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Old 9th September 2021, 11:29 AM   #81
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Originally Posted by JoeMorgue View Post
Well that and Zeno lived in the 5th Century BCE and Schrodinger in the first half of the 19th century so I'm going to assume the person who lived after we learned the Earth revolved around the Sun and that maggots didn't magically form in rotting meat knew what his was talking about more.
Since motion through time seems to have the same problem as motion through space, there was no distance actually traveled between Zeno and Schroedinger. They are in fact coterminous in time (as well as in space). This is how we know that Zeno had access to our modern mathematics. He's been here in our modern times all along (or we and our maths have been there with him all along).

On the other hand, phrases like "all along" no longer make sense. For example, "no longer make sense" is now nonsensical. As is "is now nonsensical".
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Old 9th September 2021, 12:00 PM   #82
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Zeno: For the arrow to reach the wall it must first reach halfway to the wall, correct?
Me: Sure. It goes halfway and then goes the other halfway. 1/2 + 1/2 = 1.
Zeno: Ah! But having gone halfway, it must then reach half the remaining distance to the wall.
Me: Sure, if you want to keep track of it that way, by quarters. Starting from the bow it goes a quarter of the way, then another quarter, then another quarter, then reaches the wall. 4 * 1/4 = 1
Zeno: No so! For perhaps you have failed to consider that after going three quarters of the way, it must then travel half the remaining distance.
Me: That is, an eighth. No problem. It has to go one eighth of the distance from bow to wall eight times. 8 * 1/8 = 1
Zeno: But to cross that final eighth it must still pass through the point halfway...
Me: Way ahead of you. 16 * 1/16 is still 1.
Zeno: But I can continue subdividing without limit!
Me: So can I. However far I subdivide it I still get 1. What's the problem?
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Old 9th September 2021, 12:06 PM   #83
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The sequel:

Me: Now suppose the distance the arrow is fired is the diagonal of a square whose side is 1 unit.
Zeno: I'm not going to talk to you any more if you're going to be irrational.
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Old 9th September 2021, 12:07 PM   #84
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Originally Posted by theprestige View Post
The infinite series you're talking about is 3 by definition.
Indeed. Take the series

2.9 + .09 + .009 + .0009 + ... for which

(1) None of the numbers in the series is 3
(2) The series sums to 3

Does this condemn me to AAH?
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Old 9th September 2021, 12:11 PM   #85
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Originally Posted by Myriad View Post
Zeno: For the arrow to reach the wall it must first reach halfway to the wall, correct?
Me: Sure. It goes halfway and then goes the other halfway. 1/2 + 1/2 = 1.
Zeno: Ah! But having gone halfway, it must then reach half the remaining distance to the wall.
Me: Sure, if you want to keep track of it that way, by quarters. Starting from the bow it goes a quarter of the way, then another quarter, then another quarter, then reaches the wall. 4 * 1/4 = 1
Zeno: No so! For perhaps you have failed to consider that after going three quarters of the way, it must then travel half the remaining distance.
Me: That is, an eighth. No problem. It has to go one eighth of the distance from bow to wall eight times. 8 * 1/8 = 1
Zeno: But to cross that final eighth it must still pass through the point halfway...
Me: Way ahead of you. 16 * 1/16 is still 1.
Zeno: But I can continue subdividing without limit!
Me: So can I. However far I subdivide it I still get 1. What's the problem?
I like this explanation. Turns out the half, then half-of-half, then half-of-half-of-half is just a red herring.
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Old 9th September 2021, 12:12 PM   #86
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Originally Posted by Humots View Post
Indeed. Take the series

