Deeper than primes - Continuation

realpaladin · Oct 31, 2014

doronshadmi said:
<snip>
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

--------

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

Again, this uses sleight of hand to add a convenient 'final digit' on an *in*finite series.

Since that, by definition is *WRONG*, Doron Shadmi is *WRONG*

There is no need to heed any further information divulged by him as he demonstrably has got the basics *WRONG*

As long as Doron needs linguistical mangling for his "work", no grade school mathematician need fear him...

doronshadmi · Oct 31, 2014

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

By understanding the concept of "size" as used in http://www.internationalskeptics.com/forums/showpost.php?p=10302577&postcount=4240 and by using |R| size as an observation's view of the real-line (where this view is done beyond any one of the values along the real-line), the value 0.000...1 is the complement of value 0.999... to value 1 (where 0.000...1, 0.999... and 1 are values along the real-line), the value 0.000...1 acts differently than value 0, as follows: the ".000..." is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

Again, 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, where 0.000...1, 0.999... and 1 are values along the real-line.

jsfisher · Oct 31, 2014

doronshadmi said:
0.000...1 acts differently than 0...

Assertions don't make something true.

realpaladin · Oct 31, 2014

doronshadmi said:
By understanding the concept of "size" as used in http://www.internationalskeptics.com/forums/showpost.php?p=10302577&postcount=4240 and by using |R| size as an observation's view of the real-line (where this view is done beyond any one of the values along the real-line), the value 0.000...1, which is the complement of value 0.999... to value 1 (where 0.000...1, 0.999... and 1 are values along the real-line), the value 0.000...1 acts differently than value 0, as follows: the ".000..." is used as a |N| size place value kipper that is inaccessible to "...1" that is at |R| size.

Again

Place value 'keeper'... that is algorithmic talk.

Show me an algorithm that ever arrives at the ...1 and I´ll smoke you a kipper for breakfast.

If you can not show an algorithm (let alone a mathematical proof) then this is just another wind egg.

doronshadmi · Oct 31, 2014

jsfisher said:
Assertions don't make something true.

Using the fact that |N|<|R| and by using |R| size as an observation's view of the real-line (where |N| or |R| are not values along the real-line), is not an assertion, but it is a rigorous observation of the real-line as taken from |R| size, where size (as used here) is:

The number of values that are used in a given parallel sum, in order to be, to get, to reach, etc. a given value along the real-line.

(This definition is a correction of the one that was given in http://www.internationalskeptics.com/forums/showpost.php?p=10302577&postcount=4240).

So size here has nothing to do with length, as you used it in the quote in http://www.internationalskeptics.com/forums/showpost.php?p=10302577&postcount=4240.

doronshadmi · Oct 31, 2014

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|.

Let's use what you have wrote here, not in terms of length, but in terms of values along the real-line.

The definition of size, as used by me here is:

The number of values that are used in a given parallel sum, in order to be, to get, to reach, etc. a given value along the real-line.

So, by using a finite number of values that are used in a given parallel sum, in order to be, to get, to reach, etc. a given value along the real-line, you "get" value b.

Here are examples of infinite sizes:

Here is an example of infinite size |N| (the |N| observation of the real-line):

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| observation of the real-line):

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

By using my definition of size and by using |R| size as an observation's view of the real-line (where this view is done beyond any one of the values along the real-line), the value 0.000...1 is the complement of value 0.999... to value 1 (where 0.000...1, 0.999... and 1 are values along the real-line).

The value 0.000...1 acts differently than value 0, as follows:

The ".000..." is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

--------

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

Moreover, since |R| size is uncountable, it can't be expressed by a string of notations, as used, for example, by the place value method (the best that can be done is, for example, of the form "...1").

realpaladin · Oct 31, 2014

doronshadmi said:
The ".000..." is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

You are not supposed to be reading my posts Doron...

This means you avoid confronting anything that you can not kibitz about.

Admitting defeat can also be done by ignoring someone

realpaladin · Oct 31, 2014

doronshadmi said:
The value 0.000...1 acts differently than value 0, as follows:

The ".000..." is used as a |N| size place value keeper that is inaccessible to "...1" that is at |R| size.

Let me spell it out:

Which. Steps. Lead. To. The. End. Of. Infinity. So. It. Finally. Reaches. ...1 ?

If you do not have steps, be they formulaic or be they algorithmic, you only have assertions.

