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

[matt grime]

I will concede this - I cannot state 'infinity is ...' and fill in something for the ... that is a definition in anyway that is very satisfactory. No mathematician would, or could without qualifying their statement. There was an interesting thread on sci.math about the role of infinity in mathematics, and the consensus was that mathematicians whilst using the term to illustrate concepts, would, when pressed to be rigorous, switch to another definition.

For instance, when we say there is an 'infinity of' possibilities, we actually mean, there is not a finite number of possiblities; given any finite number of options I can find another one'. When we say x(n) tends to x as n tends to infinity, what we actually mean is a statement that at no point includes the word infinity. Then there's the case of the sum to infinity, which is just the limit of a sequence as above, again with no infinity mentioned. Then the sum is 'infinite' if it is not finite, if there is no limit in the sequence of finite sums, that's all, agian we don't actually have an infinity there do we? Of course there is the point at infinity of the riemann sphere which neatly encapsulates the idea of being 'not finite', and which allows us to do many useful analytic operations. It is often called infinity, and can be related to the other examples, but is it 'infinity'? No, just like things such as multiplication it is contextual - the multiplication of real numbers isn't the multiplication of matrices is it? In short infinity is a useful concept, just as continuity is, but there is no object one can satisfactorily point to as infinity, just as there is no object one can point to and say that object is continuity.

Many cranks have this idea that infinity is actually something, something tangible, and that when we say the sum from one to infinity, we actually mean sum all the finite bits and then stop AT infinity just like we can stop at 7 or 20,445. If people learned the distinctions about these things we'd all be a lot better off. All this is compounded by the teaching that 1/0 IS infinity. It isn't, it is undefined in the ordinary arithmetic that they know, but it is true that 1/x can be made arbitrarily large, which is not the same thing at all.

 

[Deeviant]

Originally posted by Organic
Matt,

My allegory to you:

Matt: “Define a cat”.

Organic: Taking a cat and put it in front of Matt, then says “here is a cat”.

Matt: “No, define a cat”.

Organic: “Matt, it is in front of you”.

Matt: “You don’t understand, define a cat”.

Organic: ”what is define?”.

Matt: “Take a knife cut the cat to pieces and define each piece by putting it back to its place”.

Organic: ”But then you have no alive cat but pieces of flash. For me a cat is first of all alive thing in a one organic piece”.

Matt: “life is not important, definition is important, define a cat”.

A interesting analogy. Like the cat, your position would not survive a thorough disection, unlike the cat, your position wasn't a living breathing thing cute furry thing before the process.

 

[Hurkyl]

Let us assume that there exists a list of all real numbers.

I wish to create a list of intervals such that every real number is in one of these intervals.


For my first interval, I will look at the first real number in the list, and then choose an interval of length 1/4 that contains that real number.

For my second interval, I will look at the second real number in the list, and then choose an interval of length 1/8 that contains that real number.

etc...

For my n-th interval, I will look at the n-th real number in the list, and then choose an interval of length (1/2)^(n+1) that contains that real number.


So now I have a list of intervals.


Now, every real number is in one of these intervals.
Proof: Pick any real number. It appears somewhere in the list of all real numbers we selected at the beginning; let's say it appears in the n-th position. Well, the n-th interval was created so it contains the n-th real number in the list, therefore the real number we picked is in one of my intervals.


(Aside: Notice how I gave a set of instructions on how to create my list of intervals before I started proving things about my list. And notice that I'm not going back and altering those instructions or adding new instructions now that I have started proving things)


So this collection of intervals covers the entire real line.


Now, let's look at the length of the intervals:
The first one has length 1/4.
The second one has length 1/8.
The third one has length 1/16.
etc.
The n-th one has length (1/2)^(n+1) for all n.

The sum of all of these lengths is 1/2.


Now, one of the nifty things about length (and similarly about area and volume ) is that if you have a list of "shapes", then the length covered by those shapes cannot be bigger than the sum of the lengths of those shapes!

So, this means that the length of the real line cannot be bigger than 1/2!

 

[Organic]

Hi Deeviant,

unlike the cat, your position wasn't a living breathing thing cute furry thing before the process.

Why not?

 

[Organic]

Dear Hurkyl,

I hope that by this post we (maybe for the first time) will communicate between our different perceptions about the infinity concept.

First let us take the model of a line and I mean a smooth line without any points or segments (what you call intervals) included in it.

