Deeper than primes - Continuation

[jsfisher]

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) ?

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.

{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.

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?

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?

 

[doronshadmi]

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.

Thank you for this explanation.

A series is the sum of a sequence of values. It is not, itself, a sequence.

But this sum of a sequence of values is the result of at most |N| values.

Reached? You write of a series as if it were a sequence of operations. It is not. It is complete.

It is |N| complete.

That aside, though, what sequence of cardinality |R| do you mean?

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.

 

[jsfisher]

I mean that the series 0.9+0.09+0.009+0.0009+... = 0.9999... actually = 1 only if it at least |R| complete.

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.

 

[doronshadmi]

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.

Only if the sum over that sequence is at least |R| complete.

The cardinality of the continuum comes into this nowhere.

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.

 

[jsfisher]

Only if the sum over that sequence is at least |R| complete.

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?

 

[doronshadmi]

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?

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.

 

[jsfisher]

Exactly by 0.000...1

Ok, I see: You are back to meaningless nonsense. You are on your own.

 

[doronshadmi]

Ok, I see: You are back to meaningless nonsense.

jsfisher, please explain to me why

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

is a meaningless nonsense?

Thank you.

 

[realpaladin]

jsfisher, please explain to me why

is a meaningless nonsense?

Thank you.

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.

 

[doronshadmi]

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.

 

[jsfisher]

Ok, here is my summary of the last posts between jsfisher and me:

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.

 

[doronshadmi]

Don't include me in that summary.

That is why I wrote "my summery".

You've focused on the concepts you added all on your own without any foundation.

It has the same foundation that enables to conclude that |N|<|R|.

"|N| complete" and "|R| complete" are yours and yours alone,

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.

and the related "completeness" requirement for 0.999... to be exactly 1 is pure supposition.

and the related "non-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.

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).

Doron, to show that 0.999... and 1 are different, you must show how they behave differently.

Please define "behave differently" (or at least provide some useful example).

Simply saying they are different because of some difference you assert but cannot show doesn't accomplish anything.

It is not an assertion exactly as the difference between |N| and |R| is not an assertion according to actual mathematics.

 

[jsfisher]

...
Please define "behave differently" (or at least provide some useful example).

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).

 

[doronshadmi]

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.

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).

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

Without such an invention\discovery in actual mathematics |N|,|R|, etc. are undefined.

The most straightforward approach might involve showing how |1 - 0.999...| acts differently than 0.

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).

Personally, I kind of favor the sum( |1 - 0.999...|, i = 1 to infinity).

Please define the accurate size of infinity (there are infinitely many of them, so which one is it?)

 

[doronshadmi]

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).

 

[jsfisher]

0.999... - 0.999... = 0

1 - 1 = 0

1 - 0.999... = 0.000...1

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).

(This is your own private invention. It is not Mathematics.)

 

[doronshadmi]

(This is your own private invention. It is not Mathematics.)

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=10301106&postcount=4215.

 

[jsfisher]

(This is your own private invention. It is not Mathematics.)

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=10301106&postcount=4215.

Yes, that would an example of a post confirming my statement.

 

[realpaladin]

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).

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*

 

[doronshadmi]

Yes, that would an example of a post confirming my statement.

Did you try to do this thought experiment?