Assertions which are *WRONG* because, as I explained to you years ago, infinity is 'without end' or 'not finite' or 'for each position you have, I can tack another one to it' or 'there is no final digit'.

jsfisher · Nov 1, 2014

doronshadmi said:
Let's use what you have wrote here, not in terms of length, but in terms of values along the real-line.

Then you won't be using what I wrote. You will be using something completely different. Something you, as you are wont to do, made up, fabricated, pulled out of the air.

doronshadmi · Nov 1, 2014

jsfisher said:
Then you won't be using what I wrote.

Are you claiming that a, b or |a - b| are not values along the real-line?

If not, then please define what are a, b or |a - b|?

Thank you.

jsfisher · Nov 1, 2014

doronshadmi said:
Are you claiming that a, b or |a - b| are not values along the real-line?

If not, then please define what are a, b or |a - b|?

Thank you.

Just as I wrote, a and b were distinct points along the real number line, and |a - b| was the distance between them.

doronshadmi · Nov 1, 2014

jsfisher said:
Just as I wrote, a and b were distinct points along the real number line, and |a - b| was the distance between them.

EDIT:

Please pay attention that |a - b| can't be defined as a given point along the real-line, but it can be defined as a value along the real-line, and my purpose was the define a, b, or |a - b| by using the same concept (value, in this case).

After all you are the one that (for purpose of generalization) used the term value, in order to define objects along the real-line, for example:

jsfisher said:
1/2 does not have length. It is a value.

(http://www.internationalskeptics.com/forums/showpost.php?p=10297296&postcount=4196)

jsfisher · Nov 1, 2014

doronshadmi said:
EDIT:

Please pay attention that |a - b| can't be defined as a given point along the real-line, but it can be defined as a value along the real-line

Defined? That would be twisted.

...and my purpose was the define a, b, or |a - b| by using the same concept (value, in this case).

Then, just as I said before, you are not talking about anything I posted, but instead about stuff you just made up, fabricated, pulled out of thin air.

After all you are the one that (for purpose of generalization) used the term value, in order to define objects along the real-line

I did not.

...for example:

(http://www.internationalskeptics.com/forums/showpost.php?p=10297296&postcount=4196)

Precisely!

doronshadmi · Nov 2, 2014

jsfisher said:
Precisely!

Precisely? ,let's see:

As you put it, a or b are distinct points along the real line, where the length of a point is defined by value 0 (which is some value along the real-line).

It is clear that the length of |a - b| is defined by some value > 0 (which is some value along the real-line), in case that a is not identical to b.

a+|a - b| is the finite parallel-summation, in case that a<b.

a+(-|a - b|) is the finite parallel-summation, in case that b<a.

Both finite parallel-summations = b

So, your argument is some case of finite parallel-summation that does not address my argument, which is at least |R| parallel-summation.

The definition of size, in this case is:

The number of values that are used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

Finite size < |N| size < |R| size is defiantly defined by actual mathematics.

jsfisher · Nov 2, 2014

doronshadmi said:
Precisely? ,let's see:

As you put it, a or b are distinct points along the real line, where the length of a point is defined by value 0 (which is some value along the real-line).

It is clear that the length of |a - b| is defined by some value > 0 (which is some value along the real-line), in case that a is not identical to b.

Defined? No.

a+|a - b| is the finite parallel-summation, in case that a<b.

a+(-|a - b|) is the finite parallel-summation, in case that b<a.

Both finite parallel-summations = b

"Parallel-summation" is a term of your invention.

So, your argument is some case of finite parallel-summation, which does not address my argument, which is at least |R| parallel-summation.

Meaningless.

The definition of size, in this case is:

The number of values that are used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

No, it's not. That's just nonsense.

Finite size < |N| size < |R| size is defiantly defined by actual mathematics.

Defined? Definitely not, and your use of language continues to be extremely awkward.

doronshadmi · Nov 2, 2014

jsfisher said:
"Parallel-summation" is a term of your invention.

jsfisher, it is not so hard to understand that, for example, 1+1+1+1+1+1+1+1 = 8, where the sum of 1+1+1+1+1+1+1+1 is done in parallel (no step-by-step is involved here) without any kind of, so called, process.

Parallel-summation holds also in case of |N| values, and since |N|<|R|, the parallel-summation of the |N| values of, for example, sequence <0.9, 0.09, 0.009, ...> (if observed from |R|) is value 0.999... < value 1 by value 0.000...1, as very simply explained and demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=10303144&postcount=4246.

jsfisher said:
No, it's not. That's just nonsense.

