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

[Hurkyl]

There is no infinite term in a geometric series. Each of the aleph0 terms are finite.

 

[Organic]

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.

<---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

 

[Hurkyl]

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?

 

[Organic]

Hurkyl,

then what are the terms that come immediately before them?

Please give some simple example in plain English.

 

[Hurkyl]

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 ω-th step of the process (if one even exists)

 

[Organic]

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:

<---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

 

[moshek]

Gaifman lecture

Organic !

please do read the abstract of Haim Gaifman lecture at:

http://www.cs.tau.ac.il/~nachumd/semester.html

thank you
Moshek

 

[matt grime]

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.

 

[Organic]

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.

 

[matt grime]

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.

 

[moshek]

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

 

[Organic]

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.

 

[moshek]

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

 

[Hurkyl]

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.

 

[Organic]

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."

 

[Organic]

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:

<---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

 

[Organic]

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.

 

[moshek]

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

 

[Hurkyl]

Hrm, ω 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!

 

[matt grime]

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.

 

[moshek]

matt : Thank you for your long answer i feel now comftable with you.
I will study cerfully your answer and replay to it tommorow.

Best

Moshek

 

[matt grime]

IN particular Organic has never produced any evidence to back up his assertion that the countable set of rows in the diagram for N corresponds to a set of card 2^aleph-0. In the case for n columns the rows correspond to the power set of card 2^n, however this is not true in the infinite case, as has been clearly demonstrated. The orginal diagram had only rows with a finite number of non-zero entries, then there came a diagram interleaving with a second array where each row had only finitely many zero entries. The proofs of these facts are elementary and organic has continually refused to either accept these proofs or indicate where they are wrong. His only tactic has been to introduce yet more irrelavant things. If you wish to treat him as a mathematical thinker perhaps you, Moshek, could try and peresuade him to explain where these elementary proofs are wrong; they aren't as anyone can see, but Organic does not extend to us the courtesy he demands of others in reading and trying to understand what others write. We have read what he has written and do understand what he is attempting to do, and we can also see where he is going wrong.

 

[moshek]

Thank you Matt for the extension of your view and your really deep care for mathematics that teach me. I promise you to do all the best I can do for understanding of the situation were we are now !

sincerely
Moshek

 

[Organic]

Matt,

We have read what he has written and do understand what he is attempting to do...

No Matt you you do not understand what I am doing because my point of view is a paradigm change of the infinity concept used by Math language.

There are two kinds of it:

1) Actual infinity (the word "many" cannot be used) which is the unreachable limit of Math language.

2) Potential infinity (infinitely many objects) which is the only form
that can be used by mathematics language.

If you don't distinguish between these two forms of infinity concept, then you cannot say even a single word about my mathematical definitions.

Simple as that.


And for you Moshek, it looks that you have no opinion of your own and you are trying to be nice to everyone.

So, please let me tell you, this is not the way, you have to choose your side.

 

[moshek]

Well Organic you know already that I am on the side were you are since I work on this discovery by myself for the last 20 years. This is not the point here . I try to understand matt as much as I can to find some bridge among the gap now. you must understand that the fact that you don’t have many background in the tradition mathematics make it very difficult to understand you. and they may say by a mistake that it is a Garbage. but they are really care from there heart on mathematics.

Moshek

 

[Organic]

Dear Moshek,

In this case you have to show clearly what is your fundamental paradigm about the infinity concept and also clearly show what do you think we have to do if we want to show to the people of the old paradigm, how they can make Math under the new paradigm.

Can you do that?

 

[matt grime]

As I've said before you are prefectly entitled to whatever philosophical position you want. What I will not accept is you telling us that mathematics cannot cope with something that it makes no claims to explain, and is not mathematical. The only problems arise when we accept your ideas, ideas which are inconsistent and ill-defined. If you want to show mathematics is inconsistent you cannot work from outside, you must demonstrate it internally.

Noticably you don't defend yourself against the allegations that you do not read or understand other people's posts.

It is only your opinion that the rows you write down are in correspondence with a set of cad 2^aleph-0. We can prove it isn't as the property of being in bijection is an elementary equivalence relation, only in your theory is there a problem with this; it is only if one declares that the words 'many' and 'for all' are incompatible with logic that your problems with cantor arises, however that isn't mathematics.

It is now clear that you cannot refute the disproof that the rows are not in correspondence with the power set, which shows how poor your abstract thought is.

Do you want another thing to talk about?

You've demanded we provide a multiplication on N that is complementary to addition.

Let's do that.

1. What do you mean by complementary

2. What is a multiplication?Give the rules this notional operation must define.

Note yours

1. Is not a binary operation from NxN to N

2. Does not associate,

3. Either does not distribute over addition, or have 1 as the identity (if it had both then it would be commutative).

So what must something be to qualify as a 'multiplication' and be complementary to addition.

(And you claim we don't read and understand you)

 

[matt grime]

Originally posted by Organic
Hurkyl,

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.

It's very simple, Organic, and if you understood this your life would be so much better. Although there are 'aleph-0' entries, each entry is either in an odd or even position (at a finite point in the sequence), therefore it makes perfect sense to say the string

(x_i) i in N is 1 when i is even 0 when i is odd.

Aleph-0 is not a number, there is not an aleph-0'th entry in the sequence, every element in the sequence occurs at some n in N, which is either an even or an odd number.