Zoom-in 1x2, 1x3, 1x4, 1xn, 1xn+1 ... and you find the invariant self similarity 1.

In a more formal way |{__}| = 1.

Now, let us talk on what you call the "real-line" of R collection.

The real line of R collection is not {__} form but {...} form.

Shortly speaking, we can find unique elements only in {...} form.

Only in {...} form we can find a one-to-many relation , and if we take your private case of 1/4, 1/8, 1/16, ... then our one-to-many relation has the invariant 1/2 which is exactly the invariant of a Binary Tree on infinitely many scales of it.

Shortly speaking, R collection cannot be but a {...} form.

Your conceptual mistake is that you take R collection as {__} form.

But {__} is a representation of what I call an "actual infinity", and cannot be used as an available information for Math language, or in other words, it is the strong limit of Math language.

Shortly speaking, R collection cannot use the word "line" because no infinitely many points or segments(=intervals) can be a solid line.

From a symmetry point of view, the opposite of the strong limit {__} is the weak limit, which is notated as {} content.

Please look again my major theorem: http://www.geocities.com/complementarytheory/Theory.pdf

And please give your detailed remarks.

Yours,

Organic

 

[Hurkyl]

First off, length is not preserved by scaling; a scaling factor of 2 means that lengths are doubled, areas are quadrupled, volumes are octupled, et cetera.

In particular, the line segments [1, 2] and [1, 4] may be similar, but there's a scaling factor of 3, so corresponding figures do not have the same lengths.


My currest best guess as to what you're trying to say is that you're either trying to make a distinction between a geometric line and the set of points incident with that line, or you're trying to conceptualize the distinction between the ideas of the "set of real numbers" and the "topological space of real numbers."

I have to go to work so I don't have time to expound upon this at the moment.

 

[matt grime]

why do you, organic, not accept the simple assertion that there is not bijection between the Natural numbers and R? Why do none of the proofs offered satisfy you? Why? It's not difficult; these things are just defininitions, and not very hard ones at that.

 

[Organic]

I agree that |R|>|N| but both are countable because R collection cannot be a solid line.

 

[Hurkyl]

The word "countable" is defined as:

A set S is countable if and only if there exists a function from the natural numbers onto S.

The word "uncountable" is defined as:

A set S is uncountable if and only if it is not countable.

The relation > is defined as:

|S| > |T| if and only if there does not exist a function from T onto S.


So why do you accept |R| > |N| but not R is uncountable.

 

[Organic]

Hurkyl,

I am talking about the invariant self similarity of the structural property of {__} and/or {...} where quantity is not important at this stage.

First off, length is not preserved by scaling; a scaling factor of 2 means that lengths are doubled, areas are quadrupled, volumes are octupled, et cetera.

Length measurement depends on comparison between some constant value and some changeable value (I am talking about 1 obserber and one object at a time).

If the observer is the constant then we say: "the observed object is changed".

If the object is the constant then the observer is changed.

Let us say that in our case the observer is changed.

In this case observer x 1/2 --> object x 2 , observer x 1/3 --> object x 3, ... , observer x 1/(n+1) --> object x (n+1)

So, in this case the observed object length = 1.

If there are at least two different objects, the length changes between them are not depend on the observer but on one of the objects, which is used as the constant value 1.

But again the first thing here is not the length changes but the invariant structural self similarity of {___} and/or {...} forms.

My currest best guess as to what you're trying to say is that you're either trying to make a distinction between a geometric line and the set of points incident with that line, or you're trying to conceptualize the distinction between the ideas of the "set of real numbers" and the "topological space of real numbers."

My point of view disagree with Contor's point of view about a collection with cardinality aleph-1 of infinitely many objects (={...} form) that can construct a solid line (={___} form), as we can find here: http://mathworld.wolfram.com/LineSegment.html

Distinction is only the first step.

The second step is to combine between {__} and {...} forms and the result is:

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

So why do you accept |R| > |N| but not R is uncountable.

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

Definitions are only tools of meaning, if they contradict the meaning then we have to find another definitions that can express the meaning.

In this case "countable" and the "must have" connection with N is a conceptual mistake.

Shortly speaking:
"A set S is countable if and only if there exists a function from the natural numbers onto S."

Is a conceptual mistake.