Why?

jsfisher · Nov 2, 2014

doronshadmi said:
Why?

I have no idea why you posted nonsense.

Perhaps it has something to do with your insistence on replacing established meanings with your own private vocabulary and your refusal to define what your invented terms mean. Your awkward use of language plays into it, too.

jsfisher · Nov 2, 2014

doronshadmi said:
jsfisher, it is not so hard to understand that, for example, 1+1+1+1+1+1+1+1 = 8, where the sum of 1+1+1+1+1+1+1+1 is done in parallel

How would you propose to do this parallel addition? You invented the term; you need to define it. Be careful, too, in your definition, since the order of operations can be important for series.

...(no step-by-step is involved here) without any kind of, so called, process.

You have hand-waved your way to one step, not zero.

And be all that as it may, what has this to do with infinite series, since their evaluation is completely determined by limits?

doronshadmi · Nov 2, 2014

jsfisher said:
How would you propose to do this parallel addition?

In parallel, not so hard, isn't it?

jsfisher said:
You have hand-waved your way to one step, not zero.

jsfisher, talking about particular values along the real line, is done by at least one step known as distinction, which is a step from the all possible values to the particular values.

Without this step, no particular values are available for discussion.

In this case (without loss of generality) the particular values are the |N| values of sequence <0.9, 0.09, 0.009, ...> where 0.9+0.09+0.009+... = 0.999... is the parallel-summation based on this one step of distinction.

jsfisher said:
And be all that as it may, what has this to do with infinite series, since their evaluation is completely determined by limits?

The definition of size, in this case is:

The number of values that are used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

The |N| size series 0.9+0.09+0.009+..., if observed from |N| size, it is = 1.

The |N| size series 0.9+0.09+0.009+..., if observed from |R| size , it is < 1 by 0.000...1, as very simply addressed in http://www.internationalskeptics.com/forums/showthread.php?p=10305282#post10305282.

jsfisher · Nov 2, 2014

doronshadmi said:
In parallel, not so hard, isn't it?

You can't provide a definition, can you? It remains meaningless.

jsfisher, talking about particular values along the real line, is done by at least one step known as distinction, which is a step from the all possible values to the particular values.

Without this step, no particular values are available for discussion.

More hand-waving. More insertion of your private, undefined vocabulary.

...non sequitur regarding "size"...

And be all that as it may, what has this to do with infinite series, since their evaluation is completely determined by limits?

doronshadmi · Nov 2, 2014

jsfisher said:
... established meanings ...

What you call established meanings, is |N| size observation of value 0.9+0.09+0.009+... = 0.999... = 1

No less, no more.

jsfisher · Nov 2, 2014

doronshadmi said:
What you call established meanings, is |N| size observation of value 0.9+0.09+0.009+... = 0.999... = 1

No less, no more.

More of a consequence of established meanings. No special pleadings.

Not that "|N| size observation of value" part, though. That's just gibberish.

doronshadmi · Nov 2, 2014

jsfisher said:
You can't provide a definition, can you? It remains meaningless.

Look, Hilbert did not define Line or Point, which enabled to give them meanings according to its axioms.

So is the case about parallel-summation, its meaning is given by the definition of size.

The definition of size, in this case is:

The number of values that are used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

jsfisher said:
More hand-waving. More insertion of your private, undefined vocabulary.

What is not understood about distinction, as I described it?

jsfisher said:
And be all that as it may, what has this to do with infinite series, since their evaluation is completely determined by limits?

If |N| size series like 0.9+0.09+0.009+... is observed from |N| size, there is a limit.

If |N| size series like 0.9+0.09+0.009+... is observed from |R| size, there is no limit.

realpaladin · Nov 2, 2014

doronshadmi said:
jsfisher, it is not so hard to understand that, for example, 1+1+1+1+1+1+1+1 = 8, where the sum of 1+1+1+1+1+1+1+1 is done in parallel (no step-by-step is involved here) without any kind of, so called, process.

Parallel summation can not exist in the fashion you describe; if the concurrency where to be of order 8 (in your example) then there is nothing to sum each term to as each term is single without context.

So you need to have at least two steps; one breaking the example into 4 sums of 1+1 and another one of summing 2+2+2+2.

