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

[matt grime]

so |R| is in your opinion simultaneously any of n^aleph-0, for any n in N?

Seeing as you are doing arithemic on cardinals, must not |R|=|R|?

thus mustn't 2^aleph-0 = 3^aleph-0 =...?

Otherwise |R| is not equal to |R|.


As for the other questions you raise.

you write that |Q| =|N| trivially because of the counting algorithm you give.

But I can give a counting algorithm that shows |N| is equal to the cardinality of the even natural numbers - the counting goes n <--->2n

Since YOU have said the evens have cardinality aleph0/2, it must then follow if your arithemetic is correct that aleph0 =|N|=|even naturals|=aleph0/2

that is logically what you are claming if you think |Q|=|N|.


And no cardinality does not depend on what the elements in a set are, even in the finite case. I have 4 oranges, I have 4 apples, it's the same 4 in each case.


Who says that the real numbers ARE a line? They are not a line - they are, in one construction, the set of equivalence classes of cauchy sequences of rational numbers. That they are useful for measuring and drawing a line is not important. sqrt -1 is useful in electrical engineering, that doesn't mean it is a voltage or a current.

I've read page 5. It is not correct in showing |Q|=|N| as the correspondence sends both 1/3 and 3 to the same element in N, so it isn't a bijection.

 

[Organic]

Matt,

Since YOU have said the evens have cardinality aleph0/2, it must then follow if your arithemetic is correct that aleph0 =|N|=|even naturals|=aleph0/2

that is logically what you are claming if you think |Q|=|N|.

((aleph0/1)*(1/aleph0))*aleph0=1*aleph0=aleph0

And no cardinality does not depend on what the elements in a set are, even in the finite case. I have 4 oranges, I have 4 apples, it's the same 4 in each case.

So cardinality value depends on set's content.

|N| is the cardinal of any arithmetic series of infinitely many elements. therefore |Q|=|N|

|R| is the cardinal of any geometric series of infinitely many elements.

|N| or |R| have no unique values, and they are used by me like two common "family names" to infinitely many "first names".

Who says that the real numbers ARE a line?

Please explain to me how to different and unique elements have no gap between them, and the result is NOT a solid line.

'half the integers' so have card aleph0/2, yet, they are also the set {2n|n in N}, and so they have as many elements as N as well?

So aleph0/2 = aleph0 isn't it?

Which means that by your statistics you say: "1".

Also by your statistics 2^aleph0=3^aleph0 --> "1".

Now, if you have these two "1" can you tell me what created each "1"?

 

[matt grime]

1. I'm going to stop writing aleph-0 because it's just confusing to anyone who might read this. your aleph-0 is not our aleph-0 I will just say A

So let me get this right. A is |N|, now A+1 and A are different? n^A is different for every n in N? |R| is simultaneously 2^A, 3^A, 4^A... so |R| is not equal to |R|, as it has distinct 'cardinalities' depending on which base you write your decimal expansions in. A is the 'cardinality' of any arithmetic progression, so that A is the 'cardinality' of 2,4,6,8... which is an infinite arithmetic progression, it is also the even numbers and their 'card' is A/2? |Q| is A as |Q| is for some reason the same as A*(1/A)*A

2. Card(S) depends on what the elements are that are in the set too. So that the cardinality of the set of 4 oranges is different from the cardinality of the set of 4 bananas? This is the translation in english of whatever you are thinking in Hebrew.


3. Your 'card' has nothing to do with bijections of sets, and is a many valued function defined on a a set. |R| is many things that you've said are not equal), and have no place in Cantor's Diagonal argument, becuase you have just redefined all the terms in the theorem and its proof, so why should it remain valid?


4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from




5. As to the real line. What do you mean by 'different and unique [real numbers] have no element between them'? Given any two distinct real numbers there is always a third between them. Are yo attempting to say there are 'no gaps' in R and it must therefore 'be' a real line? Erm, no, the real numbers are not a line, they are formally equivalence classes of cauchy sequences of real numbers, or Dedekind cuts. They from a totally ordered field that we some times think of as a line, naively, just as the set ZxZ 'is' a lattice point set in the plane.

 

[Organic]

Matt,

First, thank you very much for this post, it is really a good one.

So let me get this right. A is |N|, now A+1 and A are different? n^A is different for every n in N? |R| is simultaneously 2^A, 3^A, 4^A... so |R| is not equal to |R|, as it has distinct 'cardinalities' depending on which base you write your decimal expansions in. A is the 'cardinality' of any arithmetic progression, so that A is the 'cardinality' of 2,4,6,8... which is an infinite arithmetic progression, it is also the even numbers and their 'card' is A/2? |Q| is A as |Q| is for some reason the same as A*(1/A)*A

YES.

