
f(n) := 0 otherwise"

Hurkyl

1: The floor function, floor(x), \lfloor x \rfloor, is the function that "rounds down" a real number. So, floor(1) = 1, floor(2.5) = 2, floor(-3.7) = -4.

PS: It's not an entirely unreasonable supposition; you disagree with the statements I make about the array as I think it's supposed to be!
 
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  • #300
I can write that Organic's musings are garbage because he does not attempt to define anything he uses and just invents some notation at random. He has claimed to have defined a multiplication on N which does not even send natural numbers to natural numbers, he makes plainly wrong statements about the axiom of infinity, he misuses the phrase binary tree, does not understand the concept of countable, thinks that mathematics implies there is a bijection from a set of cardinality aleph-0 to one of card 2^aleph-0. Very often the sentences he writes do not make sense as sentences, never mind mathematically. He also claims frequently to have a cast iron proof that is so cast iron that he has to keep rewriting it to remove errors. He has also demonstrated that he does not know what a bijection is despite claiming their use in a 'proof', he did not understand the n maps to n+1 is a bijection from N to a proper subset of itself, claiming seom bizarre things about this fact. He frequently contradicts himself and does not extend to others the courtesy he demands from them by not reading and understanding their counter arguments.

Moreover he does not seem to grasp that it is not mathematics that cannot handle infinity but his philosphy. None of the constructions and results he claims must be true are true in mathematics, and are only true in his philosophy. With these presumptions he then tells us our mathematics is faulty, when the only inconsistencies arise if we believe his unfounded assertions.

He often cites the axiom of infinity induction, yet no such thing exists without his philosophy, and it is something he refuses to explain.One needs only to see that he believes that axioms cannot be proven true to see that he doesn't understand that which he claims is wrong - axioms are proven true trivally in an axiomatic theory.

The collatz conjecture he insisted was undecidable in ZF, despite not evidently knowing what that meant as he asserted it was equivalent to the axiom of infinity (something he was unable to prove). Of course if it is equivalent to the axiom of the infinite set then it is trivally provable in ZF. He appears now to have changed his position on that despite repeatedly berating me for being too unimaginative to see how he was correct. Another example of him contradicting himself.

Proving the Collatz conjecture is undecidable in ZF is not that big a deal really. Conway showed there are Collatz type conjectures that are undecidable. It is known the conjecture is true for all numbers less than 10^53 I believe.If you wish to see why Organic provokes much ire from mathematicians just read all his amny posts where he passionately argues against the blindingly obvious, and cannot even understand the simple objections raised against his argument. One needs only read some of his respsonses to the challenge to prove his construction has the properties he claims to see that he doesn't understand what's going on.
 

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