It follows therefore that 'parallel summation' does not hold in the most basic of cases.

doronshadmi said:
Parallel-summation holds also in case of |N| values, and since |N|<|R|, the parallel-summation of the |N| values of, for example, sequence <0.9, 0.09, 0.009, ...> (if observed from |R|) is value 0.999... < value 1 by value 0.000...1, as very simply explained and demonstrated in http://www.internationalskeptics.com/forums/showpost.php?p=10303144&postcount=4246.

Yes, were the demonstration and explanation of a rigorous nature, you would have caught the errors in your thinking.

realpaladin · Nov 2, 2014

doronshadmi said:
In parallel, not so hard, isn't it?

As demonstrated in my previous post; Doron you are *WRONG*, both in your concept of parallel summation and in that silly claim here.

jsfisher · Nov 2, 2014

doronshadmi said:
Look, Hilbert did not define Line or Point, which enabled to give them meanings according to its axioms.

Nor does ZFC Set Theory define "membership". Key point, though, the axioms don't depend on the definition.

So is the case about parallel-summation, its meaning is given by the definition of size.

The definition of size, in this case is:

The number of values that are used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

Nice circle you have going there. And why do you feel compelled to define "size" anyway?

What is not understood about distinction, as I described it?

You used the term; you didn't describe it.

If |N| size series like 0.9+0.09+0.009+... is observed from |N| size, there is a limit.

If |N| size series like 0.9+0.09+0.009+... is observed from |R| size, there is no limit.

If only Mathematics worked like that, then you'd be all set. Too bad, though, 'cause limits don't work as you might want.

doronshadmi · Nov 2, 2014

jsfisher said:
Nor does ZFC Set Theory define "membership". Key point, though, the axioms don't depend on the definition.

So is the case about concepts within definition, their meanings are given according to the context of the definition, where parallel-summation is an example of such concept, within the context of the definition of size.

jsfisher said:
Nice circle you have going there.

It is actually straightforward, also definitions can use concepts even if they are not already defined, and in this case their meanings are given by the context of the definition.

jsfisher said:
And why do you feel compelled to define "size" anyway?

In order to accurately be understood by others, about this particular concept.

jsfisher said:
You used the term; you didn't describe it.

It is called a relevant example. Not every concept must be defined in order to be understood, sometimes relevant examples do the job, and the case of distinction is such one.

jsfisher said:
If only Mathematics worked like that, then you'd be all set.

I don't understand this part, please write is in a straightforward way.

realpaladin · Nov 2, 2014

doronshadmi said:
In order to accurately be understood by others, about this particular concept.

The fact that the three of us are at this for over 7 years does not clue you in that you are grandly failing in that effort?

jsfisher · Nov 2, 2014

doronshadmi said:
So is the case about concepts within definition, their meanings are given according to the context of the definition, where parallel-summation is an example of such concept, within the context of the definition of size.

You have (your own version of) "size" defined (if I may stretch meaning here) in terms of "parallel-summation" and "parallel-summation" in terms of "size".

As I said, nice circle.

Good thing set theory and axiomatic geometry don't make that silly mistake.

doronshadmi · Nov 2, 2014

jsfisher said:
You have (your own version of) "size" defined (if I may stretch meaning here) in terms of "parallel-summation" and "parallel-summation" in terms of "size"

The number of values and the parallel-summation of this number of values are not the same thing in this definition, for example:

The number of values in 100+1 is 2, where the parallel-summation of these 2 values is 101.

Since you have missed it, let's make it clearer:

The definition of size:

The number of values, which is used in a given parallel-summation, in order to be, to get, to reach, etc. a given value along the real-line.

This size can be finite, |N| or |R|, such that finite size < |N| size < |R| size.

You don't have to stretch meaning here, because this size is identical to what is known as cardinality, in case that you have missed it.

jsfisher · Nov 2, 2014

doronshadmi said:
...nonsense that has nothing to do with series...

If you want to discuss Mathematics, stop making stuff up. Instead, work from something more like this:

Definition for limits:
L = Limit(F(j), j = 1 to infinity) if and only if for all e > 0 there exists M such that for all k > M, |F(k) - L| < e.

Definition for series:
Let S be the series Sum(a_i, i = 1 to infinity). Then, S = L if and only if Limit(Sum(a_i, i = 1 to N), N = 1 to infinity) exists and is equal to L. If the limit exists, S is said to be convergent; otherwise, S is said to be divergent.

