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

[matt grime]

But what is your definition of aleph-0 and aleph-0 + 1?

Two sets with an isomorphism between them no longer have the same cardinality

Now suppose S and T are sets and there is an injection from S to T. Does your system imply |S|<|T|?

 

[Organic]

Matt,

Because by my point of view the cardinality of infinitely many elements is unknown (no collection of infinitely many elements is completed) I use aleph0 as the notation of this open (non completed) state.

By doing this I can use any operation that we use between finite collections for example aleph0/2 < aleph0.

For example: the number of odd numbers in N is aleph0/2.

 

[matt grime]

But then what is the cardinality of an infinite set that isn't a set of natural numbers, or a subset of them?

For instance, what is the cardinality of the set of finite groups, what is the cardinality of the set of algebraic integers, what is the cardinality of the set of functions from projective n space to projective m space? What is the cardinality of the underlying field of the rank one free module of D_2n over an algebraically closed field of characteristic 2?

There is more to life than just the natural numbers. Unless you about to invent a different cardinal for every set you ain't getting much.

And I can differentiate between the set of integers and the set of even integers because they are different (but isomorphic) sets.

 

[Organic]

Matt,

Aleph0 is only the basis of a collection of infinitely many elements.

For example 2^aleph0 = |R|

For example (that you already know):

<---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

Also please pay attention that 3^aleph0 > 2^aleph0

 

[matt grime]

But why is the card of R 2^aleph-0?

Every real number has an integer part, and there are aleph-0 of those, for each of these surely there are 10^aleph-0 possible decimal expansions of the non-integer part? so |R| is aleph-0*10^aleph-0. But wait what if I chose base 2,3,4 or 7 expansions, surely then |R| is aleph-0*3^aleph-0, so they must be the same, cancelling aleph-0 surely 2^aleph-0=3^aleph-0...

so, what's your justification for defining |R|=2^aleph-0

you said any cosntruction I could do with finite sets I could do with infinite ones, so I just did.So, what's the cardinality of the set of all finite groups?

 

[matt grime]

I've got an even better one.

What's the cardinality of the rationals?

 

[Hurkyl]

By my definition I can distinguish between aleph0+1 and aleph0.

By your definition you cannot distinguish between them.

So, where is the advantage of your system, that we have to keep?

There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

 

[Organic]

Matt,

so, what's your justification for defining |R|=2^aleph-0

My mistake, I mean |R|=base_value>1^aleph0.

surely 2^aleph-0=3^aleph-0...

surely 2^aleph-0<3^aleph-0...

because in 3 notations we have much more combinations in both width and length of the 0,1,2 matrix, then in the width and length of the 0,1 matrix.

Shortly speaking, my system does not ignore these differences between 2^aleph0 and 3^aleph0 matrixes and can use them to make math.

Your system cannot do that.

What's the cardinality of the rationals?

Please look at page 5 in this paper:

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

 

[Organic]

Hurkyl,

There is an (essentially) unique statistic about sets that tells us when sets have bijections between each other.

My definition of cardinality is that statistic.

Whatever sort of thing aleph0 is supposed to be, it is not that statistic.

Therefore, your definition is entirely useless if I want to know a statistic that tells us when sets have bijections between each other, however my definition works.

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

But I can be much more sensative then you and find an interesting information that exists between your 0 1 statistic results, for example:

2^aleph0 < 3^aleph0 because base value of 3 creates more information (that can be explored) then base 2.

The same is about aleph0 < 2^aleph0 or aleph0+1 > aleph0.

The finite part of each operation here can be used to give us statistical results, which are much more interesting then any general statistics, which is reduced to and based on 0 1 results.

Your definitions work, but what a price you pay (by ignoring information that can be very important)?

Shortly speaking, your word is a synthetic 0 1 digital world.

My world is an analogical world that can use details and also can give statistical results, which are much more accurate then your 0 1 digital statistical world.

For example, from my point of view the CH problem is an artificial problem that was forced on infinitely many elements that can be understood only by 0 1 digital point of view.

Shortly speaking, from my analogical point of view Cantor's world is nothing but a collection of shortcuts that do not distinguish between the simple and the trivial.

Let me give you some example:

We have this inifintely long periodic patterns list:

0
0
1
0
0
1
0
0
1
.
.
.

Let us say that we want to know the ratio between notations '1' and '0' in this infinitely long list.

By your system we can find a bijection between '1' and '0' notations:

0 <--> 1
0 <--> 1
0 <--> 1
...

So the ratio value does not exist when we deal with infinitely many elements (and it cannot be used to make Math)


By my system the ratio r1=aleph0/3 or aleph0*(1/3), the ratio of r0=aleph0*(2/3) and r1+r0=aleph0.

