Deeper than primes - Continuation

[doronshadmi]

... 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]

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]

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.

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

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

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]

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.

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]

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]

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]

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.

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.

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

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

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.

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]

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]

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]

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]

...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]

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]

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:

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

I did. How could you have missed it?

 

[doronshadmi]

I did.

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

 

[doronshadmi]

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]

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]

By using |R| size observation:

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

 

[doronshadmi]

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]

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]

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.