Cont: Deeper than primes - Continuation 2

[jsfisher]

"for all real numbers ε > 0" is not satisfied in case of (0,0,0,...) (ε = 0).

What are you on about now? Have you decided ε must be an element of the sequence? It doesn't.

You really, really need to take the time to understand the actual definition, and stop making up all this extra stuff.

 

[doronshadmi]

What are you on about now? Have you decided ε must be an element of the sequence? It doesn't.

You are right in this case, please answer to http://www.internationalskeptics.com/forums/showpost.php?p=10650757&postcount=520.

 

[jsfisher]

a_n is the n-th element of a sequence. Full stop. Period.

So what

So, it is correct. Adding the extra appendages you insist on adding to things only highlights your lack of understanding of the thing.

...after all you are the one who wrote:

However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1

Sure did. And in that case, the thing being discussed was the limit of a sequence of partial sums. It is to be expected that when discussing sequences of partial sums, on occasion it may be mentioned that the sequences are of partial sums.

On the other hand, when discussing limits of sequences, it is highly inappropriate to thrown in additional qualifiers that played no part in the discussion.

 

[doronshadmi]

So, it is correct. Adding the extra appendages you insist on adding to things only highlights your lack of understanding of the thing.



Sure did. And in that case, the thing being discussed was the limit of a sequence of partial sums. It is to be expected that when discussing sequences of partial sums, on occasion it may be mentioned that the sequences are of partial sums.

On the other hand, when discussing limits of sequences, it is highly inappropriate to thrown in additional qualifiers that played no part in the discussion.

The discussion is about this case:

Mathematics gives meaning to infinite series by defining it to be the limit of the sequence of partial sums.

In this case the n-th term (also written as a_n) is some partial sum of a sequence of partial sums (that is why such a sequence is called "sequence of partial sums" in the first place), so when I wrote "which is a partial sum of a given sequence" I meant that a_n is some partial sum of a sequence of partial sums, so no additional quantifiers were used in this case (in the case of (1,0,0,0,...) the n-th term (also written as a_n) = L).

Now please answer to:

http://www.internationalskeptics.com/forums/showpost.php?p=10650300&postcount=510

http://www.internationalskeptics.com/forums/showpost.php?p=10650650&postcount=516

Thank you.

 

[jsfisher]

please answer to:

http://www.internationalskeptics.com/forums/showpost.php?p=10650300&postcount=510

http://www.internationalskeptics.com/forums/showpost.php?p=10650650&postcount=516

Thank you.

If there is something interesting in those posts (which have no doubt been edited multiple times since you first hit the SUBMIT button), then please, feel free to repost the interesting something.

Please, also, be kind enough to repost the interesting something with your corrected understanding of the definition of limit of a sequence and without irrelevant qualifiers and without irrelevant requirements.

Thank you.

 

[doronshadmi]

If there is something interesting in those posts (which have no doubt been edited multiple times since you first hit the SUBMIT button), then please, feel free to repost the interesting something.

Please, also, be kind enough to repost the interesting something with your corrected understanding of the definition of limit of a sequence and without irrelevant qualifiers and without irrelevant requirements.

Thank you.

I have finished to edit http://www.internationalskeptics.com/forums/showpost.php?p=10650892&postcount=524 including its links.

Please air your view about its content (including its links).

Thank you.

 

[jsfisher]

I have finished to edit http://www.internationalskeptics.com/forums/showpost.php?p=10650892&postcount=524 including its links.

Please air your view about its content (including its links).

Thank you.

I have no interest in encouraging your rude habit of exploiting the edit window to rewrite your posts and then expecting everyone to go back to re-read them to see what they might say now.

If there is something interesting in that post or any of the posts to which it links that you feel deserves discussion, then please, feel free to repost the interesting something.

Please, also, be kind enough to repost the interesting something with your corrected understanding of the definition of limit of a sequence and without irrelevant qualifiers and without irrelevant requirements.

Thank you.

 

[doronshadmi]

If there is something interesting in that post or any of the posts to which it links that you feel deserves discussion, then please, feel free to repost the interesting something.

Only you, by reading all of http://www.internationalskeptics.com/forums/showpost.php?p=10650892&postcount=524 including its links, can decide if you find some interesting stuff in it.

Since I think that all of it is interesting, I avoid cut-n-paste-spam of it in new post.

So you are invited to air your view about it (including its links).

Thank you.

 

[Little 10 Toes]

Only you, by reading all of http://www.internationalskeptics.com/forums/showpost.php?p=10650892&postcount=524 including its links, can decide if you find some interesting stuff in it.

Since I think that all of it is interesting, I avoid cut-n-paste-spam of it in new post.

So you are invited to air your view about it (including its links).

Thank you.

So you don't want to cut and paste spam, but you're willing to go back and edit posts multiple times even when people have quoted the original post and the first edit?

Freakin' classic.

 

[doronshadmi]

Ok, let's re-post it as follows:

...in that case, the thing being discussed was the limit of a sequence of partial sums.