0.999... is a number expressed in a base-ten positional notation with the meaning of the series, Sum(9x10^-i, i = 1 to infinity). The corresponding sequence of partial sums has a limit, and the limit is 1.

Therefore, 0.999... = 1.

doronshadmi · Nov 2, 2014

jsfisher said:
If you want to discuss Mathematics, stop making stuff up. Instead, work from something more like this:

Definition for limits:
L = Limit(F(j), j = 1 to infinity) if and only if for all e > 0 there exists M such that for all k > M, |F(k) - L| < e.

Definition for series:
Let S be the series Sum(a_i, i = 1 to infinity). Then, S = L if and only if Limit(Sum(a_i, i = 1 to N), N = 1 to infinity) exists and is equal to L. If the limit exists, S is said to be convergent; otherwise, S is said to be divergent.

0.999... is a number expressed in a base-ten positional notation with the meaning of the series, Sum(9x10^-i, i = 1 to infinity). The corresponding sequence of partial sums has a limit, and the limit is 1.

Therefore, 0.999... = 1.

This is simply the case of |N| size observation of value 0.9+0.09+0.009+... = 0.999... = 1

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10305555&postcount=4270.

Thank you.

jsfisher · Nov 2, 2014

doronshadmi said:
This is simply the case of |N| size observation of value 0.9+0.09+0.009+... = 0.999... = 1

No, it isn't. It is simply the case of Mathematics, not gibberish.

ETA:

doronshadmi said:
Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10305555&postcount=4270.

I did. How could you have missed it?

doronshadmi · Nov 2, 2014

jsfisher said:
I did.

In what post (please write the number of this post)?

doronshadmi · Nov 2, 2014

jsfisher said:
0.999... is a number expressed in a base-ten positional notation

By using |R| size observation:

0.999...₁₀ < 1 by 0.000...1₁₀

0.888...₉ < 1 by 0.000...1₉

The difference between 0.000...1₁₀ and 0.000...1₉ is given by direct proportionality, according to the following formula:

abs((8/9)-(9/10)))=0.12

The general formula for all n>1 natural numbers is:

base j = 2 to n
base k = 2 to n

abs( ((base j-1)/(base j)) - ((base k-1)/(base k)) )

realpaladin · Nov 2, 2014

doronshadmi said:
By using |R| size observation:

0.999...₁₀ < 1 by 0.000...1₁₀

0.888...₉ < 1 by 0.000...1₉

The difference between 0.000...1₁₀ and 0.000...1₉ is given by a proportion, according to the following formula:

abs((1/9)*(8-1)-(1/10)*(10-1))=0.12

The general formula for all n>1 natural numbers is:

base j = 2 to n
base k = 2 to n

abs((1/base j)*(base j-1)-(1/base k)*(base k-1))

Doron, does it never bother you to be so *WRONG*?

There is *NO* ...1 in whatever notation you like to put it.

Infinity means infinity, where your creativity limits you to just 'Unimaginably far away'.

Because that *IS* what you are writing here; if there ever is a ...1 it means there is a final digit. And if there is a final digit, it is not *IN*finite.

Maybe you missed the language part here, but finite and final have a common root.

By saying there is a *final* digit, you are implicitly stating your formulation is *finite* and therefore invalid in this context.

I know you read my posts because you keep reacting to them so please take a moment and understand what I am writing here because continuing the way you are doing now just makes you look silly and deluded.

jsfisher · Nov 2, 2014

doronshadmi said:
By using |R| size observation:

Let's not. Let's stick to Mathematics, please.

doronshadmi · Nov 2, 2014

jsfisher said:
Let's not. Let's stick to Mathematics, please.

If you wish to be limited to |N| size observation of the real-line, than this is your choice, not mine.

jsfisher · Nov 2, 2014

doronshadmi said:
If you wish to be limited to |N| size observation of the real-line, than this is your choice, not mine.

I wish to be limited to the consistency and rigor of Mathematics. The contradiction and invention you offer has no utility whatsoever.

doronshadmi · Nov 3, 2014

jsfisher said:
I wish to be limited to the consistency and rigor of Mathematics. The contradiction and invention you offer has no utility whatsoever.

Invention, probably yes.

Contradiction, only if |N|<|R| is contradictory.

No utility whatsoever, no jsfisher, it is |R| size number system along the real-line that is available for new mathematical developments.

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