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

 

[matt grime]

|R| = base_value>1^aleph-0?

what does that mean? base value of what? why isn't 1^aleph-0 1 as it ought to be.
Did you understand why I wrote that 2^alpeh-0=3^alpeh=0 etc based upon the construction of R?Our stastice is about sets, yours is only about sets of natural numbers

if we have 001001001001... I can tell you the ratio as it's the limit of n/3n as n tends to infinity, ie 1/3.Why, if I can see that there is a copy of Q inside N, and a copy of N inside Q, can I not conclude they have the same number of elements? Why must the cardinality of a set depend on how it's written down. For instance, the evne integers are '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?

 

[Organic]

There is a conceptual problem in the basis of the "transfinite" world.

By using the word "transfinite" we mean that we can capture the all collection of infinitely many elements where the capturer tool does not belong (transcendent) to the elements which it captures.

R set is a complete set (no gaps) that described as “given any arbitrary interval, this interval includes infinitely many points which are connected to each other”.

Now please show me how infinitely many elements can be both unique AND non-unique (connected) on the same level?

 

[matt grime]

But you've not correct th problemt that 'capturer tool does not belong to the element it captures'

your aleph0 is still not a natural number.Don't you in newdigaonlpdf state clearly the rationals are countable, that is |Q|=|N|?

but the even numbers are countable too, so |evens|=|N|, implies aleph0=aleph0/2?

Or would you like to clarify what countable means.

first you state that |R| is 2^aleph-0, then you change that to something that doesn't make sense (just put real numbers in there to see why), and now when asked to clarify that, you come up with something about uniqueness.What do you mean points of R are connected to each other?
The interval [x,x] contains exactly one point in it.
Q also has these properties, or at least the best guess I can make from what you actually write.

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

 

[Organic]

Matt,

'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"?

 

[moshek]

I see that it is very exiting here ,
I will join you in few days.

Have fun with mathematics!

Moshek

 

[Organic]

Matt,

What do you mean by an infinite set whose points are unique and non-unique? where did that come from.

As I clearly show here:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

No infinitely many unique elements can construct (can be or can use a model of) a solid line.

So you have to deside between infinitely many unique elements XOR solid line.

If you choose infnitely many elements, then their cardinality is the unknown value of infinitely many elements that cannot be completed by definition ("infinite" means no end therefore no completeness).

If you choose solid line then you have no iput that can be used to make math or shortly speaking |R|=?^aleph0.

 

[Organic]

Matt,

(just put real numbers in there to see why),

base_value is any n>1


aleph0 is the cardinal of any arithmetic series of infinitely many elements.

n>1^aleph0 is the cardinal of any geometirc series of infinitely many elements.

 

[matt grime]

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?


The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

Take N, take NU{-1}, and NU{-2}

They both have card, in your world, of aleph-0+1, but they are different sets.

The card of {1,2...,n} is the same as {2,3,...,n+1} yet they are different sets. What makes you think cardinality is in anyway a measure of what the elements of the set are?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

 

[Hurkyl]

So, all you have is: there is(=1) or there is not(=0) a bijection between two sets.

That is not all I have; I have other things like well-order types, topologies, algebras, and measures.

But it is frequent that cardinality is what I want. In fact, the idea of there being a map from the natural numbers onto a set is so important that it was given a special name.

 

[matt grime]

To echo Hurkyl, two sets having the same cardinaltiy is by definition saying there is a bijection between them, that is all it says. You seem to think that different sets cannot ave the same 'cardinality', at least that is how i interpret you assertion that I can't distinguish between aleph-0 and aleph-0+1, well, how can you distinguish between them? what is aleph-0? You assert |Q| and |N| are the same but surely from the way you construct them |Q| = aleph0*log(aleph0) approx.

we have a very good way of distinguishing between N and Q - they are not equal. They are in 1-1 correspondence, though.

 

[Organic]

Matt,

Please answer to each part of this post.

part 1:

'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"?


part 2:

n>1^aleph0 makes no sense, why is there a "greater than" sign in there, what does it mean?

By n>1 I mean any n value greater than 1.

The real numbers arre equivalence classes of cauchy sequences of rationals, it is a well defined construction.

It is not well defined construction because it uses simultanuasly two different models that contradict each other (existing on the same level), which are:

1) A model of inifintly many elements.
2) A model of solid line with on gaps.

What makes you think cardinality is in anyway a measure of what the elements of the set are?

dependency Matt,

Can your body exist without the atoms of it?

You claim |N|=|Q| in your cardinal system, and they are different sets too, so what's the point of it?

Again, please read page 5 in:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

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


Another important thing:

When I use ", ..." notation in {1, 2, 3, ...} I mean that a set with infinitely many elements cannot be completed, because the meaning of the word "infinite" is "has no end".

 

[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?