Indeed the last discussion was always about this case:

Mathematics gives meaning to infinite series by defining it to be the limit of the sequence of partial sums.

In this case the n-th term (also written as a_n) is some partial sum of a sequence of partial sums (that is why such a sequence is called "sequence of partial sums" in the first place), so when I wrote "which is a partial sum of a given sequence" I meant that a_n is some partial sum of a sequence of partial sums, so no additional quantifiers were used in this case (in the case of (1,0,0,0,...) the n-th term (also written as a_n) = L).

So jsfisher's remark about my "which is a partial sum of a given sequence"

Why do you persist in inserting this bogus qualifier? The definition of 'limit' in play here is of a sequence (period), not of a sequence of partial sums derived from a sequence.

has no basis.

Moreover,

Really? Exactly where does my version use partial sums? For your convenience, here's the definition again. Please just highlight were I make any mention of partial sums whatsoever.

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.​

So by his own evidence jsfisher's definition is irrelevant to the last discussion.

Now let's move on by not forgetting that the last discussion is always about sequences of partial sums.

However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1

Let's check these claims by using (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) as the particular sequence of partial sums.

However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity

jsfisher, please provide a definition that does not avoid sequences of partial sums, and by using it please prove that there is no number that is greater than any term of the sequence of partial sums (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) AND smaller than 1.

Without this proof you can't reject the claim that 0.999...₁₀ is greater than any term of the sequence of partial sums (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) AND < 1.

Thank you.

 

[jsfisher]

Ok, let's re-post it as follows:

...in that case, the thing being discussed was the limit of a sequence of partial sums.

Indeed the last discussion was always about this case

No, it was not. Just like this thread is not entirely about "Deeper than primes."

We were discussing 0.999... and that we can discuss without the conversation being anything about "deeper than primes".

The meaning of 0.999... gets into infinite series. We can discuss infinite series without the conversation being anything about "deeper than primes" nor specifically 0.999....

The meaning of infinite series gets into infinite sequences of partial sums. We can discuss that without the conversation being anything about "deeper than primes" nor 0.999... nor series.

The valuation of infinite sequences of partial sums gets into limits of sequences. We can discuss limits of sequences without the conversation being anything about "deeper than primes" nor 0.999... nor series nor partial sums.

Yet, you insist on introducing irrelevancies. When you are prepared to enter into a discussion without them, let me know.

 

[doronshadmi]

No, it was not. Just like this thread is not entirely about "Deeper than primes."

I have said the last discussion, not all the discussions that can be found in the thread called "deeper that primes".

The meaning of 0.999... gets into infinite series.

Please provide this meaning without using sequences of partial sums.

The meaning of infinite series gets into infinite sequences of partial sums. We can discuss that without the conversation being anything about "deeper than primes" nor 0.999... nor series.

Please provide the meaning of infinite series without using series.

Generally you simply avoided any detailed reply of the content of http://www.internationalskeptics.com/forums/showpost.php?p=10651216&postcount=530.

So please try again.

 

[jsfisher]

The meaning of 0.999... gets into infinite series.

Please provide this meaning without using sequences of partial sums.

Your demand is irrelevant to what I posted.

The meaning of infinite series gets into infinite sequences of partial sums. We can discuss that without the conversation being anything about "deeper than primes" nor 0.999... nor series.

Please provide the meaning of infinite series without using series.

Another irrelevancy. Do you even read what you are replying to? I said we could discuss infinite sequences of partial sums without the conversation being about series.

 

[Slings and Arrows]

Please forgive my intrusion, but has it been shown or demonstrated that ‘0.000…1’ is indeed a valid number?

I can’t find an example of how it's used, or even how it's defined, anywhere in the literature. My best guess is that it's an attempt to assign an actual number to an infinitesimal; in other words:

infinitesimal = 0.000...1

Is it a hyperreal number?

 

[doronshadmi]

Your demand is irrelevant to what I posted.



Another irrelevancy. Do you even read what you are replying to? I said we could discuss infinite sequences of partial sums without the conversation being about series.

Will you please reply in details to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10651216&postcount=530?

Thank you.

 

[jsfisher]

Please forgive my intrusion, but has it been shown or demonstrated that ‘0.000…1’ is indeed a valid number?

I can’t find an example of how it's used, or even how it's defined, anywhere in the literature. My best guess is that it's an attempt to assign an actual number to an infinitesimal; in other words:

infinitesimal = 0.000...1

Is it a hyperreal number?

No, it is not a number at all. It is a notational crutch Doronshadmi has invented so he can pretend there is a difference between 0.999... and 1. The difference is 0.000...1, or so he claims, but he will provide no further clarification other than to complete his circular of defense.

 

[jsfisher]

Will you please reply in details to the content of http://www.internationalskeptics.com/forums/showpost.php?p=10651216&postcount=530?

Thank you.

I have responded already to its contents more than it deserves. As I have previously indicated, if you can get yourself past the irrelevant and the misrepresentations upon which you frequently dwell, we may have something to discuss. You also need to show some basic understanding of the material, which "So by his own evidence jsfisher's definition is irrelevant to the last discussion" demonstrates your lack.

