Deeper than primes - Continuation

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doronshadmi said:
jsfisher, is the abstract concept length depends on the abstract concept number, but not vice versa (which means that there is an hierarchy of dependency, where number is more fundamental than length) ?
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Numbers are not required for length, just like it is not required for cardinality. You may recall that Cantor's Theorem deals with relative cardinality with a greater-than relationship. No numbers. Length can be treated the same way in the appropriate setting.

doronshadmi said:
{0.9, 0.09, 0.009, 0.0009, ...} is a one abstract mathematical object that has |N| members.

The series 0.9+0.09+0.009+0.0009+ ... is a one abstract mathematical object of convergent sequence of |N| values.
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A series is the sum of a sequence of values. It is not, itself, a sequence.

My question is this:

How a one abstract mathematical object of convergent sequence of |N| values is equal to a given value on the real line (known as the limit of that convergent sequence of |N| values) if this given value actually can be reached only by a convergent sequence of at least |R| values?
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Reached? You write of a series as if it were a sequence of operations. It is not. It is complete. That aside, though, what sequence of cardinality |R| do you mean?
 
jsfisher said:
Numbers are not required for length, just like it is not required for cardinality. You may recall that Cantor's Theorem deals with relative cardinality with a greater-than relationship. No numbers. Length can be treated the same way in the appropriate setting.
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Thank you for this explanation.


jsfisher said:
A series is the sum of a sequence of values. It is not, itself, a sequence.
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But this sum of a sequence of values is the result of at most |N| values.


jsfisher said:
Reached? You write of a series as if it were a sequence of operations. It is not. It is complete.
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It is |N| complete.

jsfisher said:
That aside, though, what sequence of cardinality |R| do you mean?
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I mean that the series 0.9+0.09+0.009+0.0009+... = 0.9999... actually = 1 only if it is at least |R| complete.
 
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doronshadmi said:
I mean that the series 0.9+0.09+0.009+0.0009+... = 0.9999... actually = 1 only if it at least |R| complete.
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Why do you say that?

The cardinality of the sequence <0.9, 0.09, 0.009, ...> is |N|, and the sum over that sequence is 1.

The cardinality of the continuum comes into this nowhere.
 
jsfisher said:
Why do you say that?

The cardinality of the sequence <0.9, 0.09, 0.009, ...> is |N|, and the sum over that sequence is 1.
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Only if the sum over that sequence is at least |R| complete.

jsfisher said:
The cardinality of the continuum comes into this nowhere.
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You are right. This is exactly the reason of why the series 0.9+0.09+0.009+0.0009+... = 0.9999... (which is at most |N| complete) actually < value 1.
 
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doronshadmi said:
Only if the sum over that sequence is at least |R| complete.
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Repeating the same comment doesn't help. Why are you saying this? The number expressed as 0.999... is exactly equal to 1.

If they be different values, as you seem to be claiming (again), then there must be some way to distinguish the two. If they be different, then of necessity there must be a difference. How are they different?
 
jsfisher said:
Repeating the same comment doesn't help. Why are you saying this? The number expressed as 0.999... is exactly equal to 1.

If they be different values, as you seem to be claiming (again), then there must be some way to distinguish the two. If they be different, then of necessity there must be a difference. How are they different?
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Exactly by 0.000...1

.000... is at most |N| complete where ...1 is the complement to value 1, where this complement is at least |R| complete.
 
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doronshadmi said:
jsfisher, please explain to me why

is a meaningless nonsense?

Thank you.
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Because ... in your example denotes infinity. There is NO ...1 because that would signify a FINAL digit.

There is no final digit. the sequence 0.999... never has a last digit ...9

I explained this using the concept of limit some years ago. In this thread. Look it up.
 
Ok, here is my summary of the last posts between jsfisher and me:

Without loss of generality, the sum 0.9+0.09+0.009+ 0.0009+… of the sequence of values <0.9, 0.09, 0.009, 0.0009 …> is the result of at most |N| values, which means that 0.9+0.09+0.009+0.0009+… is at most |N| complete.

