Concerning Infinity (of course)

[wtf]

28 minutes ago, studiot said:

The question is not whether the sequences 'can reach 1' but whether you can.

 

Why is that a question? What do my own personal capabilities have to do with this? All that matters is that I can prove that the limit of the sequence 1/2, 3/4, 7/8, ... is 1. I truly don't understand why, having acknowledged this point, you are still making up meaningless questions.

 

28 minutes ago, studiot said:

Of course neither you nor any other being following along the sequence can ever get to the end.

 

"Getting to the end," a meaningless concept, has been rendered irrelevant by the formal definition of a limit. 

 

28 minutes ago, studiot said:

I hope you agree that the numbers 1, 1.5 2, 22 million along with every other conceivable number, all existed ie were available to be discovered, long before Man evolved in the universe.

 

I agree to no such thing. It's a philosophical question. Before the universe existed, were there numbers? Where were they?

 

28 minutes ago, studiot said:

Luckily numbers take up zero space so there is plenty of room in our universe for all of them.

 

In the universe? Can you point to them? Identify their location? You are stating as a fact a matter of philosophy that you can't possibly hope to prove. 

 

28 minutes ago, studiot said:

So the entire sequence was there before we were and has effectively always been there.

According to some, yes. According to others, no. How about the game of chess? It's a formal system with precise rules. Did it exist before there were people?

You've entirely changed the subject, I don't know why. 

Your belief is called mathematical Platonism.

https://plato.stanford.edu/entries/platonism-mathematics/

[Genady]

2 hours ago, Boltzmannbrain said:

Because my issue in the OP has essential come down to the sequence from the partial sums of 1/2^n, namely (n-1)/n. 

You mean rather (2ⁿ-1)/2ⁿ.

 

2 hours ago, Boltzmannbrain said:

If this actually reaches 1, ...

But we know that it does not.

[Boltzmannbrain]

1 hour ago, studiot said:

 

The question is not whether the sequences 'can reach 1' but whether you can.

Of course neither you nor any other being following along the sequence can ever get to the end.

 

I hope you agree that the numbers 1, 1.5 2, 22 million along with every other conceivable number, all existed ie were available to be discovered, long before Man evolved in the universe.

Luckily numbers take up zero space so there is plenty of room in our universe for all of them.

 

So the entire sequence was there before we were and has effectively always been there.

Yes, I agree that time/humans have nothing to do with what numbers exist.  

But I still need to know if it reaches 1 or not with or without humans existing. 

[studiot]

13 hours ago, wtf said:

Why is that a question? What do my own personal capabilities have to do with this? All that matters is that I can prove that the limit of the sequence 1/2, 3/4, 7/8, ... is 1. I truly don't understand why, having acknowledged this point, you are still making up meaningless questions.

My comment was not addressed to you and the reason why was answered in the next line.

 

Really I am just trying to help the op get his head around the idea that assembling the infinite sequence or series is a thought experiment than cannot be completed, but nevertheless has its uses.

I note that this still hasn't clicked for him despite all your proofs.

13 hours ago, wtf said:

13 hours ago, studiot said:

I hope you agree that the numbers 1, 1.5 2, 22 million along with every other conceivable number, all existed ie were available to be discovered, long before Man evolved in the universe.

 

I agree to no such thing. It's a philosophical question. Before the universe existed, were there numbers? Where were they?

Since I have no knowledge of before the universe existed I wisely said nothing about that subject.

As you say existence is a philosophical question, as is the approach I offered.

 

13 hours ago, wtf said:

Did it exist before there were people?

Can I remind you that mathematical existence means simply that it is consistent with the adopted axioms, nothing more.

Physical existance is another subject. Since you apparantly believe that numbers do not occupy zero volume (space) perhaps you can supply a formula for the volume occupied by a shadow ?

[studiot]

12 hours ago, Boltzmannbrain said:

Yes, I agree that time/humans have nothing to do with what numbers exist.  

But I still need to know if it reaches 1 or not with or without humans existing. 

So often in Mathematics there is more than one way of looking at things.

So far in this thread we have been all been looking at what might be described as 'the assembly' method.

Nothing wrong with that.

But we should perhaps also consider the dissection method.

 

I take it that you are happy that the counting process 1.2.3.4.... can be carried on indefinitely.

So if you are not happy with the process of listing first, second third and placing a term in 1 - 1 correspondence with each then try this thought experiment.

 

Take a perfect line, exactly 1 unit long

Using a perfect scalpel cut off an amount exactly equal to the fraction of the line given by your first term. Call this the first slice.

Then cut off a second slice exactly equal to your second term. call this your second slice.

 

and so on and so on

 

You know that all your slices started off adding up to exactly 1

But you will also find that you, or any being,  can never reach an end to this process.

 

Does this help?

 

By the way, you initially complained that your series had a start and an end, against how you perceived infinity, but you never responded to me showing you that there is nothing wrong with an infinite sequence having a start and going on indefinitely.