 

[matt grime]

Right, you say we must be open to your ideas, but you won't listen to us.
Definitions must not contradict what we know, yet in your newdiagonalpdf you define an array using some (undefined) construction that I've shown doesn't have the properties you define it to have.

Listen to your own advice.

So, where does Hurkyl's measure theoretic proof go wrong for showing there is no bijection from N to R? And don't start this invariant self similarity crap again because that isn't what the proof is using. We can dress it up some more if you don't understand measure theory, and use Baire's Category theorem - a one point set is nowhere dense, hence the countable union of them is nowhere dense and thus cannot be R.

 

[Hurkyl]

The thing is, sets do not have any structural quantities.

The only thing that matters about a set is the elements it contains.

And up to bijection¹, the only property of a set is its cardinality².



So, basically, when you're using set theory to describe things, at the ground floor, quantity is the only thing that matters³. Once you have a set, you then impose structure on that set. For instance, you can impose an order on a set, and the result is called an ordered set So, for example:

(R, <) is an ordered set⁴
(where R stands for the set of real numbers, and < stands for the usual ordering on them)

Now, if we're being entirely proper, we should say "(R, <) is an ordered set" or "< is an ordering for R", not "R is an ordered set" or "R is an ordered set with ordering <" The reason is that the structure of order is something that is beyond the meaning of the term 'set'. That being said, mathematicians often abbreviate themselves by using "R" when they should be using "(R, <)" for economy of notation.



Another structure we can impose on a set is that of being a field. For example

(R, +, *) is a field⁴.
(Where R stands for the set of real numbers, and + and * stand for the usual addition and multiplication operation on them)

Again, a mathematician will often abbreviate himself by saying "The field R" instead of saying "The field (R, +, *)". Anyways, the important thing is that the structure of + and * are not part of the set of real numbers; they are additional structure imposed beyond that of being a set.


For one more example, there is the ordered field (R, +, *, <). This is yet another structure that is different from the previous ones. The ordered set (R, <) does not have arithmetic, and the field (R, +, *) is not ordered.


To say it again, the point I'm making is that sets do not have structure! All sets have are elements! If you are using a structure on some set, that is something that is beyond its definition as a set!

To state my point in a different way, if you don't the notion of "quantity" and "elements" to be there from the very beginning, you can't use a set theoretic description of your ideas.

Hurkyl



1: In laymen terms, this means that if we consider two sets to "the same" if there is a bijection between them.

2: and any property that can be derived knowing nothing about the set but its cardinality, such as the property of being finite (cardinality equal to a natural number) or infinite (cardinality not equal to a natural number)

3: More precisely, it is cardinality that matters.

4: I am being somewhat narrow here for clarity. This is the set theoretic way to write a field; the theory of fields does not have things like ordered triples or sets of elements! Also, even when using ZF, there exist fields for which a set of all elements does not exist, such as the surreal numbers. Also, instead of what I wrote, one would often write (R, +, *, 0, 1), putting the additive and multiplicative identities into the notation.

 

[Hurkyl]

My point of view disagree with Contor's point of view about a collection with cardinality aleph-1 of infinitely many objects (={...} form) that can construct a solid line (={___} form)

The idea that lines have points on them goes back at least to Euclid; I imagine much further.

I imagine the concept that lines are made out of points is just as old, but modern mathematics dosen't require this point of view. In fact, depending on the circumstance, mathematicians can be downright pedantic about insisting that you keep the ideas of "a line" and "the points on a line" seperate.

Definitions are only tools of meaning, if they contradict the meaning then we have to find another definitions that can express the meaning.

In this case "countable" and the "must have" connection with N is a conceptual mistake.

Shortly speaking:
"A set S is countable if and only if there exists a function from the natural numbers onto S."

Is a conceptual mistake.

Mathematics need a phrase that means "There is a function from N onto S". The word countable was chosen. Thus, when a mathematician means "there is a function from N onto S", we use the phrase "S is countable", and when a mathematician uses the phrase "S is countable", we mean "there is a function from N onto S".

Mathematicians are using the definition of the phrase "S is countable" as a tool to express the meaning "There is a function from N onto S".


So, I'm not sure what your objection is. Do you just not like the word mathematicians have chosen for this purpose? If we used the term "S is gazorninplat" to mean "there is a function from N onto S", would it be acceptable to you? If so, then R is not gazorninlpat, and anytime you see a mathematician use the word "countable" you should mentally substitute the word "gazorninplat", and be happy!