2.9 + .09 + .009 + .0009 + ... for which

(1) None of the numbers in the series is 3
(2) The series sums to 3

Does this condemn me to AAH?
Luckily for you, moves to AAH aren't real.
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Old 9th September 2021, 12:13 PM   #87
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Originally Posted by Myriad View Post
Zeno: For the arrow to reach the wall it must first reach halfway to the wall, correct?
Me: Sure. It goes halfway and then goes the other halfway. 1/2 + 1/2 = 1.
Zeno: Ah! But having gone halfway, it must then reach half the remaining distance to the wall.
Me: Sure, if you want to keep track of it that way, by quarters. Starting from the bow it goes a quarter of the way, then another quarter, then another quarter, then reaches the wall. 4 * 1/4 = 1
Zeno: No so! For perhaps you have failed to consider that after going three quarters of the way, it must then travel half the remaining distance.
Me: That is, an eighth. No problem. It has to go one eighth of the distance from bow to wall eight times. 8 * 1/8 = 1
Zeno: But to cross that final eighth it must still pass through the point halfway...
Me: Way ahead of you. 16 * 1/16 is still 1.
Zeno: But I can continue subdividing without limit!
Me: So can I. However far I subdivide it I still get 1. What's the problem?
Me:. Hey Zeno. Stand over against that wall for a minute.
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Old 9th September 2021, 12:16 PM   #88
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I wonder what Robin imagines happens instead of motion. Large-scale quantum tunneling? The moon doesn't really orbit the Earth, it just executes a never-ending series of infinitesimal teleportations?
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Old 9th September 2021, 01:55 PM   #89
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Lets start again.

Say we have a convergent series that sums to 3.

But if you were ​to actually do that infinite sum it would never reach 3.

But these things are somehow the same?

Doing a sum which, by definition can't reach 3 infinitely many times it will make it reach 3?

Even though the maths says that doing it infinitely many times will never make it reach 3?

A sum which behaves differently under the assumption of change?

This is the resolution to Zeno Paradox? Really?

It converges to 3, but that depends on the fact that infinitely many iterations of the sum will never reach 3?

So when those infinitely many operations that can't get to 3 complete, where exactly does the come from?

How do you get 3 from a set of numbers that doesn’t contain 3?

Can someone give me a sensible answer to that?

Solving the Zeno Paradox with a paradox?

Let’s face it, the convergent sum would actually agree with Zeno if it was considered an actual sum performed.

Oh and let's not forget the absurd stonewalling when I asked them if they really believed it, that you can find the number 3 in a set which by definition does not contain the number 3. Where does the 3 come from?

I can judge your certainty that I am wrong by the amount of stonewalling.
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Old 9th September 2021, 02:27 PM   #90
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Originally Posted by Robin View Post
Lets start again.

Say we have a convergent series that sums to 3.

But if you were ​to actually do that infinite sum it would never reach 3.
The convergent series that sums to 3 isn't an arithmetic problem you sit down and do. It's a single logical symbol that represents 3.

Myriad's little dialog above probably puts it best. If you divide 3 into 8 pieces, those pieces still add up to 3. If you divide 3 into 16 pieces, those pieces still add up to 3. If you divide 3 into infinite, infinitesimal pieces, those pieces still add up to 3. This is a logical conclusion from certain arbitrary axioms, not an arithmetic problem anyone is supposed to, one sum at a time. The whole point is that logically, the sum is already done. It's an interesting mathematical footnote, not a model of motion in the real world.
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Old 9th September 2021, 02:29 PM   #91
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Originally Posted by Robin View Post
Lets start again.

Say we have a convergent series that sums to 3.

But if you were ​to actually do that infinite sum it would never reach 3.

But these things are somehow the same?

Doing a sum which, by definition can't reach 3 infinitely many times it will make it reach 3?

Even though the maths says that doing it infinitely many times will never make it reach 3?

A sum which behaves differently under the assumption of change?

This is the resolution to Zeno Paradox? Really?

It converges to 3, but that depends on the fact that infinitely many iterations of the sum will never reach 3?

So when those infinitely many operations that can't get to 3 complete, where exactly does the come from?

How do you get 3 from a set of numbers that doesnít contain 3?

Can someone give me a sensible answer to that?

Solving the Zeno Paradox with a paradox?

Letís face it, the convergent sum would actually agree with Zeno if it was considered an actual sum performed.

Oh and let's not forget the absurd stonewalling when I asked them if they really believed it, that you can find the number 3 in a set which by definition does not contain the number 3. Where does the 3 come from?