2. Card(S) depends on what the elements are that are in the set too. So that the cardinality of the set of 4 oranges is different from the cardinality of the set of 4 bananas?

No, cardinals depends on quantity or on arithmethic/geometric progression+operations.

3. Your 'card' has nothing to do with bijections of sets, and is a many valued function defined on a a set. |R| is many things that you've said are not equal), and have no place in Cantor's Diagonal argument, becuase you have just redefined all the terms in the theorem and its proof, so why should it remain valid?

By this model:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf
I don't see how the transfinite universes can exists between what you call progressions (which I call intersections, in the above model) and
a "solid-line" state of what I call "the strong limit of Math language" (or "actual infinity").

4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from

Cardinality is only one parameter, the other parameter is the "structural properties" of |Q| members:
http://www.geocities.com/complementarytheory/UPPs.pdf
that can help us to distinguish between Q and N.

5. As to the real line. What do you mean by 'different and unique [real numbers] have no element between them'? Given any two distinct real numbers there is always a third between them. Are yo attempting to say there are 'no gaps' in R and it must therefore 'be' a real line? Erm, no, the real numbers are not a line, they are formally equivalence classes of cauchy sequences of real numbers, or Dedekind cuts. They from a totally ordered field that we some times think of as a line, naively, just as the set ZxZ 'is' a lattice point set in the plane.

So why standard math uses the word "line" and connect it to a collection of infinitely many elements?

 

[moshek]

Dear Matt,

I am really exited
from the development
of a real dialog

We only ... try to follow
the real vision of Hilbert
from his famous lecture
at Paris on 1900.

Not the solution of the 23 problems
was his target !
he said that on a very clear way
at the end.

Waiting to join you.

Moshek


p.s My paper on 01 laws is almost complete now.

 

[matt grime]

You have a different definition and a different set of rules relating to cardinals. Therefore, Cantor's argument and its proof, which uses the same symbols but with different meanings, need not remain true.


Example. A function f is unifomly continuous if, for all e greater than zero there is d greater than zero such that |x-y|<d => |f(x)-f(y)| <e.

Now suppose I change the meaning of e 'greater than' 0 so it corresponds in the old terminology to 'e=-1'

then the proof that f(x)=x is uniformly continuous will not remain valid because I've just changed all the meanings around, and infact nothing is uniformly continuous because |a|<-1 is impossible (| | here means modulus, absolute value, not cardinality)

A mathematical quantity is what it does, the aleph-0 of cantorian set theory are an indactor of the isomoprhism type of the set. Two sets have the same cardinality iff they are bijective. This is not true in your meaning of the word cardinality.

you have assigned different meanings and properties to symbols as used by other people to mean radically different things, you cannot expect the translation of every result to hold if you do not translate the meanings.

So you saying the transfinite universe does not exist is the same as saying there are no such things as uniformly continuous functions, when you've redefined all the terms to mean something entirely different.



When elementary mathematics speaks of the number line it is speaking of a mental picture you can draw to learn how to handle the real numbers. It is called a 'line' in higher mathematics because it allows us to do (idealized) geometry, where things behave like the imperfect world of drawing on paper with a pen. We say there are 'no gaps' in the 'line' because by definition the metric space R is complete and therefore every cauchy sequence converges, unlike Q, where, say, the sequence, 3, 3.1, 3.14, 3.141, 3.1415,... is cauchy but does not converge in Q as pi is not rational.

As such R is a totally well ordered complete field, and is unique (up to order-field-isomorphism) and it can be visualized as a line of elements.

There are other properties we use to distinguish sets in mathematics other than cardinality. Q is not (naturally) well ordered (it can be well ordered with order type w^2 I think), N is.

 

[Organic]

Matt,

4. You claim that your 'statistics' distinguish sets, well, N and Q have the same statistics, or at least one in each class of objects |N| and |Q| is the same, thus you don't distinguish between where the cardinalities come from

You know what? you a right about |N| and |Q|.
|N| = A
|Q| = Ak/Aj where k or j are independed n's.

 

[Organic]

Matt,

So, once more why do you think my 01 matrix does not have width aleph0 and length 2^aleph0 from a Cantorian point of view?

For example:

<-------------------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

 

[matt grime]

When you ask me why i think the number of rows is not 2^aleph-0 in cantorian theory, are you asking me that with MY concept of cardinality? The rows are in bijection with N, since they are formed by a sequence indexed by N, N does not have card 2^aleph-0 in my theory because that statement is equivalent to saying there is a bijection from N to P(N), when no such exists as is easily demonstrated by one of at least 7 proofs that I've seen over the years.

By definition two sets have the same cardinality IFF they are isomorphic as sets.