 

[matt grime]

R is not countable. Please prove it is. None of the things you've claimed so far is a proof.

You say complementary multiplication is not commutative. I would question the assertion that it is a multiplcation. Not because it is not commutative, but because it is not a binary operation from NxN to N. It is an operation on certain combinatorial structures. I might call them trees but you don't believe in the usual definition of tree. For instance, you calculate 2*3. What is the outcome? Which number is it? Moreover, as you've challenged us to produce complementary additive and multiplicative operations (on N) would you mind stating what complementary means in this sense, and why we must define it on N when you don't define it on N.

And if you dismiss Hurkyl's use of the word structure as unimportant, how come you use the same word in your quote from one of your writings?

You don't like N in countable? Why not? the idea of bijection with N and the reason you have the word 'count' in there is because then you can label the elements with the counting numbers.

If you don't like that then we can say a set is countable if it is bijection with:

1. Q

2. Z

3. The set of primes

4. The set of even integers

5. The set of polynomials with coefficients in F_2.

6. The set of spheres in R^3 whose centres lie at rational coordinates and whose radius is an integer.

7. The set of all finite groups. (Up to isomorphism. Or the set of all finite sets up to isomorphism in the category SET)
Anyway, here's another question you've still not answered.

Why does the fact that the arrays with columns labellled 1 to n and rows labelled 1 to 2^n forms a complete description of the power set of 1,..,n imply that the case for an infinite countable set also has a countable number of rows? You keep saying 'by construction'. What construction? I don't see one in your latest version. You had one once, and using that we've proved again and again that you've made a mistake. Now you don't even attempt to describe the construction. Why not? What is it?

 

[Organic]

Hurkyl,

The thing is, sets do not have any structural quantities.

There is no such a thing structuarl quantitiy, and I never wrote such a thing.

I wrote structoral property, and a set has structural property, please take for example the Von Neumann Hierarchy:

[b][i]0[/i][/b] = |{ }| (notation = {}) 

[b][i]1[/i][/b] = |{[b]{[/b] [b]}[/b]}| (notation = {0})
               
[b][i]2[/i][/b] = |{[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b]}| (notation = {0,1})
  
[b][i]3[/i][/b] = |{[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b],[b]{[/b]{ },{{ }}[b]}[/b]}| (notation = {0,1,2})

[b][i]4[/i][/b] = |{[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b],[b]{[/b]{ },{{ }}[b]}[/b],[b]{[/b]{ },{{ }},{{ },{{ }}}[b]}[/b]}| (notation = {0,1,2,3})

and so on.

A set is only a framework to explore our ideas.

The concept of oreder set does not depend on the quantity concept as can clearly shown here:

[b]
By Complementary Logic multiplication is noncommutative,
but another interesting result is the fact that multiplication 
and addition are complementary opreations that can be ordered 
by different symmetry degrees where quantity remains unchanged 
for example:

A Number is anything that exists in ({},{__})

Or in more formal definition:

({},{_}):={x|{} <-- x(={.}) AND x(={._.})--> {_}}

Where -->(or <--) is ASPIRATING(= approaching, but cannot become closer to).

If x=4 then number 4 example is:

Number 4 is a fading transition between multiplication 1*4 and 
addition ((((+1)+1)+1)+1) ,and vice versa. 

This fading can be represented as:
 

(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's 
(((+1)+1)+1*2)     ={{{1},1},1,1}            clarity-degree
((1*2)+(1*2))      ={{1,1},{1,1}}            (no uniqueness) 
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                            Maximum information's  
                                            clarity-degree                                            
                                            (uniqueness)


============>>>

                Uncertainty
  <-Redundancy->^
    3  3  3  3  |          3  3             3  3
    2  2  2  2  |          2  2             2  2
    1  1  1  1  |    1  1  1  1             1  1       1  1  1  1
   {0, 0, 0, 0} V   {0, 0, 0, 0}     {0, 1, 0, 0}     {0, 0, 0, 0}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    (1*4)            ((1*2)+1*2)      (((+1)+1)+1*2)   ((1*2)+(1*2))
 
 4 =                                  2  2  2
          1  1                        1  1  1          1  1
   {0, 1, 0, 0}     {0, 1, 0, 1}     {0, 0, 0, 3}     {0, 0, 2, 3}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1))  ((1*3)+1)        (((1*2)+1)+1)

   {0, 1, 2, 3}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this [i]no-redundancy_no-uncertainty_symmetry[/i])
    |_____|  |
    |        |
    |________|
    |    
    ((((+1)+1)+1)+1)
 

Multiplication can be operated only among objects with structural identity, 
where addition can be operated among identical and non-identical 
(by structure) objects.

