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.
  • #361
Damn it iI just wiped my reply.

OkIf you are stating that the number of rows has card 2^aleph-0 and that you are using these terms in the conventional way that mathematics DEFINES them, then you are stating that there is a bijection between N and its power set. Now you know there isn't one in the case of finite sets, and the same proof states the same in the infinite case if written properly, as does the observation that the reals do not have measure zero.

However, we have established that you are not using aleph-0 in its conventional sense, so in your theory who knows what happens. The important thing is to realize that the things you are talking about are not the things a mathematician talks about.

It is by definition that two sets have the same cardinality iff they are bijective, and by definition that 2^aleph-0 is the cardinality of the power set of N. We could have declared aleph-1 to be the cardinality of the power set of N, but we didn't because the statement that the cardinality of the power set of N is the 'smallest' uncountable cardinal is independent of ZF! (Cohen et al, the continuum hpothesis). There is no bijection between N and P(N) therefore we DECLARE them to have different cardinalities. Cardinality is not some abstract concept independent of alephs that we 'model' with alephs, they are inextricably bound. So it is because you refuse to accept a definition that you are apparently contradicting mathematics. You cannot contradict a definition, only state that it does not do what you want.

You should then offer a different label for a different object. Your aleph-0 is not the aleph-0 of mathematics, it does not behave the same way and does not encode the same information, which is simply the isomorphism class of the set
 
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  • #362
Matt,

In this post aleph0 is the Cantorian aleph0, so in this case langth magnitude cannot be but 2^aleph0, as we clearly can see here:
Code: 
<-------------------Width magnitude = aleph0
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]    Length magnitude = 2^aleph0
 ...,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

Please tell me what do you thing?
 
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  • #363
So you are saying that you believe there is a bijection between P(N) and N? Which is what you mean if you are saying there is a bijection between a set of rows that are in 1-1 correspondence with N If you are using aleph-0 properly. This is wrong. It is easily and variously proven to be wrong. As any person with the most basic understanding of mathematical convetion knows.
 
  • #364
No Dear Matt,

This is the beautiful thing in Math, you don't have to believe in anything and nothing is wrong or right.

All we have is consistent(=interesting) system or non-consistent(=non-interesting) system, no less no more.

As you can see by the list below, there are infinitely many information structures which are beyond the scope of N members, but the structural arrangement of them give us the possibility to construct a list of infinitely many unique sequences.

The length of this list has a magnitude of 2^aleph0(the Cantorian one)
and as we can clearly see, it is enumerable.

Code: 
<----------------------Width magnitude = aleph0
  {...,3,2,1,0} = Z*
      2 2 2 2
      ^ ^ ^ ^
      | | | |
      v v v v
 [b]{[/b]...,1,1,1,[b][i]1[/i][/b][b]}[/b] <--> [b][i]1[/i][/b]  Length magnitude = 2^aleph0
  ...,1,1,1,                   |
  ...,1,1, ,                   |
  ...,1,1, ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...                          |
  ...,0,0,0,[b][i]0[/i][/b] <--> [b][i]2[/i][/b]           | 
  ...,0,0,0,                   |
  ...,0,0, ,                   |
  ...,0,0, ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...                          |
  ...,1,1,1,[b][i]1[/i][/b] <--> [b][i]3[/i][/b]           | 
  ...,1,1,1,                   |
  ...,1,1, ,                   |
  ...,1,1, ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
 ...                           |
  ...,0,0,0,[b][i]0[/i][/b] <--> [b][i]4[/i][/b]           | 
  ...,0,0,0,                   |
  ...,0,0, ,                   |
  ...,0,0, ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...,0, , ,                   |
  ...                          |
  ...,1,1,1,[b][i]1[/i][/b] <--> [b][i]5[/i][/b]           | 
  ...,1,1,1,                   |
  ...,1,1, ,                   |
  ...,1,1, ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
  ...,1, , ,                   |
                               |
  ...                          V

Let us see what is the connaction between a Binary list and the above list.

