Deeper than primes

[The Man]

It does not work that way The Man.

This time please read all of http://www.internationalskeptics.com/forums/showpost.php?p=6577818&postcount=12392 any try to reply to each part of it, in details.

If you can't do that, we have a concrete proof of your inability to deal with this fine subject.

You claimed “the most right (Red or Blue) segment must be reduced into a single 0-dimensional space” for your purported “proof” if you can’t even show that then we have concrete proof that you have no poof for your purported “proof”.

That’s not even the worst of it Doron, and we have been over this before. Your claim is that in order for the limit to be reached it requires the addition of a zero length segment in the series. However, as already shown to you before (even though it should have been trivially obvious to you before) the addition of the previous series sum and a zero length segment equaling the limit means that the previous series sum already equaled the limit. So your whole premise of not reaching the limit without adding a zero length segment actually requires the series to have already reached the limit before the addition of your zero length segment. Your so called “proof” explicitly requires the refutation of what you claim you’re trying to prove, that the limit is not reached.

 

[The Man]

I think confronting that reasoning put Cantor in a mental institution, though I'm by no means an authority.

Infinitesimals are perfectly legitimate based on Robinson's hyperreal set as long as you use Zermelo-Fraenkel set theory which pretty much everyone does on the blackboard I think.

However, in real life, we can't deal with actual infinity so we just adopt Kroneker's intuitionism and Gauss' views usually. Would you actually ever contemplate slicing a three dimensional sandwich into an infinite number of two dimensional cross sections?

You seem smart, but you're fatigued and bewildered by a Cantorian view. Avoid his fate and adopt Constructivism.

Well put Astrodude, welcome to the forum and the thread. Unfortunately, I fear Doron has invested so much time and effort in his endeavor, not to mention staking the continued existence of our civilization on it as he does, that there may be no turning back for him now.



Welcome to the forum and thread as well HatRack

 

[jsfisher]

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.

You continue to ascribe that as our position, Doron. It is not. Please stop with these silly assertions.

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.

That is your fallacy, not ours. As has been stated repeatedly, but only because you persist in ignoring it, between any two distinct points along a line are an infinite supply of more points.

 

[doronshadmi]

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).

zooterkin,

"gap" is "distance > 0" is "1-dimensional space".

 

[doronshadmi]

You continue to ascribe that as our position, Doron. It is not. Please stop with these silly assertions.



That is your fallacy, not ours. As has been stated repeatedly, but only because you persist in ignoring it, between any two distinct points along a line are an infinite supply of more points.

No jsfisher, your reasoning is only a top to bottom (macro) notion of this fine subject.

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?

 

[epix]

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.

Doron, you've buried yourself under a pile of misconceptions about points. There is a gap between "points" Joe and Bill until Mary steps between them. Nothing is "covered by points." This terminology is misleading. See, there is a difference between "Mary" and "someone." Can you complete this?

3-dim = volume
2-dim = area
1-dim = length
0-dim = ?

 

[doronshadmi]

You claimed “the most right (Red or Blue) segment must be reduced into a single 0-dimensional space” for your purported “proof” if you can’t even show that then we have concrete proof that you have no poof for your purported “proof”.

That’s not even the worst of it Doron, and we have been over this before. Your claim is that in order for the limit to be reached it requires the addition of a zero length segment in the series. However, as already shown to you before (even though it should have been trivially obvious to you before) the addition of the previous series sum and a zero length segment equaling the limit means that the previous series sum already equaled the limit. So your whole premise of not reaching the limit without adding a zero length segment actually requires the series to have already reached the limit before the addition of your zero length segment. Your so called “proof” explicitly requires the refutation of what you claim you’re trying to prove, that the limit is not reached.

In other words, you are unable to reply in details to each part of http://www.internationalskeptics.com/forums/showpost.php?p=6577818&postcount=12392 .

Since you can't do it, at least try to reply in details to this:

The Man, your reasoning is only a top to bottom (macro) notion of this fine subject.

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]

Doron, you've buried yourself under a pile of misconceptions about points. There is a gap between "points" Joe and Bill until Mary steps between them. Nothing is "covered by points." This terminology is misleading. See, there is a difference between "Mary" and "someone." Can you complete this?

3-dim = volume
2-dim = area
1-dim = length
0-dim = ?

3-dim = volume
2-dim = area
1-dim = length
0-dim = locality

Emptiness is strictly below collections.

Fulness is strictly above collections.

1-dim is the space that is strictly above collection of 0-dim spaces w.r.t it.

2-dim is the space that is strictly above collection of 0-dim and 1-dim spaces w.r.t it.

3-dim is the space that is strictly above collection of 0-dim, 1-dim and 2-dim spaces w.r.t it.

... etc ...

 

[epix]

3-dim = volume
2-dim = area
1-dim = length
0-dim = locality

Emptiness is strictly below collections.

Fulness is strictly above collections.

1-dim is the space that is strictly above collection of 0-dim spaces w.r.t it.

2-dim is the space that is strictly above collection of 0-dim and 1-dim spaces w.r.t it.

3-dim is the space that is strictly above collection of 0-dim, 1-dim and 2-dim spaces w.r.t it.

... etc ...

Now, how about...

3-dim = gallon
2-dim = acre
1-dim = mile
0-dim = ?

 

[epix]

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.

What do you mean by "finite sum?" Setting the limit to 1/2ⁿ doesn't cancel n → ∞ for the whole series, it merely cancels it in the final part of the computation of the limit.