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) , ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) , ( ((1,1),1),((1,1),1) ) , ( (((1),1),1),(((1),1),1) )
[/b]

 

[Hurkyl]

If there is a function from N to S, then there certainly must be a function from N to S...

I don't see what the problem is. Anyways, I'll give a better response later!

 

[matt grime]

The huge great problem that you have never overcome is that you at no point prove that the rows you enumerate are in bijection with the power set of N.

So.

1. Write down here in this forum the construction you use to generate the r'th row of the array for the ones labelled 1,2,3,, no need to do it for the 1',2' etc2. Explain where the proof that this enumerates only the finite and cofinite sets is wrong. I've posted it at least 4 times in this thread and you've not said what's wrong with it beyond asserting that the rows mus by construction be the power set, yet you've not defined the construction.
3. Here we go again, let's od it one step at a time.

Just take the diagram as in Newdiognal.

the first column is the alternating sequence 0101010101...
the second goes 001100110011...
the rth goes in alternating blocks of r 0s and r 1s.Is that correct? and all the rows so produced are in correspondence with the power set of N, aren't they?

Just answer this for now

 

[matt grime]

Take the array as written in Newdiagonal.

is it true that the construction you allude to is to write the right hand column as 01010101... alternating 0s and 1s, the next column is 00110011.. alternating pairs of 0s and 1s, and that the rth row is r 0s then r 1s then r 0s and so on?

We can ignore the rows yo've now added in as they only correspond to the countable set of cofinite subsets.

 

[Organic]

Hurkyl,

"There is a function from N onto S". The word countable was chosen.

My problem is not the word "countable" but the "must have" connection between this word and N.

R is also countable and it is defenetly not N because |R|>|N|.

The proof that R is countable can be found here:

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

If your problem is that I use the notations of N to mark each unique 01 row sequence,
then you can use notations tricks, for example:

By using this trick 0 0' 1 1' 2 2' 3 3' 4 4' ( please see http://home.ican.net/~arandall/abelard/math12/Cantor.html )
we can build this list:

    3 2 1 0
   2 2 2 2
   ^ ^ ^ ^
   | | | |
   v v v v
...1 1 1 1 <--> 1
...1 1 1 0 <--> 1'
...1 1 0 1 <--> 2
...1 1 0 0 <--> 2'
...1 0 1 1 <--> 3 
...1 0 1 0 <--> 3'

0 0' 1 1' 2 2' 3 3' 4 4'... is good for base 2.

For base 3 the trick is: 0 0' 0'' 1 1' 1'' 2 2' 2'' 1'' 3 3' 3'' 4 4' 4''...

For base 4 the trick is: 0 0' 0'' 0''' 1 1' 1'' 1''' 2 2' 1'' 3 3' 3'' 3''' 4 4' 4'' 4'''...

And so on ...

Shortly speaking cardinality is first of all based on unique structural differences between elements that have to be notated by equivalence structural periodic repetitions of notations that related to these structural differences.

It means that from structural point of view this mapping

(structural periodic notations = 1)
1 <--> 1 
2 <--> 2  
4 <--> 3 
3 <--> 4 
5 <--> 5 
8 <--> 6 
7 <--> 7 
6 <--> 8 
...

can be done, where this mapping

(structural periodic notations = 1 <--> structural periodic notations = 2)
1 <--> 1
2 <--> 1' 
3 <--> 2
4 <--> 2'
5 <--> 3
6 <--> 3'
7 <--> 4
8 <--> 4'
...

cannot be done.

Shortly speaking, structural property is reacher and much more interesting then quantity property.

I'll say it again, mapping between elements must be checked first of all by the structural propery that exists (or does not exist) between these elements.

By structual point of view aleph0 < 2^aleph0 < 3^aleph0 ... < n^aleph0 < (n+1)^aleph0

By quantitative point of view aleph0 < 2^aleph0 = 3^aleph0 ... = n^aleph0 = (n+1)^aleph0

 

[matt grime]

Is there any reason why you will not give a direct answer to a direct question?