First let us look at this list:
Code: 
<---arithmetic magnitude
 {...,3,2,1,0} = Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]... 1 1 1 1[b]}[/b]   geometric magnitude 
 ... 1 1 1                    |
 ... 1 1   1                  |
 ... 1 1  /                   |
 ... 1   1 1                  |
 ... 1   1 /                  |
 ... 1   //1                  |
 ... 1  // /                  |
 ...   1 1|1                  |
 ...   1 1|                   |
 ...   1 ||1                  |
 ...   1 //                   |
 ...   /|1 1                  |
 ...  / |1                    |
 ... |  || 1                  |
 ... |  ||                    |
 ... 1  ||                    V

[b]the same can be done with '0' notations[/b]
Shotly speaking the main structure here is:
Code: 
<---arithmetic magnitude
 {...,3,2,1,0} = Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]... 1 1 1 1[b]}[/b]   geometric magnitude
 ... 1 1 1                    |
 ... 1 1                      |
 ... 1 1                      |
 ... 1                        |
 ... 1                        |
 ... 1                        |
 ... 1                        |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          |
 ...                          V

[b]again, the same can be done with '0' notations[/b]
Now we shall show thet this information is greather then aleph0 magnitude.

Step 1: we will show again our list in this way:
Code: 
<---arithmetic magnitude
 {...,3,2,1,0} = Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]   geometric magnitude 
 ...,1,1,1,0                  |
 ...,1,1,0,                   |
 ...,1,1,0,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,1,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          V
step 2: to make it clearer we shall show it now in this way:
Code: 
<---arithmetic magnitude
 {...,3,2,1,0} = Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b] <--> 1 geometric magnitude 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...                          |
 ...,0,0,0,0 <--> 2           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,1 <--> 3           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
...                           |
 ...,0,0,0,0 <--> 4           | 
 ...,0,0,0,                   |
 ...,0,0, ,                   |
 ...,0,0, ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...,0, , ,                   |
 ...                          |
 ...,1,1,1,1 <--> 5           | 
 ...,1,1,1,                   |
 ...,1,1, ,                   |
 ...,1,1, ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
 ...,1, , ,                   |
                              |
 ...                          V
 
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  • #365
I'm not entirely sure what the picture you draw now is meant to be, or what's going on with it, but that isn't really important.

consistent doesn't mean interesting.

your use of alephs is inconsistent (and thus uninteresting...?) with the correct mathematical usage. And there are conventions that we must keep to.

you have just stated that there is a bijection from N to P(N); that the whole of measure theory is wrong; that Baire's category theory needs rethinking; that the principle of least upper bound makes no sense.
 
  • #366
Matt,

Please forget about my point of view on aleph0.

In my last post and in this post I am talking about the standard Cantorian meaning of aleph0.

I made a pdf of my last post, maybe you will find it clearer:

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

you have just stated that there is a bijection from N to P(N); that the whole of measure theory is wrong; that Baire's category theory needs rethinking; that the principle of least upper bound makes no sense.
I know that for more than 20 years.
 
  • #367
Then you have been mistaken about mathematics for more than 20 years.
 
  • #368
Can you tell us what your construction is, instead of showing a tiny piece of it?
 
  • #369
Matt,

Prove that my matrix does not have a length with 2^aleph0 magnitude.
 
  • #371
I repeat, can you tell us what your construction is, rather than showing a small corner of the array and assuming the rest is obvious?
 
  • #372
Originally posted by Organic
Matt,

Prove that my matrix does not have a length with 2^aleph0 magnitude.


Suppose that there is a set, R of cardinality 2^aleph-0, and that there is a function from N to R.

We may replace R with P(N) as by definition R is in bijective correspondence with it, and thus we have a map f from N to P(N).

Define T in P(N) by t in T iff t not in f(t)

T is by the usual argument not in ran(f), hence there is no suejective set map from N to P(N). Thus R is not countable.
 
  • #373
Matt,

Please don't repeat again on your MANTRA.

You can prove that my list does not have a length of magnitude 2^aleph0 iff you can prove thet Z* /= Z* .
 
  • #374
Why mantra? It is a proof.

Two sets have the same cardinality IFF they are in bijective correspondence, that is the definition of what it means for two sets to have the same cardinality. 2^aleph-0 is, by definition, the cardinality of the power set of N. Cardinality is purely a statement about isomorphisms, not what ever you have in mind. Thus you are stating there is a bijection from N to P(N) if you are stating that the set of rows simultaneously has cardinality aleph-0 (is in bijection with N) AND 2^aleph-0 (is in bijection with P(N). If there is a bijection from A to B and a bijection from A to C there is a bijection from B to C.)