 

[moshek]

Dear Matt !

I am afraid that i will not be able to answer you today i am writing a paper on 01 laws and i have to finish it until thursday.

But i will certanly do that in few days.

take care
Moshek

 

[moshek]

and also to you Dear Organic

Sorry

Moshek

 

[Organic]

Hurkyl,

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"

Can you please show us how we can use "even" and/or "odd" when there are infinitely many '1' notations and infinitely many '0' notations in some 01 infinitely long sequence?

If you ask me then if there is a way to define 01 infinitely long sequence where "even" and/or "odd" cannot be defined, then this sequence is bijective to N members.

And this is exactly what I mean when I speak about aleph0.

In aleph0 magnitude "even" and/or "odd" can't be used.

If we examine again the structure of the most right column of the matrix which is drawn below we find that the length of the list has at least aleph0 magnitude.

<---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

Now I am going to show my new point of view about the potential infinity.

First let us look at this list:

<---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                    |
 ... 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:

<---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                    |
 ... 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:

<---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,                   |
 ...,1,1,0,                   |
 ...,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:

<---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] <--> 1 geometric series 
 ...,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

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

Therefore we can conclude that the length magnitude of our list > aleph0 magnitude.

Please pay attention that in this case ...0101 most right column length has 2^aleph0 magnitude.

Therefore the width and length of my 01 matrix are both countable.

    3 2 1 0                   3 2 1 0
   2 2 2 2                   2 2 2 2
   ^ ^ ^ ^                   ^ ^ ^ ^
   | | | |                   | | | |
   v v v v                   v v v v
...0 0 0 0 <--> 1    or   ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2    or   ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3    or   ...1 1 0 1 <--> 3    
...0 0 1 1 <--> 4    or   ...1 1 0 0 <--> 4    
...0 1 0 0 <--> 5    or   ...1 0 1 1 <--> 5   
...0 1 0 1 <--> 6    or   ...1 0 1 0 <--> 6    
...0 1 1 0 <--> 7    or   ...1 0 0 1 <--> 7    
...0 1 1 1 <--> 8    or   ...1 0 0 0 <--> 8   
...1 0 0 0 <--> 9    or   ...0 1 1 1 <--> 9   
...1 0 0 1 <--> 10   or   ...0 1 1 0 <--> 10  
...1 0 1 0 <--> 11   or   ...0 1 0 1 <--> 11  
...1 0 1 1 <--> 12   or   ...0 1 0 0 <--> 12  
...1 1 0 0 <--> 13   or   ...0 0 1 1 <--> 13  
...1 1 0 1 <--> 14   or   ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15   or   ...0 0 0 1 <--> 15  
...1 1 1 1 <--> 16   or   ...0 0 0 0 <--> 16
...                  or   ...

or a mixing of them

 

[matt grime]

your view on infinity means that the rows are simultaneously countable, and that they are NOT in bijection with N. Thus your theory proves a statement that is simultaneous true and false. And you think our is the faulty system?

 

[Organic]

No Matt,

It is true and false iff countability is related to N (as it understood by standard Math).

My point of view clearly shows that this is not the case, and standard Math does not understand the infinity concept.

 

[Hurkyl]

As we can simply and clearly see, there is no bijection between the natural numbers and the infinitely long geometric series of my list.

It looks to me like there's one block of length 8 for every natural number. (and nothing else)

Thus, I can make a list such that there's one row for each natural number (and nothing else) through the following mapping:

For any natural number n, write it as 8p + q for p and q integers and 1 < q <= 8. Then, I map n with the q-th row in the p-th block.

    3 2 1 0                   3 2 1 0
   2 2 2 2                   2 2 2 2
   ^ ^ ^ ^                   ^ ^ ^ ^
   | | | |                   | | | |
   v v v v                   v v v v
...0 0 0 0 <--> 1    or   ...1 1 1 1 <--> 1
...0 0 0 1 <--> 2    or   ...1 1 1 0 <--> 2
...0 0 1 0 <--> 3    or   ...1 1 0 1 <--> 3    
...0 0 1 1 <--> 4    or   ...1 1 0 0 <--> 4    
...0 1 0 0 <--> 5    or   ...1 0 1 1 <--> 5   
...0 1 0 1 <--> 6    or   ...1 0 1 0 <--> 6    
...0 1 1 0 <--> 7    or   ...1 0 0 1 <--> 7    
...0 1 1 1 <--> 8    or   ...1 0 0 0 <--> 8   
...1 0 0 0 <--> 9    or   ...0 1 1 1 <--> 9   
...1 0 0 1 <--> 10   or   ...0 1 1 0 <--> 10  
...1 0 1 0 <--> 11   or   ...0 1 0 1 <--> 11  
...1 0 1 1 <--> 12   or   ...0 1 0 0 <--> 12  
...1 1 0 0 <--> 13   or   ...0 0 1 1 <--> 13  
...1 1 0 1 <--> 14   or   ...0 0 1 0 <--> 14
...1 1 1 0 <--> 15   or   ...0 0 0 1 <--> 15  
...1 1 1 1 <--> 16   or   ...0 0 0 0 <--> 16
...                  or   ...

or a mixing of them

Both of these are countable (you've even "counted" them!), and so is any "mix" of these.

 

[Hurkyl]

It is true and false iff countability is related to N.

And countability is related to N by definition.