The series 0.9+0.09+0.009+0.0009+... = 0.9999... actually = value 1 only if it is at least |R| complete.

One asks: "In that case, what is the difference between 0.9999... and value 1?

My answer:

It is exactly 0.000…1, as follows:

.000... is at most |N| complete, where ...1 is the complement to value 1, where this complement is at least |R| complete.

This complement (notated here as ...1) is the all R members along the real line that are beyond the range of (in this particular example, which is without loss of generality) the series 0.9+0.09+0.009+0.0009+... = 0.9999... (because it is at most |N| complete), where 1 it the largest value of that inaccessible range of R members.
 
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doronshadmi said:
Ok, here is my summary of the last posts between jsfisher and me:
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Don't include me in that summary.

You've focused on the concepts you added all on your own without any foundation. "|N| complete" and "|R| complete" are yours and yours alone, and the related "completeness" requirement for 0.999... to be exactly 1 is pure supposition. And then there is "0.000...1" -- notation that pretends to have meaning, yet doesn't.

Doron, to show that 0.999... and 1 are different, you must show how they behave differently. Simply saying they are different because of some difference you assert but cannot show doesn't accomplish anything.
 
jsfisher said:
Don't include me in that summary.
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That is why I wrote "my summery".

jsfisher said:
You've focused on the concepts you added all on your own without any foundation.
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It has the same foundation that enables to conclude that |N|<|R|.

jsfisher said:
"|N| complete" and "|R| complete" are yours and yours alone,
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Again it is what you call actual mathematics, which according to it infinite collections have accurate (or complete) sizes that are strictly different of each other (for example: |N|<|P(N)|=|R|<|P(P(N))|< ... etc. ad infinitum.

jsfisher said:
and the related "completeness" requirement for 0.999... to be exactly 1 is pure supposition.
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and the related "non-completeness" requirement for 0.999... to be exactly 1 is pure supposition.

jsfisher said:
And then there is "0.000...1" -- notation that pretends to have meaning, yet doesn't.
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And then there is "0.999...=1" -- notation that pretends to have meaning, yet doesn't, exactly because the series 0.9+0.09+0.009+0.0009+... = 0.9999... actually = value 1 only if it is at least |R| complete, but actually it doesn't (it is at most |N| complete).

jsfisher said:
Doron, to show that 0.999... and 1 are different, you must show how they behave differently.
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Please define "behave differently" (or at least provide some useful example).

jsfisher said:
Simply saying they are different because of some difference you assert but cannot show doesn't accomplish anything.
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It is not an assertion exactly as the difference between |N| and |R| is not an assertion according to actual mathematics.
 
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doronshadmi said:
...
Please define "behave differently" (or at least provide some useful example).
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If 0.999... and 1 be difference values, then under some sort of mathematical manipulation, 0.999... and 1 will end of at demonstrably different results.

Inventions such as those involving "completeness" do not qualify.

The most straightforward approach might involve showing how |1 - 0.999...| acts differently than 0. Personally, I kind of favor the sum( |1 - 0.999...|, i = 1 to infinity).
 
jsfisher said:
If 0.999... and 1 be difference values, then under some sort of mathematical manipulation, 0.999... and 1 will end of at demonstrably different results.
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0.999... - 0.999... = 0

1 - 1 = 0

1 - 0.999... = 0.000...1 (where .000... is at |N| size and ...1 is at |R| size).

jsfisher said:
Inventions such as those involving "completeness" do not qualify.
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Without such an invention\discovery in actual mathematics |N|,|R|, etc. are undefined.

jsfisher said:
The most straightforward approach might involve showing how |1 - 0.999...| acts differently than 0.
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0.000...1 acts differently than 0 (where in this value .000... is used as a |N| size place kipper that is inaccessible to ...1 that is at |R| size).

jsfisher said:
Personally, I kind of favor the sum( |1 - 0.999...|, i = 1 to infinity).
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Please define the accurate size of infinity (there are infinitely many of them, so which one is it?)
 