I can judge your certainty that I am wrong by the amount of stonewalling.
Gibberish.

What are you going to do now start a 3rd thread claiming you were "dogpiled" in this one too?
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Old 9th September 2021, 02:30 PM   #92
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Originally Posted by Robin View Post
Doing a sum which, by definition can't reach 3 infinitely many times it will make it reach 3?
You are profoundly wrong on this point. The convergent series which sums to 3 is, by definition a sum that can - and must! - reach 3. When I say that the axioms of the system are arbitrary, I didn't mean you get to pick and choose which axioms to apply within the system.
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Old 9th September 2021, 02:36 PM   #93
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Originally Posted by Robin View Post
How do you get 3 from a set of numbers that doesnít contain 3?

Can someone give me a sensible answer to that?
You've been given several sensible answers, most of which you have ignored. But let's try again.

You are wrong about your set of numbers. The set as a whole represents 3. You won't find a human body among the set of cells in a human body. But if you find the set of cells in a human body, you have indeed found a human body.

Quote:
Letís face it, the convergent sum would actually agree with Zeno if it was considered an actual sum performed.
The only people who consider it an actual sum performed are you and Zeno. The rest of us understand that it is the conclusion of a logical proof that even an infinite number of subdivisions of a whole must add up to the whole, given certain arbitrary axioms.

Quote:
Oh and let's not forget the absurd stonewalling when I asked them if they really believed it, that you can find the number 3 in a set which by definition does not contain the number 3. Where does the 3 come from?
It comes from the complete set of its parts.

Quote:
I can judge your certainty that I am wrong by the amount of stonewalling.
I'm certain you're wrong because for all your bluster, eppur si muove.
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Old 9th September 2021, 02:36 PM   #94
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Originally Posted by Robin View Post

How do you get 3 from a set of numbers that doesnít contain 3?

Can someone give me a sensible answer to that?
2+1?
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Old 9th September 2021, 02:38 PM   #95
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Originally Posted by Robin View Post
Lets start again.
And finally: Fringe reset!

You've received several substantive replies since your last post. More than one by me. Rather than engage with those replies and continue making progress, you want to go back to square one. Why is that?
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Old 9th September 2021, 02:39 PM   #96
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Originally Posted by Robin View Post
Lets start again.

Say we have a convergent series that sums to 3.

But if you were ​to actually do that infinite sum it would never reach 3.
Yes, it would. That's the definition of a convergent series that sums to 3.

Dave
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Old 9th September 2021, 03:18 PM   #97
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Originally Posted by Robin View Post
Lets start again.

Say we have a convergent series that sums to 3.

But if you were ​to actually do that infinite sum it would never reach 3.

.
You are using "series" in some places where mathematicians would use "sequence". They use two words because they mean two different things. The sequence does not contain three, but it approaches 3 as a limit. The series sums to 3.

I haven't looked it up, but I am confident Wikipedia has an article that explains the meaning of the terms.

Eta:. If you put Sequence vs. series into Google, it will probably take you right where you want to go.

Or at least, right where it would explain the difference.
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Old 9th September 2021, 03:53 PM   #98
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I read about Zeno's paradoxes in Wikipedia.

They're thoroughly wrong, of course. Infinite series is one of many ways to explain the error. However, if they thought motion was an illusion, what did they think the underlying reality was?

I think they must have had some sort of Tralfamadorian-type view, in which all events for all time were fixed in space. How else could there be no motion?

Well, even if that were true, Zeno's paradoxes wouldn't prove it. He was just wrong.
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Old 9th September 2021, 04:03 PM   #99
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Originally Posted by JoeMorgue View Post
Gibberish.

What are you going to do now start a 3rd thread claiming you were "dogpiled" in this one too?
Let us sum up this situation shall we.

I was stonewalled endlessly, and you can't deny this because it is there, stonewalled in order to avoid admitting that a set of numbers that does not contain the number 3 does not contain the number 3

All because they couldn't bear to admit that something I said before was true.

