Deeper than primes

[epix]

For finite members, yes. For infinite, it's exactly 1. I'm sorry you can't grasp infinity doron, but that's not really our problem, now is it?

The problem is that for every n that belongs to N where n → ∞, aⁿ/aⁿ = 1 with the following implication:

Since 1/2+1/4+1/8+1/16+1/32+1/64+ ... = (2ⁿ - 1)/2ⁿ where n → ∞, the numerator would be always less 1 than the denominator and therefore the sum of the series cannot reach it's limit 1 as n goes to infinity. We are no longer talking finite or infinite members, coz the series is identical to function f(x) = (2^x - 1)/2^x, but we are talking the difference between

1) f(x) = (2^x - 1)/2^x for 1 ≤ x < ∞

and

2) f(x) = (2^x - 1)/2^x for x → ∞

Since

(2^x - 1)/2^x = 2^x/2^x - 1/2^x
it follows that

2^x/2^x - 1/2^x = 1 - 1/2^x
and again, f(x) cannot reach its limit 1, unless

[x → ∞] 1/2^x = 0

But that would contradict the existence of the limit

[lim x → ∞] N/x = 0 for any real N greater than 0.

And so, the traditional math must prove that absolute convergent series can reach their limits, or at least "claim it" somewhere.

 

[epix]

Furthermore, persons like The Man and jsfisher have to explicitly demonstrate (by using bottom to top techniques) that two distinct points define a line segment, such that there is no gap between these two points.

Lemmetrylemmetry...

A. .B






A__B

Aha. Thanks, Doron.

 

[jsfisher]

For all epsilon that are greater than zero and belong to no particular set, there exists N that belongs to no particular set as well, but is greater than n and therefore epsilon is greater than the absolute value of 1 minus the given sum.


Can you post a link to that implication?

For any arbitrarily small neighborhood (epsilon) of 1, the sequence of partial sums (1/2+1/4+1/8+...+1/2^n) eventually (n>N) stays within that neighborhood (|1-sum(...)|).

It expresses formally what it means to say:

[latex]$\displaystyle 1 = \lim_{n \rightarrow \infty} \sum_{i=1}^n {1 \over 2^i}$[/latex]

ETA: For completeness, I probably should have indicated epsilon was a real number and N, an integer.​

Since Doron continues to pretend he knows what "traditional mathematics says", I thought I'd give him something closer to what is actually said.

 

[laca]

No Laca, you are using two points which exist along different lines of a given plan.

I didn't express myself clearly. I was talking about coordinates, not points. Sorry.

I am talking about different points along the same line.

The Man and jsfisher claim that this single line is completely covered by points.

Yes, me too.

In that case they have no choice but to formally prove that there exist two distinct points along the same line, whiteout any gap between them.

Let's see how they are formally prove it

No, all anyone has to prove is that there are no gaps on any line. In order to have gaps on a line, there has to be an x coordinate for which there is no corresponding point on the line. There is no such coordinate. Hence, no gaps on lines. You fail. It's not our problem you can't grasp uncountable infinity.

 

[jsfisher]

jsfisher http://www.internationalskeptics.com/forums/showpost.php?p=6577818&postcount=12392 is different.

Not in any substantive way. You still weave in your misrepresentations and bogus premises. In that regard, it is the same.

 

[jsfisher]

Furthermore, persons like The Man and jsfisher have to explicitly demonstrate (by using bottom to top techniques) that two distinct points define a line segment, such that there is no gap between these two points.

Why would we do such a nonsensical thing?

According to their reasoning, such a thing is possible, because they claim that a line segment is completely covered by points, which means that there is no gap between these two points along a given line segment.

Were Aristotle alive today, he'd rap you on the head with a ruler for that bogus bit of logic. Between any two points along a line there are...wait for it...wait for it...more points. Is that really so hard for you to comprehend?

 

[doronshadmi]

Were Aristotle alive today, he'd rap you on the head with a ruler for that bogus bit of logic. Between any two points along a line there are...wait for it...wait for it...more points. Is that really so hard for you to comprehend?

This is a top to bottom (macro) reasoning jsfisher.