The expression 1/2 + 1/4 + 1/8 + 1/16 + ... has its equivalent in 1 - 1/2ⁿ where n → ∞. The equivalent doesn't specifically ask for any limit and as such the infinite sum is always less than 1 as n approaches infinity. The proof is simple

[lim n → ∞] 1 - 1/2ⁿ = 1.

1 is the limit of the infinite sum.

 

[doronshadmi]

Now, how about...

3-dim = gallon
2-dim = acre
1-dim = mile
0-dim = ?

3-dim = gallon
2-dim = acre
1-dim = mile
0-dim = dot

 

[HatRack]

What do you mean by "finite sum?"

The notation 1/2 + 1/4 + 1/8 + 1/16 + ... is, loosely speaking, alternative notation for the limit as n approaches infinity of the finite sum of the series 1/2 + 1/4 + ... + 1/2^n. This finite sum is of course 1 - 1/2^n. Hence, 1/2 + 1/4 + 1/8 + ... = lim n->inf (1 - 1/2^n) = 1. Doronshadmi's assertion that 1/2 + 1/4 + 1/8 + ... < 1 is complete nonsense for that reason. For any natural number n, it is true that 1/2 + 1/4 + 1/8 + ... + 1/2^n < 1. But, having a "+ ..." at the end indicates that the limit is being taken.

as such the infinite sum is always less than 1 as n approaches infinity.

No. The partial sums of the series are always less than 1 for any given natural number n. The infinite sum is defined as the limit of the sequence of partial sums, and the value of that limit is 1 in this particular case.

 

[laca]

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

Allow me to bother nonetheless.

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

Sorry, must have been confused by all the talk about infinity during my classes.

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.

I see you mastered the spontaneous generation of gibberish. Kudos!

This beauty is clearly beyond your mind.

Oh, how cute! Beauty is the color teal and comic sans! Who knew!

 

[laca]

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.

No doron, try to read for comprehension. What I clearly said is that between any two distinct points there are only points. Just as many points as on the whole line. No gap. Are you suffering from some kind of condition that makes you skip words in sentences? Let me try again in caps: NO GAPS. ALL POINTS.

I think that there is no cure in your case.

Projection, much?

 

[laca]

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

Of course they don't have gaps between them. You what they have instead? More points. Magnificent, isn't it?

So where is the formal proof that supports your assertion?

The formal proof is trivial and left as an exercise for sixth graders. You may try too doron, since you're special. I'll even give you some pointers, in case you get stuck.

Start with the definition of a line and assume that there is a gap on it somewhere. If you get to a contradiction, then your assumption is false, leaving you with the formal proof that lines do not have gaps on them.

 

[doronshadmi]

I think confronting that reasoning put Cantor in a mental institution, though I'm by no means an authority.

Infinitesimals are perfectly legitimate based on Robinson's hyperreal set as long as you use Zermelo-Fraenkel set theory which pretty much everyone does on the blackboard I think.

However, in real life, we can't deal with actual infinity so we just adopt Kroneker's intuitionism and Gauss' views usually. Would you actually ever contemplate slicing a three dimensional sandwich into an infinite number of two dimensional cross sections?

You seem smart, but you're fatigued and bewildered by a Cantorian view. Avoid his fate and adopt Constructivism.

Welcome Astrodude,

Cantor did not deal with actual infinity, because the very notion of collection (which is the framework of Cantor's actual infinity) is strictly below actual infinity, where actual infinity is not less than the non-locality of 1-dim space w.r.t any given collection of 0-dim spaces along it, such that given an arbitrary 0-dim space along 1-dim space, 1-dim space is at AND beyond the location of the considered 0-dim space.

Traditional Math, which is mostly based on Hilbert's Formalism, asserts that a 1-dim space is completely covered by a collection of 0-dim spaces.

If you assert that a given line is completely covered by points (as Traditional Math asserts), then you must prove this 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.

... which pretty much everyone does on the blackboard I think.

By using this screen as your blackboard, please use Standard or Non-standard Analysis, in order to prove this assertion by using bottom to top (micro) reasoning.

 

[doronshadmi]

No doron, try to read for comprehension. What I clearly said is that between any two distinct points there are only points. Just as many points as on the whole line.

No laca, try to read for comprehension.

You actually assert that 1-dim space is actually a collection of 0-dim spaces.

In that case you have to provide the proof that two points are both distinct AND there is no gap between them AND they are able to construct a 1-dim space (a line).

So please provide the concrete proof about your assertion that a 1-dim space is constructable by a collection of 0-dim spaces.

 

[zooterkin]

No jsfisher, your reasoning is only a top to bottom (macro) notion of this fine subject.

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.

If there is no gap between them, then the points are not distinct.

 

[doronshadmi]

If there is no gap between them, then the points are not distinct.

Exactly!

But since Traditional Math asserts that a 1-dim space is completely covered by a collection of distinct 0-dim spaces, then (according to Traditional Math) two points along a 1-dim space are both distinct AND there is no gap between them.

In that case Traditional Math has to provide the formal proof, which is resulted by the ability of two 0-dim spaces along a 1-dim space, to be both distinct AND gap-less.

Let's see how Traditional Math formally proves it

 

[doronshadmi]

Doronshadmi's assertion that 1/2 + 1/4 + 1/8 + ... < 1 is complete nonsense for that reason. For any natural number n, it is true that 1/2 + 1/4 + 1/8 + ... + 1/2^n < 1. But, having a "+ ..." at the end indicates that the limit is being taken.

Welcome HatRack,

Do you assert that a 1-dimensional space is completely covered by a collection of 0-dimensional spaces?

Please answer only by yes or no.