[wtf]

5 hours ago, studiot said:

My comment was not addressed to you and the reason why was answered in the next line.

 

Oops I'm terribly sorry. You were replying to BB. For some reason I got confused and thought that was BB replying to me. My bad. Nevermind whatever I wrote. 

[studiot]

1 hour ago, wtf said:

Oops I'm terribly sorry. You were replying to BB. For some reason I got confused and thought that was BB replying to me. My bad. Nevermind whatever I wrote. 

🙂

 

Hopefully we we wprk it out for BB in the end

 

I admire your steadfastness so far.

[Boltzmannbrain]

19 hours ago, Genady said:

But we know that it does not.

I was watching some professor on YouTube talking a little while ago (I only watch lectures from top universities), and he said something like "we never reach the limit but of course in some sense it does".  Now of course I cannot find the video.

Anyways, I also want to say that the definition of the infinite sum seems to imply that the sequence (2^n-1)/2^n reaches its limit.  If it doesn't reach its limit, then that is in conflict with the definition.

9 hours ago, studiot said:

I note that this still hasn't clicked for him despite all your proofs.

What proofs?  If there is a proof that the infinite sum of 1/2^n does not equal 1, then wouldn't that be proving the definition wrong?

 

7 hours ago, studiot said:

Take a perfect line, exactly 1 unit long

Using a perfect scalpel cut off an amount exactly equal to the fraction of the line given by your first term. Call this the first slice.

Then cut off a second slice exactly equal to your second term. call this your second slice.

 

and so on and so on

 

You know that all your slices started off adding up to exactly 1

But you will also find that you, or any being,  can never reach an end to this process.

 

Does this help?

No it doesn't.  As you have said, there is not someone cutting each slice one by one in a limited amount of time.  Clearly that wouldn't end.  What if all infinite (all n of N) slices were cut in, say, 1 second.  What would be left?  

 

Quote

By the way, you initially complained that your series had a start and an end, against how you perceived infinity, but you never responded to me showing you that there is nothing wrong with an infinite sequence having a start and going on indefinitely.

That is not an accurate interpretation of my OP.  My perception of infinite would not have an end or a final point.

[Genady]

36 minutes ago, Boltzmannbrain said:

I also want to say that the definition of the infinite sum seems to imply that the sequence (2^n-1)/2^n reaches its limit.  If it doesn't reach its limit, then that is in conflict with the definition.

The (2^n-1)/2^n does not reach 1 and the definition does not imply that it does.

[Boltzmannbrain]

28 minutes ago, Genady said:

The (2^n-1)/2^n does not reach 1 and the definition does not imply that it does.

How can that be?  The definition says that the infinite sum of 1/2^n = 1.  This would mean that the partial sums total 1. 

 

[Genady]

1 hour ago, Boltzmannbrain said:

How can that be?  The definition says that the infinite sum of 1/2^n = 1.  This would mean that the partial sums total 1. 

 

No, it does not mean that the partial sums total 1. It means that the limit of the partial sums is 1.

I emphasize again: partial sums do not total 1; limit of partial sums is 1.

[Boltzmannbrain]

11 minutes ago, Genady said:

No, it does not mean that the partial sums total 1. It means that the limit of the partial sums is 1.

I emphasize again: partial sums do not total 1; limit of partial sums is 1.

This is how I am getting that the partial sums would total 1.

(2^n-1)/2^n = sum of 1/2^n 

sum of 1/2^n (when n goes to infinity) = 1

-> (2^n-1)/2^n (when n goes to infinity) = 1

 

[Genady]

1 minute ago, Boltzmannbrain said:

This is how I am getting that the partial sums would total 1.

(2^n-1)/2^n = sum of 1/2^n 

sum of 1/2^n (when n goes to infinity) = 1

-> (2^n-1)/2^n (when n goes to infinity) = 1

 

That exactly means that the limit of the partial sums is 1. It does not mean that a partial sum is 1.

[Boltzmannbrain]

16 minutes ago, Genady said:

That exactly means that the limit of the partial sums is 1. It does not mean that a partial sum is 1.

I did not have to put a limit anywhere to get 1.  

[Genady]

1 minute ago, Boltzmannbrain said:

I did not have to put a limit anywhere to get 1.  

Yes, you did. That what "when n goes to infinity" means. It just another phrase for "limit".

[Boltzmannbrain]

4 hours ago, Genady said:

Yes, you did. That what "when n goes to infinity" means. It just another phrase for "limit".

Wiki just shows the infinite sum as being 1 without the limit notation, (at the top) 1/2 + 1/4 + 1/8 + 1/16 + ⋯ - Wikipedia

[Genady]

3 hours ago, Boltzmannbrain said:

Wiki just shows the infinite sum as being 1 without the limit notation, (at the top) 1/2 + 1/4 + 1/8 + 1/16 + ⋯ - Wikipedia

The "+ ..." is yet another way to indicate "limit" as they in your link explain:

              

[studiot]

15 hours ago, Boltzmannbrain said:

Quote

By the way, you initially complained that your series had a start and an end, against how you perceived infinity, but you never responded to me showing you that there is nothing wrong with an infinite sequence having a start and going on indefinitely.