Since you assert that a given line is completely covered by points, then you must prove your assertion by bottom to top (micro) reasoning.

By bottom to top (micro) reasoning two distinct points have no gap between them, if the line is completely covered by points.

So where is the formal proof that supports your assertion?

 

[doronshadmi]

In order to have gaps on a line, there has to be an x coordinate for which there is no corresponding point on the line.

You are missing it laca.

Please prove how two points along the same line are distinct AND there is no gap between them.

 

[zooterkin]

This is a top to bottom (macro) reasoning jsfisher.

Since you assert that a given line is completely covered by points, then you must prove your assertion by bottom to top (micro) reasoning.

By bottom to top (micro) reasoning two distinct points have no gap between them, if the line is completely covered by points.

I think your English is letting you down again. What are you meaning by 'gap'? You're the one who needs to show that there is a gap, in the sense of a place on the line where there is no point, i.e. a gap between points.

 

[epix]

For any arbitrarily small neighborhood (epsilon) of 1, the sequence of partial sums (1/2+1/4+1/8+...+1/2^n) eventually (n>N) stays within that neighborhood (|1-sum(...)|).

It expresses formally what it means to say:

[latex]$\displaystyle 1 = \lim_{n \rightarrow \infty} \sum_{i=1}^n {1 \over 2^i}$[/latex]

ETA: For completeness, I probably should have indicated epsilon was a real number and N, an integer.​

Since Doron continues to pretend he knows what "traditional mathematics says", I thought I'd give him something closer to what is actually said.

I think that he is accepting the existence of the limit L=1, but he feels that the traditional math speaks through Wiki where the series in question can actually reach its limit:
http://en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_·_·_·

So I side with him: no way that the series -- the sum of the sequence -- in question can reach its limit 1. But I don't go by his reasoning, which is again nothing but intuitive -- you know what I mean.

 

[doronshadmi]

I think your English is letting you down again. What are you meaning by 'gap'? You're the one who needs to show that there is a gap, in the sense of a place on the line where there is no point, i.e. a gap between points.

No zooterkin, since you assert that a given line is completly covered by points, then you have no choice but to prove that two points along the same line are distinct AND there is no gap between them.

 

[doronshadmi]

I think that he is accepting the existence of the limit L=1, but he feels that the traditional math speaks through Wiki where the series in question can actually reach its limit:
http://en.wikipedia.org/wiki/1/2_+_1/4_+_1/8_+_1/16_+_·_·_·

So I side with him: no way that the series -- the sum of the sequence -- in question can reach its limit 1. But I don't go by his reasoning, which is again nothing but intuitive -- you know what I mean.

You are wrong epix, it is both supported by Logic AND intuition under a one framework, such that a given point is logically at XOR not at a given location w.r.t a given line, and a line is at AND not at a given location w.r.t a given point.

In other words, there is an essential logical difference between being a point and being a line, which "goes hand by hand" with the intuition that an infinite collection of distinct points can't completely cover a given line.

 

[jsfisher]

This is a top to bottom (macro) reasoning jsfisher.

Since you assert that a given line is completely covered by points, then you must prove your assertion by bottom to top (micro) reasoning.

By bottom to top (micro) reasoning two distinct points have no gap between them, if the line is completely covered by points.

So where is the formal proof that supports your assertion?

You logic continues to fail you. Your conclusion does not follow from the premise. While a line is completely covered by points, leaving no gap on the line not covered by points, it is also true that between any two distinct points there is line segment (and that segment is completely covered by...wait for it...wait for it...more points).

 

[laca]

You are missing it laca.

Please prove how two points along the same line are distinct AND there is no gap between them.

I don't have to. I just proved that there is no gap on any line. As jsfisher already said, between any two distinct points on a line there are more points. Just as many as on the whole line. Funky, isn't it? Too bad you will most probably never be able to get it. But don't worry. You don't have to. You are already enjoying the results of applied "traditional" math. That's the beauty of it. It just works.

 

[doronshadmi]

Ok, jsfisher and laca,

It is clear know as a high noon sun that your reasoning is so weak until it has no ability to get the fallacy of the assertion that two points along a given line are distinct of each other AND also there is no gap between them.