Now, rather that actually address the issue you attempt rather feebly to claim it is gibberish.

Something that is quite true and for which I have only received meaningless responses.

Hmmm.
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Old 9th September 2021, 04:06 PM   #100
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Originally Posted by Meadmaker View Post
You are using "series" in some places where mathematicians would use "sequence". They use two words because they mean two different things.
Well yes I know and I have taken care to make sure that I said "sequence" when I meant tha and "series" when I meant series.

I need to refer to a sequence because I am talking about the case where a step by step summing.

Perhaps I have slipped up, can you please point out where you think I have done this.
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Old 9th September 2021, 04:17 PM   #101
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Originally Posted by Dave Rogers View Post
Yes, it would. That's the definition of a convergent series that sums to 3.

Dave
People stonewalled me endlessly in this thread in order to avoid admitting that, in the case that it was actually summed step by step, that it would, by definition, never reach 3.

Summing up for some finite number of steps can never be 3, obviously.

But the infinite sum, when considered as a step by step task iterated infinitely many times, contains only finite parts of the sum.

Therefore no part of this sum can actually be 3. I don't know why nobody will admit that.

Now we have the idea of a step by step summing iterated infinitely many times, which would obviously never produce the number 3, and the process stops in finite time and then, somehow, it is 3.

Nobody will explain where the three came from because the sum was run for infinitely many iterations and would never have had the number 3.

But now, somehow there is a number 3. Where did it come from?

If you could actually sum the underlying sequence for infinitely many iterations in a finite time, which is what the convergent series "solution" to the Paradox suggests, then you would not get the sum. The sequence would never have reached the sum by definition and now that all those infinite steps were completed you would just have. Well, nothing really because there is no end to an infinite process.

Why won't anyone admit that the infinite set of partial sums will never contain the sum?

Could someone acknowledge that?
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Old 9th September 2021, 04:21 PM   #102
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Originally Posted by Robin View Post
Well yes I know and I have taken care to make sure that I said "sequence" when I meant tha and "series" when I meant series.

I need to refer to a sequence because I am talking about the case where a step by step summing.

Perhaps I have slipped up, can you please point out where you think I have done this.
You are talking about series as if they contain numbers. (Specifically, the number three). Series don't contain numbers. Sequences do.
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Old 9th September 2021, 04:24 PM   #103
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Originally Posted by Robin View Post
Why won't anyone admit that the infinite set of partial sums will never contain the sum?

Could someone acknowledge that?
Yes. It's true. Such a set will never contain the sum.

Now what?
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Old 9th September 2021, 04:25 PM   #104
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Originally Posted by Robin View Post
Let us sum up this situation shall we.

I was stonewalled endlessly, and you can't deny this because it is there, stonewalled in order to avoid admitting that a set of numbers that does not contain the number 3 does not contain the number 3

All because they couldn't bear to admit that something I said before was true.

Now, rather that actually address the issue you attempt rather feebly to claim it is gibberish.

Something that is quite true and for which I have only received meaningless responses.

Hmmm.

The countably infinite set {2, 2.9, 2.99, 2.999, 2.9999, ...} contains the number 3. Specifically, it contains 2.999...

That's equivalent to saying the series (2 + .9 + .09 + .009 + ...) = 3.

What might be leading you to think otherwise is the often-garbled distinction between computation and mathematics. A lot of conventional usage including grade school arithmetic tends to instill the notion that "a = b" means "b is the result of the computation specified by a." Which makes (.9 + .09 + .009 + ...) = 1 seem questionable to some, because the specified computation would never halt.

But in mathematics, "a = b" means that a and b are different ways of writing the same thing. It doesn't matter whether you can transform a into b by some finite number of computations, especially if you can prove the equivalence in some other way.

You've seen infinite series expressions for irrational numbers like pi. Do you object to 4 (1 - 1/3 + 1/5 - 1/7 + 1/9 - ...) = pi because none of the partial sums computed from a finite number of terms is exactly pi, and a machine computing the exact value would never halt?
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Old 9th September 2021, 04:33 PM   #105
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The infinite set of all partial sums of a convergent series will never contain the sum.

