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

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give the method of construction of your infinite list.

how do you enumerate the t'th row?

it looks as though

you send the string (x_0,x_1,x_2...

with the ordering going backwards from the article's as you read right to left there

to the expansion [tex](\sum x_i2^i) -1[/tex]

is that what you're doing?
 
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Hurkyl,

until you've completely specified your list L.
This is what so beautiful in fractals, the quantity does not matter but the invariant self similarity over scales.

Any form of a^b is a fractal.

My list has a^b form, therefore it is satisfied by its own self similarity (again quantity does not matter).
And, incidentally, for your latest attempt, I choose the binary sequence:
...010101
Contors diagonal is an aleph0 width of my list, therefore ...010101 is already in the list, which its length is 2^aleph0.
 
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My list has a width of arithmatic row of (if base 2) 0,1 notations,
and a length of geometric column of 0,1 notations.

Width and length are non-finite.

but i have disproved your attempted enumerable construction of it.
You disproved nothing.
 
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so let's suppose 01010101... (the even numbers in N) is on the enumerated portion of the list.it must be at some point r for some r in N, say: the r'th row.r cannot be one of the rows you've now put in with all 1's eventually, so it must be one of the other ones. look how they're constructed.. oh, see that every entry is eventually zero so 010101... can't be on of the enumerated rows.so we can just consider the diagram without the infinite row of ones, and go back to the first diagram you have.

ever column after the rth has at least r zeros at the beginning and therefore cannot contribute a 1 to the r'th row in this diagram.

brick and wall spring to minddo you even accept that given a string of 0s and 1s and that if the list is enumerable we can just consider the row it corresponds to?
 
How do you deal with the fact that if the real numbers are assumed to be a countable set, that I can prove their total length is less than 1/2?


Contors diagonal is an aleph0 width of my list, therefore ...010101 is already in the list, which its length is 2^aleph0.

What is the number of the row which contains it?

(Incidentally, due to the way you wrote it, I don't even know what is the number of the row that contains the all ones sequence!)
 
Originally posted by Organic
My list has a width of arithmatic row of (if base 2) 0,1 notations,
and a length of geometric column of 0,1 notations.

Width and length are non-finite.You disproved nothing.

show me where the proof goes wrong? quote it word for word and point out exactly where the proof is wrong for the list as defined in your article.

Better yet in this forum explicitly define how the list is formed?I cannot offer to show where your proof that the list is both countable and contains all the elements of the power set goes wrong because you do not prove that it eumerates all of the power set, just state it must.Come on, quote that proof, and explain the flaw you think there is in it. simple, couldn't be simpler?
 
Hurkyl,
What is the number of the row which contains it?
First Please give me the number (the index if you like) of each prime number.


Please explain me again without professional notations the argument about 1/2.

Thank you.
 
,
Come on, quote that proof, and explain the flaw you think there is in it. simple, couldn't be simpler?
Please understand this, the invariant self-similarity of a fractal does not depend on quantity, which means that any part of it (its local level) is equal to "all" of it (the global level of it).

So in a fractal finite and infinite are satisfied by the invariant structural property, and quantitative property is not important.

This is the deep meaning of cardinality in a fractal, not its size but its invariant self similarity.

If you understand this then you have a gate to my world.
 
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This thread is sooooo funny!

If I didn't know better, I'd say that Matt and Hurkyl take turns pretending to be Organic and posting nonsence just for fun.
 
suyver,

You are invited to add your joke (please say some funny thing on fractal-like nanocrystals).
 
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and those counter claims state what? apart from that you don't seem to able to either understand the concept of proof or a simple request to demonstrate where you think the proof offered to you goes wrong?here is the proof based upon the array as written in your article the first column is on the right, then going to the left we number the columns accordingly.

the first column is 'based on 2^1' and goes 0101010101...

the second colummn reads, downwards, 00110011...

and so on for each column.

the r^th column starts with r zeroes.
then r ones then r zeroes...
Yes?