You have to play by my rules on this one; you said so yourself.
 
  • #375
Matt,

|Z*|<|P(Z*)| but both are contable.
 
  • #376
No, there is no bijection from N to P(Z*), which I believe you define to be {0,1,2...}.

An (infinite) set is COUNTABLE is, by _definition_, stating that the is a bijection from N to that set. There is no bijection from any set to its power set.

You do know what countable means?
 
  • #377
Matt,

You are playing with words (definitions) I show a concrete proof
that |Z*|<|P(Z*)| and both ( Z* and P(Z*) ) are enumerable.
 
Last edited:
  • #378
You cannot do that if you are using the definitiions correctly.

All your constructions start with some finite portion of a picture, and then claim that in the ellipsis everything hangs together.

How have you shown P(N) is countable? it was by those diagrams in newdiagonal.pdf, the ones that I proved had exactly enumerated the finite subsets of N.

Or have you got another 'proof' that there is a bijection from N to P(N)

write it here then, and let's start the same old tired arguments all over again...
 
  • #380
You give no reason as to why the rows form a set of cardinality 2^aleph-0 - that is that they are in bijection with P(N), indeed you aren't even saying what the rows are, and how you have constructed them. You merely state that they must be a set of cardinality 2^aleph-0. As far as I can see there *might* be a countable set of elements for each n in N, and that you are taking their union. A countable union of countable sets is countable.

Who knows what horrors your ellipsis hides. Apparently not even you because you cannot/do not explain it.


Your picture is very unclear - where is the bijection with N? I see some rows are labelled with elements of N, most aren't, what are these rows, how are you constructing them, why are they countable - give the injection to N, why are they of cardinality 2^aleph-0 or greater? give an injection from P(N).

Now, if your constructions, whatever they are are true (they aren't) then my proof that there is no bijection from N to P(N) is incorrect. where is it wrong? Where do all the proofs of this fact go wrong? Mine is a clear simple proof well known and easily checked with all its terms defined. Your pictures are what? they are fragments of something larger that you refuse to explain how to construct, and its properties are not verified, merely stated.



here, let me do this, consider the set {1,2,4,67,84...} this is clearly a countable set of real numbers and contains a brown fox. prove me wrong.
 
  • #381
Matt,

You protect yourself from being hurt by the consequences of my proof that R is enumerable.

So please don’t ask me because for me what I did is clear as a middle noon sun.

Please print my proof, and ask your colleges about it.
 
  • #382
Originally posted by Organic
Matt,

You protect yourself from being hurt by the consequences of my proof that R is enumerable.

So please don’t ask me because for me what I did is clear as a middle noon sun.

Please print my proof, and ask your colleges about it.

At what point is this considered trolling?
 
  • #384
Originally posted by suyver
At what point is this considered trolling?

can cranks be trolls? for a while he was in danger of making sense (the sense being 'I've completely rewritten all the rules and look at the problems that causes!'), though the last few posts have gone back to being just irrational assertions and refusal to acknowledge that defintions are somehow, well, erm, defined.

The average response to organic's 'proof' (ie a picture with a couple of labels) by my 'colleges' (one of the best typos I've seen) would be 'idiot' probably preceded by one of several old fashioned expletives. I myself favour a simple, muttered, 'tit' when confronted with these things in the best Peter Kay as Brian Potter way. (Apologies to everyone outside of England/UK who has no idea who that is. try getting hold of Phoenix Nights it's not half bad).

The odd thing is his diagrams keep getting more and more elaborate, each time it is easily pointed out where he's going wrong; yet he never addresses these problems, just comes back with yet another one with more things going on and even fewer explanations. And the latest one always proves mathematics wrong! Ignore the proof that it's rubbish, and that each previous one was given just the same fanfare on arrival and was just as easily dismissed as the ravings of an idiot. If he is a troll he's got a lot of free time to devise these things. I'm a professional mathematician, this is a distraction from research and each rebuttal takes about a minute to devise; he must spend hours coming up with these pretty pdfs.
 
  • #386
Nope, there's still no reason at all to conclude that the list is in anyway a set of card 2^aleph-0, there is no reason to suppose it contains all the elements of the set of all infinite strings of 0s and 1s, indeed the list STILL contains only those strings that have a finite number of 0s on them as has been proven to you. Your only proof is that it can be no other thing... erm, not true. As I've asked, and Hurkyl, where is the string ..1010101 of alternating 0s and 1s?