That is not an accurate interpretation of my OP.  My perception of infinite would not have an end or a final point.

 

In what way is that not accurate ?

Your very first line introuces doubts about ends of infinity.

 

So I tried to deal with them one at a time

 

On 9/3/2023 at 9:46 PM, Boltzmannbrain said:

My new issue in my journey to try to understand infinity concerns the "ends" of infinity.

...........

I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1. 

 

To which I replied about the starting point

 

On 9/4/2023 at 12:10 PM, studiot said:

Yes you are considering the right things this time.

As regards the start point, even simple counting has a start point 1,2,3........

But it has no end point unless you run out of numbers to count., which of course you will not.

That is what is meant by infinity in this case.  (remember there are other cases)

 

Here I tried to point you at something I believe you already know.

Namely that there are infinite counting numbers and you can start from any of them.

 

Later posts deal with the end point because

Infinity is so strange that you can have an infinity between a definite start point and a definite end point;

or you can have a definite start point but no end point

or you can have a definite end point but no start point

or you can have neither end point nor start point.

 

Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about?

 

  

15 hours ago, Boltzmannbrain said:

Quote

By the way, you initially complained that your series had a start and an end, against how you perceived infinity, but you never responded to me showing you that there is nothing wrong with an infinite sequence having a start and going on indefinitely.

That is not an accurate interpretation of my OP.  My perception of infinite would not have an end or a final point.

 

In what way is that not accureate ?

Your very first line introuces doubts about ends of infinity.

 

So I tried to deal with them one at a time

 

On 9/3/2023 at 9:46 PM, Boltzmannbrain said:

My new issue in my journey to try to understand infinity concerns the "ends" of infinity.

...........

I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1. 

 

To which I replied about the starting point

 

On 9/4/2023 at 12:10 PM, studiot said:

Yes you are considering the right things this time.

As regards the start point, even simple counting has a start point 1,2,3........

But it has no end point unless you run out of numbers to count., which of course you will not.

That is what is meant by infinity in this case.  (remember there are other cases)

 

Here I tried to point you at something I believe you already know.

Namely that there are infinite counting numbers and you can start from any of them.

 

Later posts deal with the end point because

Infinity is so strange that you can have an infinity between a definite start point and a definite end point;

or you can have a definite start point but no end point

or you can have a definite end point but no start point

or you can have neither end point nor start point.

 

Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about?

On 9/3/2023 at 9:46 PM, Boltzmannbrain said:

What could n possible equal to get to this point?

If this last point really is a solution to the equation, then wouldn't it have to be 1/infinity, or in  other words, the "infinity-ith" point?  If so, how can it be said that the natural numbers can numerate all points of a set of size aleph-null? 

 

Edited September 7, 2023 by studiot

[Boltzmannbrain]

10 hours ago, Genady said:

The "+ ..." is yet another way to indicate "limit" as they in your link explain:

              

It says "the sum", not the limit.  

It also shows more examples of it being the sum and not the limit.  

 

Here is another example,

"When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted S, it can be seen that

 

Therefore,

 

[Genady]

They clearly say that the sum of infinite series is defined to mean the limit. Everywhere when you see something + ..., it means the limit.

[Boltzmannbrain]

6 hours ago, studiot said:

 

In what way is that not accurate ?

 

You skipped a very important part.  If you want to understand me, you have to include the quote that you did not include in your last post.

"I was told on here that the infinite sum of 1/2^n = 1, and not just gets close but actually equals 1." 

And then I said,

"I can't help but notice that we are giving infinity a definite beginning point at 1/2 and a definite end point at 1."

4 minutes ago, Genady said:

They clearly say that the sum of infinite series is defined to mean the limit. Everywhere when you see something + ..., it means the limit.

Well, when they swap the word "mean" with a equals sign, I take that to mean "equals" also.  

[Boltzmannbrain]

For anyone interested, I did some digging around and found that the infinite sum of 1/2^n is actually 2, and not just arbitrarily close to 2.

[phyti]

Boltzmannbrain;

s=Σ (1/2n), for n=1 to ∞

s is a series of partial sums, a variable and a geometric progression for this example.

The sum s will converge toward a limit.

The limit of s is a constant c=1, also denoted as an asymptote.

s approaches c, but by definition of 'infinite' (without limit/last term),

never equals s.

s=(1/2+1/4+1/8+...+1/2n)

for n terms.

2s-1=(1/2+1/4+1/8+...+1/2n-1)=s-1/2n

for n terms.

s=1-1/2n

1/2n>0 for all n.

For large values of n, 1/2n is judged insignificant for the purpose

and the limit is used to represent the value of s.

This would also apply to the series .999R converging to its limit 1.