The reason that there is no gap between them is derived from the assertion that a line is completely covered by points, which is exactly the assertion of Traditional Mathematics.

Your pathetically weak reasoning is going to rest in peace very soon.

 

[laca]

Ok, jsfisher and laca,

It is clear know as a high noon sun that your reasoning is so weak until it has no ability to get the fallacy of the assertion that two points along a given line are distinct of each other AND also there is no gap between them.

No. We specifically stated that between them there are more points. An infinite number of points. So many, that one can actually build a function to generate a whole line, space or whatever. No gap. None whatsoever. It's all in your imagination doron. Take your meds.

The reason that there is no gap between them is derived from the assertion that a line is completely covered by points, which is exactly the assertion of Traditional Mathematics.

The line is completely covered by points, true. This means that there is no gap, instead, where you expect to see a gap, there'll be more points. Quite amazing, isn't it?

Your pathetically weak reasoning is going to rest in peace very soon.

I don't know about jsfisher, but I can accept that my reasoning is pathetically weak. Compared to real mathematicians and scientists, that is. But hey, at least I actually have some kind of reasoning. You got diddly squat.

 

[doronshadmi]

No. We specifically stated that between them there are more points. An infinite number of points. So many, that one can actually build a function to generate a whole line, space or whatever. No gap. None whatsoever. It's all in your imagination doron. Take your meds.



The line is completely covered by points, true. This means that there is no gap, instead, where you expect to see a gap, there'll be more points. Quite amazing, isn't it?



I don't know about jsfisher, but I can accept that my reasoning is pathetically weak. Compared to real mathematicians and scientists, that is. But hey, at least I actually have some kind of reasoning. You got diddly squat.

Don't bother laca, you don't have it.

But don't worry. You don't have to. You are already enjoying the results of applied "traditional" math. That's the beauty of it. It just works.

Laca your applied "traditional" math does not deal with Infinity.

In order to really deal with Infinity, the opposite of Emptiness (that has cardinality 0), which is Fullness (that has cardinality ∞), must be a part of the Mathematical science.

Only then it is realized that any given collection is incomplete w.r.t Fullness, such that the cardinality of any given collection < ∞ exactly because it is incomplete w.r.t Fullness.

1-dimensional space is the minimal form of Fullness, such that it is at AND not at a given location of any arbitrary 0-dimensional space along it.

This beauty is clearly beyond your mind.

 

[doronshadmi]

No. We specifically stated that between them there are more points. An infinite number of points. So many, that one can actually build a function to generate a whole line, space or whatever. No gap. None whatsoever. It's all in your imagination doron. Take your meds.

So by your "reasoning", between given two points there is a gap for an infinite number of points AND also there is no gap between the given two points. I think that there is no cure in your case.

 

[zooterkin]

So by your "reasoning", between given two points there is a gap for an infinite number of points AND also there is no gap between the given two points. I think that there is no cure in your case.

Yep, it's your English that is letting you down. By "gap", we are meaning a space between two points where there are no other points, not the distance between two arbitrary points (which is filled with points).

 

[HatRack]

As a result 1/2+1/4+1/8+1/16+1/32+1/64+ ... < 1.

No, that is incorrect. The notation 1/2 + 1/4 + 1/8 + 1/16 + ... is defined as

[latex]\lim_{n \to \infty}\displaystyle\sum\limits_{i=1}^n \frac{1}{2^i}[/latex]

The finite sum of this type of series is well known, and easily provable. In fact,

[latex]\lim_{n \to \infty}\displaystyle\sum\limits_{i=1}^n \frac{1}{2^i}=\lim_{n \to \infty}(\frac{1-(1/2)^{n+1}}{1/2}-1)=\lim_{n \to \infty}(2-\frac{1}{2^n}-1)=1[/latex]

Every theorem used in this proof is ultimately deducible from nothing more than the Peano Axioms along with some elementary set theory. If you have a problem with one of the basic axioms from which every property of the real line can ultimately be derived, including this one, then you need to clearly and precisely state which axiom is wrong.