This means that if that summing was done iteratively then the process would never end up getting to the sum, by definition.

Iteratively summing the underlying sequence for a series that converges on 3 can never possibly, even after infinitely many iterations get to the number 3. The maths says that.

So where, exactly would the 3 come from if this had completed in a finite time?
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Old 9th September 2021, 04:36 PM   #106
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Originally Posted by Myriad View Post
The countably infinite set {2, 2.9, 2.99, 2.999, 2.9999, ...} contains the number 3. Specifically, it contains 2.999...

That's equivalent to saying the series (2 + .9 + .09 + .009 + ...) = 3.

What might be leading you to think otherwise is the often-garbled distinction between computation and mathematics. A lot of conventional usage including grade school arithmetic tends to instill the notion that "a = b" means "b is the result of the computation specified by a." Which makes (.9 + .09 + .009 + ...) = 1 seem questionable to some, because the specified computation would never halt.

But in mathematics, "a = b" means that a and b are different ways of writing the same thing. It doesn't matter whether you can transform a into b by some finite number of computations, especially if you can prove the equivalence in some other way.

You've seen infinite series expressions for irrational numbers like pi. Do you object to 4 (1 - 1/3 + 1/5 - 1/7 + 1/9 - ...) = pi because none of the partial sums computed from a finite number of terms is exactly pi, and a machine computing the exact value would never halt?
I don't know why you said any of that, don't see how it relates. The point is that the infinite sequence of partial sums doesn't contain the sum of the series by definition. People don't seem to understand that.

Tell me one thing. If the underlying sequence for a series that converges on 3 was summed iteratively for the required infinitely many steps and could complete in real time what value would result from that?

Not of course that I expect anyone to give the obvious answer.
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Old 9th September 2021, 04:40 PM   #107
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Originally Posted by Robin View Post
The infinite set of all partial sums of a convergent series will never contain the sum.

This means that if that summing was done iteratively then the process would never end up getting to the sum, by definition.

Iteratively summing the underlying sequence for a series that converges on 3 can never possibly, even after infinitely many iterations get to the number 3. The maths says that.

So where, exactly would the 3 come from if this had completed in a finite time?
How do "iteratively" and "time" get involved?

It looks like Myriad's discussion of mathematics and computation is relevant here.
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Old 9th September 2021, 04:49 PM   #108
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Originally Posted by Myriad View Post
The countably infinite set {2, 2.9, 2.99, 2.999, 2.9999, ...} contains the number 3. Specifically, it contains 2.999...
I don't think so.

The set contains an infinite number of terms that are of the form 2.9, followed by more 9s, but it doesn't contain a term with an infinite number of 9s.
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Old 9th September 2021, 04:54 PM   #109
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Originally Posted by Meadmaker View Post
How do "iteratively" and "time" get involved?

It looks like Myriad's discussion of mathematics and computation is relevant here.

The first sentence of my previous post is in error. The sequence of partial sums doesn't contain the sum because the sum isn't a partial sum.

The rest remains true. Mathematics can determine the sum of the series without having to perform the computation, by means of other forms of proof.

No proven series expression of pi was proven by summing the infinite series and checking to see that the result comes out to the same infinite string of digits as pi. That would be impossible for more than one reason. Similarly, .9999... = 1 is not proved by adding up successive decimal fractions, and doesn't need to be.
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Old 9th September 2021, 04:56 PM   #110
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Originally Posted by Meadmaker View Post
How do "iteratively" and "time" get involved?

It looks like Myriad's discussion of mathematics and computation is relevant here.
We are talking about a set of numbers that will, by definition not contain the sum that would be calculated. We are talking about a specific case here and not about generalities.

Now here is the scenario again.

An infinite series converges on 3.

The infinite sequence of partial sums for this series can never, by definition, contain 3.

You could, in principle, take the underlying sequence and sum it step by step for an infinite number of times. which would give you the same values as the sequence of partial sums and so also can never by definition, contain the number 3.