Let t be any row, the entry in the s'th place (reading right to left) is the entry from the s'th column.

whenever s >t this entry must be zero.

therefore any row is eventually all zeroes, and every element you enumerate in the power set is finite.

Read that argument carefully and don't dismiss it simply because it is me. as you keep telling people they must be open to new ideas, well, so must you.edit: also recall the worked examples i gave for the case of the first to 5th rows

so go through it and at each point tell me what you think is wrong with the deduction there.
you've now interleaved the confinite sets, but it's trivial to show that they are countable and contain only sets whose complement is finite. these two observations combined demonstrate your enumerated list of elements of the power sets contains none of the uncountable number of subsets of N which are infinite and whose complement is infinite.so quote all this and go through this step by step and write at each stage whether you agree or disagree, and if you disagree, why.you do not need to cite fractals or prabability, these have no bearing on this result. And just saying 'but they do, and you're wrong' only demonstrates your frailty of position.and i wish you were right suyver - i don't know why i spend so much intellectual energy on this topic.

a goood thing to note is that organic never posts in other people's threads unless it is to tell them about how his theory applies there. surely if he cared and knew mathematics or physics he'd want to help other people too and answer their queries about maths?
 
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Originally posted by Organic
suyver,

You are invited to add your joke (please say some funny thing on fractal-like nanocrystals).

pass.
 
Originally posted by Organic
Matt,

This is useless, you are one of the prisoners in "Plato's Allegory of the Cave" http://faculty.washington.edu/smcohen/320/cave.htm .

And the name of your jailer is Cantor.

if the strength of your position is so evident, and you are this genius that you seem to purport to be why can you not respond to the simple request to go through the proof offered step by step and explain your thoughts about it.failing that present an explicit construction of the array you imply exists, and demonstrate that every element of the power set is enumerated.

define the bijection between the rows and N. I have offered you an example and you've ignored it.

explain why it is that you make these claims about the case for N based upon the cases for the finite sets without citing the axiom of infinity induction.

you are attempting to prove that mathematics is inconsistent, and thus you must do so from within mathematics, that is why i do not need to consider your assertions about fractals and probability. they are irrelevant to the discussion in hand as you've not proved that there is a problem within mathematical treatments of N that don't deal with them. moreoever, you have not defined fractals or probability without using the mathematics that you consider to be flawed.
 
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”.
 
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i didn't think you liked allegories.ok, put in front of me the array you claim has countably many rows that enumerate the power set of N.

Explain the rule for generating it. You say it must have a certain property by construction, what is the construction?

explain how you are bijecting to N, explain why then you manage to ignore the fact that that implies it does not enumerate the power set. remember you are only allowed to use my mathematics because that is what you are claiming is incomplete and cannot handle infinity.

Putting a cat in front of me doesn't define a cat - it gives me an example of a cat. If it were an abyssinian, would I then have to only accept that abyssinians were cats, and that, say, a siamese weren't a cat because it doesn't look like an abysinian? Now take the proof of mine and carefully go through it and explain where it is wrong.

i've done that for you - you're argument is wrong because you're basing *assumptions* about it on the finite case, that are meaningless in the infinite.
 
Matt,

You don't know what is infinity, no more no less.


When you know it then and only then we shall communicate.


Bye.
 
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so you can't find a flaw in the proof then?

or can't you explain how to generate this infinite array?

which of all the observations on the inadequacy of your mathematics is causing you the most concern?

or all of the above?
 
And the jury declares Matt Grime the winner!
The crowd goes wild!
 
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.
 
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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.
 
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!
 
Hi Deeviant,


unlike the cat, your position wasn't a living breathing thing cute furry thing before the process.
Why not?
 
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
 
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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.
 
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.
 
I agree that |R|>|N| but both are countable because R collection cannot be a solid line.
 
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.
 
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.
 
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