ETA: Discarding the definition for limit of a sequence on the basis it was not written specifically for sequences of partial sums screams of lack of understanding.

 

[Slings and Arrows]

No, it is not a number at all. It is a notational crutch Doronshadmi has invented so he can pretend there is a difference between 0.999... and 1. The difference is 0.000...1, or so he claims, but he will provide not further clarification other than to complete his circular of defense.

Oh, so it's a number that is unique to this thread only, and is not part of mainstream mathematics. Thank you for the clarification.

 

[doronshadmi]

Please forgive my intrusion, but has it been shown or demonstrated that ‘0.000…1’ is indeed a valid number?

I can’t find an example of how it's used, or even how it's defined, anywhere in the literature. My best guess is that it's an attempt to assign an actual number to an infinitesimal; in other words:

infinitesimal = 0.000...1

Is it a hyperreal number?

Welcome Slings and Arrows.

You will not find a number of the form ‘0.000…1’ in the current literature, because it is based on probably new notion that uses the difference between transfinite cardinals like |N|< P(|N|)< P(P(|N|))< ... etc. ad infinitum, and even goes beyond the concept of collection by logically define the difference between 0-dimesional space and 1-dimesional space.

If 0-dimesional space is represented by a point and 1-dimesional space is represented by a line, then a point is at the domain of the line, where the line (if not taken as a collection of points) is at AND not at the domain of the point.

By using visual spatial representation it can be addressed, for example, by __.__

From the observation of the point with respect to the line, being AND not being at the domain of the point is observed as logical contradiction, but it is not a contradiction if the observation is done from the line with respect to the point.

As can be seen by __.__ , being AND not being at the domain of the point is simply the natural property of the line (if not taken as a collection of points) with respect to the point, and by this reasoning actual infinity is at the level of the non-composed line, which is beyond any amount of collection of points along it.

By using the notion of different observations, I asked myself, how a collection with P(|N|) elements looks like from a collection with |N| elements?

In order to do such an observation I used, without loss of generality, the number of the form 0.999...₁₀ that has |N| places (filled by symbol 9) at the right side of the floating point.

By looking at it I have realized that I can construct a number that has |N| symbols (filled by symbol 0) at the right side of the floating point (represented as 0.000...) and then I used symbol ...1₁₀ that represents the uncountable level of P(|N|), which looks like a single element from the countable level of |N| elements, so by observing P(|N|) from |N|, you get number 0.000...1₁₀ that has countable part (represented by 0.000...) and uncountable part (represented by ...1₁₀).

This kind of number can be used as a complement from number 0.999...₁₀ to number 1, written as 0.999...₁₀ + 0.000...1₁₀ = 1

For more details please look at:

http://www.internationalskeptics.com/forums/showpost.php?p=10621469&postcount=348

http://www.internationalskeptics.com/forums/showpost.php?p=10621635&postcount=349

-------------------

The standard definition of convergence uses infinite sequence of partial sums (and in the case of 0.999...₁₀ that sequence is (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that is used in order to prove that 0.999...₁₀ = 1, and this proof observes the infinite from the finite).

Here is the standard definition of convergence (http://en.wikipedia.org/wiki/Sequence#Definition_of_convergence):

that can be written as follows:

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε.

As can be seen, this definition uses natural numbers as the indexes of a_n (which is some term in some infinite sequence of partial sums (and in the case of 0.999...₁₀ the used infinite sequence of partial sums is (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) that has numbers < 0.999...₁₀)) in order to prove, for example, that 0.999...₁₀ = 1.

So according to the standard definition of convergence and by using the case of infinite sequence of partial sums (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), where L=1, |a_n - L| must be > 0, or in other words, there is an unclosed gap between a_n and L, in this example, that is taken as irrelevant (from the mainstream point of view) in order to conclude that, for example, 0.999...₁₀ = 1.

-------------------

I find this gap as relevant because it is the result of forcing the finite on the infinite, and I claim that we can't use the finite in order to conclude meaningful notion about 0.999...₁₀ that has |N| places.

-------------------

So my definition of limit is as follows:

Let X be a placeholder for that is summed.

If |X - L| = 0 then L is called the limit of X.

 

[doronshadmi]

ETA: Discarding the definition for limit of a sequence on the basis it was not written specifically for sequences of partial sums screams of lack of understanding.

You are the one that wrote:

Exactly where does my version use partial sums? For your convenience, here's the definition again. Please just highlight were I make any mention of partial sums whatsoever.

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.​

isn't it?

jsfisher, in case that you have missed it, the discussion is about the (standard) definition of convergence.

Here it is (http://en.wikipedia.org/wiki/Sequence#Definition_of_convergence):

[qimg]http://upload.wikimedia.org/math/0/4/7/04779f5297965fb31cc5be0a111fa1a1.png[/qimg]

and it also can be written as follows:

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε.

So please reply only about it during our discussion.

Thank you.