Is that construction above that I sepcified the construction you are using in Newdiagonal.pdf?

The problem is that each row you've marked with an integer, depending on the version you're now using, contains either finitely many 1s or finitely many zeroes and is therefore not in bijection with the power set.

You've not offered to explain the construction. Until such time as you explain the construction you have nothing concrete to work with. Note you cannot deduce anything by induction on the finite case; that is not how induction works.Note I wish to amend my observation of the construction to state the rth column has alternating blocks of 2^r 0s and 1s. This doesn't affect the deductions about the construction since 2^r>r

 

[Hurkyl]

There is no such a thing structuarl quantitiy, and I never wrote such a thing.

It was a typo. I meant to say "structural property".

I wrote structoral property, and a set has structural property, please take for example the Von Neumann Hierarchy:

What about it?

Shortly speaking cardinality is first of all based on unique structural differences between elements that have to be notated by equivalence structural periodic repetitions of notations that related to these structural differences.

No, cardinality is based on the idea of "set" and "invertible function", nothing else. (though it is a generalization of the idea of "quantity" which, again, is not a structural idea)

The concept of oreder set does not depend on the quantity concept as can clearly shown here:

I don't see any ordering given in your example. Anyways, an ordered set is defined to have a set and an order. We can talk about "quantity" in relation to sets, so we can talk about "quantity" in relation to ordered sets.

Shortly seaking, structural property is reacher and much more interesting then quantity property.

Yes, structural properties are generally richer than the "quantity property". However, the point still stands that no matter how much structure you pile upon a set, it is still a set, and thus we can still talk about the "quantity property" of the set.

 

[matt grime]

Acutally I think you'll find that r^(aleph-0)=s^(alpeh-0) for all r and s in N.

 

[matt grime]

Originally posted by Organic
Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely mant rows?

Where on Earth do you think I state the number of rows and columns is finite? I said that any row has either a finite number of 1s in it or a finite number of 0s. That does not say that a row is finite in length

 

[Organic]

Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely many rows where it's width's magnitude=aleph0 and it's length's magnitude=2^aleph0?

 

[Hurkyl]

By structual point of view aleph0 < 2^aleph0 < 3^aleph0 ... < n^aleph0 < (n+1)^aleph0

By quantitative point of view aleph0 < 2^aleph0 = 3^aleph0 ... = n^aleph0 = (n+1)^aleph0

There is no structural point of view! Cardinalities ignore ALL "structure"!

The only point of view where 2^aleph0 < 3^aleph0 is one where you've actually changed the definition of the symbols!

 

[matt grime]

Originally posted by Organic
Matt,

How my 01 matrix can be finite if there are infinitely many columns and infinitely many rows where it's width's magnitude=aleph0 and it's length's magnitude=2^aleph0?

Erm, no sorry, at no point have I specified that the matrix is finite. I've told you how to make the matrix up using finite blocks repeated an infinite number of times.

In fact I'm the only one who's specified how to construct it. Why don't you correct that, and tell me what the entries in the r'th row are, and how they are organized.

 

[Organic]

Matt,

Do you agree that the length of my matrix has a magnitude of 2^aleph0 where its width has the magnitude of aleph0?

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that each row can be reduced to ...1010101010 or ...0101010101 data.

 

[Hurkyl]

Do you agree that the length of my matrix has a magnitude of 2^aleph0 where its width has the magnitude of aleph0?

It has aleph0 rows, because it is a list.

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data and can be reduced to infinitely many rows of ...1010101010... data.

So ...011011011 never appears in your list?
What about ...100001000100101?

 

[matt grime]

Your last post makes little sense - you state that it is possible for the rows to correspond to the power set of N and for there to be no row corresponding to the even integers.

By implication the sets of rows and columns both have cardinality aleph-0, since you enumerate them.

You also claim that the rows are in correspondence with the power set of N, and thus the rows of the matrix must be a set of cardinality 2^alpeh-0.

My contention is that if you enumerate the rows as you do, whereby for row n you take the decomposition of n-1 in binary expansion and write the coeffs going from right to left, that the rows you enumerate do not form a set that is in correspondence with the power set - rather obviously they are exactly in correspondence with the finite subsets. You then try to muddy the waters by including the cofinite sets, without noticing that you are thus contradicting your earlier held beliefs.I do not state that every row must have infinitely many 0s and 1s. But that there must be some rows (uncountably many as it happens) that do contain infinitely many 0s AND 1s, if this is to be all of the power set. There is none in the enumerated part of your 'matrix'Now, are you going to explain the construction of the 'matrix', so far you've not offered any way of constrcuting it.