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Maybe this thought experiment can be used as a draft for rigorous formal expression (where "formal expression" is used here for the rigorous linkage among notions and notations):

Let's say that you are value 1 along the real line.

If you look at some value which is not you in terms of finite glasses, it is clear to you that it can reach you by finitely many steps (you reach yourself by 0 steps).

If you look at some value which is not you in terms of |N| glasses, it is clear to you that it can't reach you by finitely many steps, but it can reach you by |N| steps.

jsfisher, if you are looking at the real line by using |N| glasses, then 0.999... = 1 (you).

Now please look at the real line by using |R| glasses.

EDIT:

In this case finite or |N| steps can't reach you, and form this point of view 0.999... (which is an |N| size value) can't reach you (and this is the meaning of ...1 in the value 0.000...1, where 0.000...1 is the result of looking at |N| size by using |R| glasses).

If you are looking at the real line by using |R| glasses, then some stronger method then the place value is need in order to reach you by |R| steps, but it does not mean that the place value is not a legitimate value of its own, on the real line (exactly as you are still looking at finite values as legitimate values of their own, on the real line).
 
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doronshadmi said:
0.999... - 0.999... = 0

1 - 1 = 0

1 - 0.999... = 0.000...1
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That's a side-step, not forward progress. You have moved the problem to having to show that 1 - 0.999... is different from 0. Merely asserting a difference by presenting a meaningless notation does not do that.

(where .000... is at |N| size and ...1 is at |R| size).
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(This is your own private invention. It is not Mathematics.)
 
doronshadmi said:
0.999... - 0.999... = 0

1 - 1 = 0

1 - 0.999... = 0.000...1 (where .000... is at |N| size and ...1 is at |R| size).
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There *is* no ...1

At no point, ever, anywhere, whatever size. Infinity means infinity, not 'unimaginably far away'. It simply means 'forever, without ending ever'.

...1 is a figment of an imagination.

Everything based on that premise is *WRONG*
 
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doronshadmi said:
Did you try to do this thought experiment?
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Doron, your thought experiment is a silly, poorly expressed farce. I can "get" from any point, a, on the real number line to any other point, b, on the real number line in one "step". The size of that step is |a - b|.
 
jsfisher said:
Doron, your thought experiment is a silly, poorly expressed farce. I can "get" from any point, a, on the real number line to any other point, b, on the real number line in one "step". The size of that step is |a - b|.
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This is only the first step of my suggested thought experiment, please take all of its steps.
 
doronshadmi said:
In this case finite or |N| steps can't reach you, and form this point of view 0.999... (which is an |N| size value) can't reach you (and this is the meaning of ...1 in the value 0.000...1, where 0.000...1 is the result of looking at |N| size by using |R| glasses).

If you are looking at the real line by using |R| glasses, then some stronger method then the place value is need in order to reach you by |R| steps, but it does not mean that the place value is not a legitimate value of its own, on the real line (exactly as you are still looking at finite values as legitimate values of their own, on the real line).
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Again, you are ignoring what limit means in Mathematics.

There is no ...1 because there is no final member. The thought experiment is invalid because it is based on a *WRONG* premisse.
 
doronshadmi said:
This is only the first step of my suggested thought experiment, please take all of its steps.
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Which part of "silly, poorly expressed farce" was unclear?

I can always get from one point to another in one step. Your bare assertion that something you call "|N| glasses" or "|R| glasses", whatever those are supposed to be--you failed to define your terms, does not change that fundamental, only one step necessary, fact.
 
jsfisher said:
What has the cardinality of the rational numbers to do with a single point along the real number line?
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jsfisher said:
... applies to single points along the real number line ...
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What you wrote is not about a single point, and so is the case of my experiment.

It is not about a single point, but about the number of steps that are needed in order to reach to a given point.

You are using only the finite glasses of my experiment, so please try the other glasses.
 