So, I've read the article AGAIN. WHy don't you explain where the counter proofs of you assertions are wrong in your opinion. Remember when we asked how to construct the diagram? And we agreed the th first column is (1010101010... ) the second (110011001100..) and so on - the nth is 2^n ones, 2^n 0s, looping again and again?

remember how we showed you that that implies that every row has only a finite number of zeroes in it? remember how that implies the string ...01010101) with an infinite number of 0s in it is not on the list? remember? come on, we;ve read the article, we've said what we consider wrong with it, and it is encapsulated in this paragraph and the previous one. so where are we wrong. come on, explain it in clear simple words for us that can't understand your maths, tell us where we 've gone wrong in the analysis of the diagram.
 
  • #387
Matt,

...01010101 or ...10101010 is in the list, for example:

Let us take again our set:
Code: 
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b]<--> 1
 ...,1,1,1,0 <--> 2
 ...,1,1,0,1 <--> 3 
 ...,1,1,0,0 <--> 4 
 ...,1,0,1,1 <--> 5 
 ...,1,0,1,0 <--> 6 
 ...,1,0,0,1 <--> 7 
 ...,1,0,0,0 <--> 8 
 ...,0,1,1,1 <--> 9 
 ...,0,1,1,0 <--> 10
 ...,0,1,0,1 <--> 11
 ...,0,1,0,0 <--> 12
 ...,0,0,1,1 <--> 13
 ...,0,0,1,0 <--> 14
 ...,0,0,0,1 <--> 15
 ...,0,0,0,0 <--> 16
 ...
Now let us make a little redundancy diet:
Code: 
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
...  [b]1[/b]-1-1-1 <--> 1
     \  \ \0 <--> 2
      \  0-1 <--> 3 
       \  \0 <--> 4 
       [b]0[/b]-[b]1[/b]-1 <--> 5 
        \ \[b]0[/b] <--> 6 
         0-1 <--> 7 
          \0 <--> 8 
 ... [b]0[/b]-[b]1[/b]-1-1 <--> 9 
     \  \ \0 <--> 10
      \  [b]0[/b]-[b]1[/b] <--> 11
       \  \0 <--> 12
       0-1-1 <--> 13
        \ \0 <--> 14
         0-1 <--> 15
          \0 <--> 16
 ...
and we get:
Code: 
 {...,3,2,1,0}=Z*
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
          /1 <--> 1
         1 
        / \0 <--> 2
       1   
       /\ /1 <--> 3 
      /  0
     /    \0 <--> 4 
 ... [b]1[/b]    
     \    /1 <--> 5 
      \  [b]1[/b] 
       \/ \[b]0[/b] <--> 6
       [b]0[/b]  
        \ /1 <--> 7
         0
          \0 <--> 8
          
          /1 <--> 9 
         1
        / \0 <--> 10
       [b]1[/b]  
       /\ /[b]1[/b] <--> 11
      /  [b]0[/b] 
     /    \0 <--> 12
 ... [b]0[/b]    
     \    /1 <--> 13
      \  1
       \/ \0 <--> 14
       0  
        \ /1 <--> 15
         0
          \0 <--> 16
 ...
 
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  • #388
That isn't one of the rows though is it? It is the set of rows that needs to have cardinality both aleph-0 and 2^aleph-0, not the number of ways of choosing entries from the rows, or the number of paths through the rows. Please answer this question.

Is the description of the construction of the diagram I gave accurate? The one I gave two posts back, the one I've given several times.

YES or NO? Can't say fairer than that; all we want in your next post is exactly on word, yes, or no, which is it?
 
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  • #389
Matt,
That isn't one of the rows though is it? It is the set of rows that needs to have cardinality both aleph-0 and 2^aleph-0, not the number of ways of choosing entries from the rows, or the number of paths through the rows.
My (aleph0 x 2^aleph0) matrix and an Infinitely (Width x Length) Binary Tree are two representations of the same thing.
 
  • #390
Is the description of the construction of the diagram I gave accurate? The one I gave FOUR posts back, the one I've given several times.

YES or NO? Can't say fairer than that; all we want in your next post is exactly on word, yes, or no, which is it?
 

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