 

[Organic]

Hurkyl,

Please this time pay attention to the word "each".

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that in ths case (which is not my case, where both forms exist) each row can be reduced to ...1010101010 or ...0101010101 data.

 

[Organic]

Matt,

I do not state that every row must have infinitely many 0s and 1s. But that there must be some rows (uncountably many as it happens) that do contain infinitely many 0s AND 1s, if this is to be all of the power set. There is none in the enumerated part of your 'matrix'

Both cases must exist in my 01 matrix.

Simply there is no other way.

 

[matt grime]

Originally posted by Organic
Hurkyl,

Please this time pay attention to the word "each".

You make a big conceptual mistake is you think thet there
must be infinitely many 0 AND 1 notations in each row, because in this case no one of these rows has its unique data.

It means that each row can be reduced to ...1010101010 or ...0101010101 data.

No one is saying (apart from you here) that there must be infinitely many 0s and infinitely many 1s in every row, indeed you don't produce a single row where that is true.

what on Earth does it mean to 'reduce' a row to ..0101010 or ..1010101?

no one of these rows has its unique data? eh? That looks like a meaningless sentence.

 

[Hurkyl]

"each" might be the only word in that sentence that makes sense! (Ok I'm exaggerating, but only slightly)


Does ...011011011 appear in your list?
Does ...100001000100101 appear in your list?
Does ...01010101 appear in your list?
Do you consider these three sequences different?

 

[matt grime]

Originally posted by Organic
Matt,

Both cases must exist in my 01 matrix.

Simply there is no other way.

Why *must* there be all of the strings with infinitely many 1s and infinitely many 0s in them? Why? Give me one reason. If you are about to say 'by cosntruction' which construction, you've not offered one? SO which construction - just tell me the r'th entry in the t'th column, that's all, even if it's by some inexplicit way like my comment that it appears to be

the r'th column is the sequence of alternating blocks of 2^r 0s then 2^r 1s, you may then read off the rows.

It's a simple request. Note you cannot say by the axiom of infinity induction on the power value' because that is not acceptable; oh you're about to cry foul and say I'm restricting you, but as you've never explained what the axiom of infinity of induction is you can't use it. Induction tells us something is true for an infinite number of cases, it does not tell us what, if anything, is true for an infinite set in the index if that even makes sense. For instance one can easily define the n'th fibonacci number and get an explicit value for it by induction. what would it mean to even talk of the aleph-0'th fibonacci number?

 

[Organic]

Matt,

In this private case of 01 matrix we can distinguish between to cases of uniquness.

Case 1: there are infinitely many 0 AND 1 notations in each row with a unique order of 01 combinations.

Csae 2: there are finitely many 0 XOR 1 notations in each row with a unique order of 01 combinations.

Only case 1 can be reduced to ..0101010 or ..1010101 where by reduced I mean that we don't care about the order of 01 combinations.

 

[matt grime]

Originally posted by Organic
Matt,

In this private case of 01 matrix we can distinguish between to cases of uniquness.

Case 1: there are infinitely many 0 AND 1 notations in each row with a unique order of 01 combinations.

Csae 2: there are finitely many 0 OR 1 notations in each row with a unique order of 01 combinations.

Only case 1 can be reduced to ..0101010 or ..1010101 where by reduced I mean that we don't care about the order of 01 combinations.

Firstly, ..010101 is not even on the list, secondly why can't you reduce both those to the same thing, thirdly, rubbish, of course you can't do that as then you don't have a bijection (do you even know what a bijection is?), and fouthly why aren't you answering the calls to write down the construction of the enumeration of the matrices rows? It shouldn't be very hard. Fifthly stop saying private when you mean particular, they are not synonyms.

 

[Organic]

Matt,

If the word induction is wrong here then we can use the word iteration instead.

 

[matt grime]

Originally posted by Organic
Matt,

If the word induction is wrong here then we can use the word iteration instead.

Why would that help? If you've made this construction surely you know how you made it? How come you get to be so sure of yourself and that Hurkyl and I are wrong and don't understand when you don't even know the meaning of the terms you use?