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doronshadmi said:
What you wrote is not about a single point, and so is the case of my experiment.
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No, what I wrote was very much about single points. Any single point at all.

It is not about a single point, but about the number of steps that are needed in order to reach to a given point.
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Any point can be "reached" in one step. If you have some non-standard meaning for "step", then you need to make that clear.

You are using only the finite glasses of my experiment, so please try the other glasses.
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...and you need to make clear what you mean by "glasses".
 
jsfisher said:
If you have some non-standard meaning for "step", then you need to make that clear.
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The sum 0.9+0.09+0.009+0.0009+… of the sequence of values <0.9, 0.09, 0.009, 0.0009 …> is the result of at most |N| values.

To each value I call step, and there are finite, |N| or |R| steps that are calculated in parallel.

jsfisher said:
...and you need to make clear what you mean by "glasses".
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This is the tool that enables you to know the number of the needed parallel steps in a given series.

Since the steps are done in parallel, they define a single result for a given series.

I am defiantly not talking about step-by-step.
 
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doronshadmi said:
The sum 0.9+0.09+0.009+0.0009+… of the sequence of values <0.9, 0.09, 0.009, 0.0009 …> is the result of at most |N| values.

To each value I call step, and there are finite, |N| or |R| steps that are calculated in parallel.
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No, 0.999... has no steps. It is a value, not the end of an infinite sequence of partial sums (which doesn't have an end).
 
jsfisher said:
No, 0.999... has no steps. It is a value, not the end of an infinite sequence of partial sums (which doesn't have an end).
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You are still think in terms of step-by-step.

Exactly as {1,2,3,4,5,...} is not taken step-by-step, so is the case about 0.999...
 
jsfisher said:
Let me know when you stop contradicting yourself.
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Let me know when you are able to think in terms of a single parallel steps, which identical to a single value along the real line.

0.999... is a single value along the real line in terms of |N| parallel steps if |R| glasses are used, and 0.000...1 is a single value along the real line in terms of |R| parallel steps if |R| if glasses are used.
 
jsfisher said:
No, 0.999... has no steps. It is a value, not the end of an infinite sequence of partial sums (which doesn't have an end).
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0.999... is the parallel sum (also called series) of the convergent |N| values <0.9, 0.09, 0.009, ...>. I define step as identical to a given value of that parallel sum. Since the sum is done in parallel, no process of any kind is involved here (the illusion of "process" is the result of a step-by-step observation, which is avoided by using parallel observation).

In order to actually do my suggested thought experiment, concepts like step, series or sequence must be taken by parallel observation, otherwise, one misses the thought experiment as given in http://www.internationalskeptics.com/forums/showpost.php?p=10301106&postcount=4215.
 
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jsfisher said:
Doron, your thought experiment is a silly, poorly expressed farce. I can "get" from any point, a, on the real number line to any other point, b, on the real number line in one "step". The size of that step is |a - b|.
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You are using here the term "size" in order to define a given length, but we are not talking here about length, but about a given value along the real-line.

After all, you are the one who wrote in http://www.internationalskeptics.com/forums/showpost.php?p=10297296&postcount=4196:
jsfisher said:
[0, 1/2] and 1/2 are completely different things.
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A given value can be the parallel sum of finitely or infinitely many values, where the term size, as I use it here, is the number of values (by omitting value 0) that are used in a given parallel sum, in order to be, to get, to reach, etc. a given value along the real-line.

Here is an example of a finite size:

value 1 = finite number of parallel sum (called by me the finite glasses) of value 1 + value 0 = value (-1) + value 1.9 + value 0.1 = ... etc. ...


Here is an example of infinite size |N| (the |N| glasses):

The series (the |N| parallel sum) value 0.9 + value 0.09 + value 0.009 +... = value 0.999... = value 1


Here is an example of infinite size |R| (the |R| glasses):

The series (the |N| parallel sum) value 0.9 + value 0.09 + value 0.009 +... = value 0.999... < value 1 by value 0.000...1

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|N| or |R| are not values along the real-line.
 
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