 

[Organic]

Matt,

It is very nice what you say, but now there is no ZF axiom of infinity in my new construction, so the matrix in my previous post must have a length magnitude of 2^aleph0, isn't it?

 

[matt grime]

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

 

[Organic]

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:

<-------------------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?

 

[matt grime]

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.

 

[Organic]

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.

<----------------------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:

<---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:

<---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:

<---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:

<---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

 

[matt grime]

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.

 

[Organic]

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.

 

[matt grime]

Then you have been mistaken about mathematics for more than 20 years.

 

[Hurkyl]

Can you tell us what your construction is, instead of showing a tiny piece of it?

 

[Organic]

Matt,

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

 

[Organic]

Hi Hurkyl,

Please look at this pdf:
http://www.geocities.com/complementarytheory/Countable.pdf

read it carefully and when you undestand it please try to address it in standard Math notations.

Thank you,

Organic

 

[Hurkyl]

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?

 

[matt grime]

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.

 

[Organic]

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* .

 

[matt grime]

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.

 

[Organic]

Matt,

|Z*|<|P(Z*)| but both are contable.

 

[matt grime]

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?

 

[Organic]

Matt,

You are playing with words (definitions) I show a concrete proof
that |Z*|<|P(Z*)| and both ( Z* and P(Z*) ) are enumerable.

 

[matt grime]

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...

 

[Organic]

Matt,

You still don't understand do you?

To contradict my last version, that can be found here:

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

YOU HAVE TO PROVE THAT Z* IS NOT EQUAL TO Z* !


kappish?

 

[matt grime]

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.

 

[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.

 

[suyver]

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?

 

[Organic]

Dear Matt,


Please look again at my paper, I put more details that can make
it clearer:

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


Yours,

Organic

 

[matt grime]

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.

 

[Organic]

Ok Matt,

After you air your view, please look again at page 3 (step 1):

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

 

[matt grime]

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.

 

[Organic]

Matt,

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

Let us take again our set:

 {...,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:

 {...,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:

 {...,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
 ...

 

[matt grime]

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?

 

[Organic]

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.

 

[matt grime]

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?

 

[Organic]

Matt,

1) 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?

2) 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?

1) Yes I agree that this is the redundant way to show how the columns are constructed, but:

2) You showed nothing because the three representations below are one and only one thing:

 {...,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:

 {...,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:

 {...,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
 ...

 

[matt grime]

So we have absolutely nailed the construction of the array.

Now, are you claiming the the cardinality of the rows is 2^aleph-0 because for every element in the power set of N, there is a row which corresponds to the indicator function of that element in the power set? Yes, or No is again all that is required. Note this is equivlaent to saying that the set of rows has card 2^aleph-0 because the rows form a 'complete' list of every string of 0s and 1s, ie *ANY* possible string of 0s and 1s occurs as one of the rows.

 

[Organic]

Matt,

If we use your terminology, then the answer is yes.

 

[matt grime]

But we agreed that we are using the cantor view point on cardinality, didn't we?

So, let z be some arbitrary element in P(N), it must if your conjecture is true be one of the rows. Let the number of the row (they're enumerated by N) be R.

Are we ok so far? Yes or no?

 

[Organic]

Matt,

(they're enumerated by N)

No, they're enumerated.

This is what you don't understand, the list of unique notations that related to R is longer then the list of unique notations that related to N.

In your termenology the complete R list is longer than the complete N list.

 

[matt grime]

So the rows are not of cardinality aleph-0 as you've been claiming all through this, then?

 

[Organic]

Matt,

You simply don't read what I write.

So here it is again:

1) Both R list and N list are enumerable.

2) |R|>|N|

 

[matt grime]

So, when I asked you if the rows could be enumerated by N, and you said no, were you lying or mistaken? because that's what enumerable means in this context. so we ask again, are the set of rows in bijective correspondence with N as you claim they are? So picking an element z in P(N) it occurs at row r for some r in N (this r was the R last time, i didn't mean R as in real number, sorry if that's the confusion), that is what you mean by the rows are enumerable, which they are by the construction we've agreed on. And you are claiming that each element of P(N) corresponds to some row.

 

[matt grime]

Incidentally, the word enumerable is the same as countable. So, remembering that we are usgin real mathematics here, your last post states there is a bijection between R and N, and that there isn't a bijection between R and N.

 

[Organic]

Matt,

are the set of rows in bijective correspondence with N as you claim they are?

You know what? let us play your game.

1) I constructed a list with unique sequences of 0 1 combinations each, which its length has 2^aleph0 magnitude.

2) I also showed a bijection from N to P(N).


Conclusion:

The transfinite univereses do not exist.