 

[Organic]

Matt,

..010101 is not even on the list, secondly why can't you reduce both those to the same thing, thirdly, rubbish, of course you can't do that as then you ..

Because I have the right side of my matrix then case 1 can be reduced to ...0101010 or ...1010101

 

[matt grime]

Originally posted by Organic
Matt,

Because I have the right side of my matrix then case 1 can be reduced to ...0101010 or ...1010101

right side of the matrix? what? What's case 1. why won't you answer the relevant questions and produce the alleged construction of the enumerated list. stop obfuscating and produce the proof of the result that *must* state why the enumerated list is in bijection with the power set, and why you are allowed to identify different elements of the power set when you are looking for a bijection!

Are you implying that the alternating sequence ...010101010 is on the right hand of the two lists you're now using? despite the fact that every row in the right hand list is eventually one 'by construction' that is the cofinite list. of course for a while you were insisting that it was on the left list too cos that was a complete enumeration of the power set as well.

 

[matt grime]

Originally posted by Organic
Matt,

Is it beyond you ability to understand that my matrix power_value = |N|?

Would it be beyond you to define what that sentence means? Apparently, yes.

You state the case for 2^n, then make a statement about 2^|N|. Fine but you cannot conclude anything about that case by induction or iteration from the finite case, not that you even bother to do that. In particular you cannot conclude that the information can be written encoded in a matrix with countable sets of rows and columns. One does not logically follow from the other (it is false, we have proved it).

 

[Organic]

Matt,

Why would that help? If you've made this construction surely you know how you made it? How come you get to be so sure of yourself and that Hurkyl and I are wrong and don't understand when you don't even know the meaning of the terms you use?

Is it beyond your ability to understand that my matrix power_value = |N|?

It is very easy because there is a bijection between the power_values and N members.

 

[matt grime]

So there is a bijection from the set

{2^n | n in N} and N.

So?

Or do you mean there is a bijection from a set of cardinality 2^|N| and N? Well, there isn't. Check the above post where I ask you if it is beyond you te even define what it means ot have power value |N|

so for the god knows how manyth time, write down the construction that enables you to write these matrices out and make the claims you do.

 

[Organic]

Matt,

Maybe this will help you:

Please tell me if you understand the set that I wrote below

      3 2 1 0
     2 2 2 2
     ^ ^ ^ ^
     | | | |
     v v v v
[b]{[/b]...,1,1,1,1[b]}[/b] 
 ...,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  
 ...

It is also can be written as: {10... ,1100... ,11110000... ,...}

 

[matt grime]

That does not help because you do not explain what the ellipsis of the ... means.here is what I think it means (again, for pity's sake, why do you refuse to answer simple direct questions and observations)

we start with the two diagrams in your newdiagonal pdf that you eunmerate, the left hand one

the columns go
..0000
..0001
..0010
..0011
..0100
..0101
...
...
...
...where the first column is based on repeatin the pattern 01, the second by repeating the pattern 0011 the rth by repeating 2^r 0s then 2^r 1s (each pattern repeating infinitely many times in the column, so that you cannot claim I think it's finite, and for r goes from 1 to infinity)

now the second diagram is the first but every entry above the diagonal is a 1you now interleave these rows alternating one from each.Is that accurate?

 

[Organic]

Matt,

It is also can be written as: {10... ,1100... ,11110000... ,...}

... means that each 0 1 starting repeats on itself forever.

 

[matt grime]

But is my characterization equivalent to yours?

 

[Hurkyl]

I think he's looking at the set of columns, and assuming it's obvious that when he writes "..." he means "repeat that string of digits indefinitely"

 

[matt grime]

Originally posted by Organic
Matt,

It is also can be written as: {10... ,1100... ,11110000... ,...}

... means that each 0 1 starting repeats on itself forever.

the array you wrote out - clearly it satisfies every row is eventually 1 continually (ie every entry in row r is 1 after point reading right to left - the point changing as r changes) therefore it corresponds to sets whose complement is finite. The cofinite sets again. now what about them.

 

[matt grime]

I don't understand it. I've proved one diagram corresponds only to the finite subsets, another corresponds only to the cofinite subsets, their union is not the power set, that organic doesn't know the meaning of the terms he uses, as proven by the fact that he thinks it's ok for a bijection not to be injective. what more do i need to do?