Are you with me so far.

The suggestion is that the underlying sequence is summed step by step for an infinite number of times and is able to halt in a finite time.

So the question is, what would be the value of the sum? If it is 3, then where did the 3 come from?
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Old 9th September 2021, 04:59 PM   #111
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Originally Posted by Myriad View Post
The rest remains true. Mathematics can determine the sum of the series without having to perform the computation, by means of other forms of proof.
Well yes, obviously I said so myself.

But I am saying that if the underlying sequence was summed iteratively, step by step, infinitely many times then it would never reach the sum of the series. Do you agree?
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Old 9th September 2021, 05:02 PM   #112
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So no-one will admit that it would never even in principle be possible to arrive at the sum of any convergent series by iteratively summing the underlying sequence even infinitely many times?

I wonder why.
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Old 9th September 2021, 05:08 PM   #113
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You all do realise that the convergent series solution to the paradox in question depends on the ability to iteratively sum the underlying sequence of an infinite convergent series and arrive at the sum of the series don't you?

Can't even in principle be done, even if you seem not to understand that.
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Old 9th September 2021, 05:09 PM   #114
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Originally Posted by Robin View Post
Tell me one thing. If the underlying sequence for a series that converges on 3 was summed iteratively for the required infinitely many steps and could complete in real time what value would result from that?
The result of an impossible occurrence (an infinite number of computations performed in a finite time) can be impossible to predict. The value could be 3, or 0, or negative 5 googol, or banana. (See the Thomson's Lamp and the Ross-Littlewood paradoxes.)

The sum of the series, as can be proven based on the relevant axioms of mathematics, is 3.
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Old 9th September 2021, 05:13 PM   #115
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It's like Jabba but with the "Beffudled old man" routine swapped out with a sad attempt at an internet tough guy.
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Old 9th September 2021, 05:14 PM   #116
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Originally Posted by Myriad View Post
The first sentence of my previous post is in error. The sequence of partial sums doesn't contain the sum because the sum isn't a partial sum.

The rest remains true. Mathematics can determine the sum of the series without having to perform the computation, by means of other forms of proof.

No proven series expression of pi was proven by summing the infinite series and checking to see that the result comes out to the same infinite string of digits as pi. That would be impossible for more than one reason. Similarly, .9999... = 1 is not proved by adding up successive decimal fractions, and doesn't need to be.
Yep. (But you knew that)
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Old 9th September 2021, 05:19 PM   #117
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Originally Posted by Myriad View Post
The result of an impossible occurrence (an infinite number of computations performed in a finite time) can be impossible to predict. The value could be 3, or 0, or negative 5 googol, or banana. (See the Thomson's Lamp and the Ross-Littlewood paradoxes.)
Actually there could be no possible fact of the matter. It couldn't be any of those because, well where would they come from? It wouldn't be random or chaotic or anything at all.

There just simply could be no fact of the matter.

However I am pleased that someone could finally admit that the iterative summing of an infinite sequence is an impossible occurrence and not something that could result in the sum of the series.

Quote:
The sum of the series, as can be proven based on the relevant axioms of mathematics, is 3.
Yes, again as I said. But it could never even in principle be got at by an iterative process. Not even if it were possible to run infinitely many iterations in a finite time, which you can't.

That is just the point I have been making. But some have been claiming that you could do that.
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Old 9th September 2021, 05:20 PM   #118
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The strange thing is this has turned into the classic Internet argument ".999... can't really be exactly equal to 1" in disguise.
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Old 9th September 2021, 05:20 PM   #119
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Originally Posted by JoeMorgue View Post
It's like Jabba but with the "Beffudled old man" routine swapped out with a sad attempt at an internet tough guy.
Oh don't be so hard on yourself.
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Old 9th September 2021, 05:22 PM   #120
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Originally Posted by Myriad View Post
The strange thing is this has turned into the classic Internet argument ".999... can't really be exactly equal to 1" in disguise.
So maybe you could all stop turning it into that and stick to the point.
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