Deeper than primes

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doronshadmi said:
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.
Click to expand...

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.
 
Astrodude said:
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.
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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
 
doronshadmi said:
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.
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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.
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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.
 
zooterkin said:
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).
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zooterkin,

"gap" is "distance > 0" is "1-dimensional space".
 
jsfisher said:
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.
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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?
 
doronshadmi said:
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.
Click to expand...
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 = ?
 
The Man said:
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.
Click to expand...

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?
 
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epix said:
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 = ?
Click to expand...

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 ...
 
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doronshadmi said:
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 ...
Click to expand...
Now, how about...

3-dim = gallon
2-dim = acre
1-dim = mile
0-dim = ?
 
HatRack said:
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.
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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.
 
epix said:
What do you mean by "finite sum?"
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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.

epix said:
as such the infinite sum is always less than 1 as n approaches infinity.
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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.
 
doronshadmi said:
Don't bother laca, you don't have it.
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Allow me to bother nonetheless.

doronshadmi said:
Laca your applied "traditional" math does not deal with Infinity.
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Sorry, must have been confused by all the talk about infinity during my classes.

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

doronshadmi said:
This beauty is clearly beyond your mind.
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Oh, how cute! Beauty is the color teal and comic sans! Who knew!
 
doronshadmi said:
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.
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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.

doronshadmi said:
I think that there is no cure in your case.
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Projection, much?
 
doronshadmi said:
By bottom to top (micro) reasoning two distinct points have no gap between them, if the line is completely covered by points.
Click to expand...

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

doronshadmi said:
So where is the formal proof that supports your assertion?
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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.
 
Astrodude said:
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.
Click to expand...
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.

Astrodude said:
... which pretty much everyone does on the blackboard I think.
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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.
 
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laca said:
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.
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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.
 
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doronshadmi said:
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.
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If there is no gap between them, then the points are not distinct.
 
zooterkin said:
If there is no gap between them, then the points are not distinct.
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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 :popcorn1
 
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HatRack said:
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.
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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.
 
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doronshadmi said:
No laca, try to read for comprehension.
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What are you, 8?

doronshadmi said:
You actually assert that 1-dim space is actually a collection of 0-dim spaces.
Click to expand...

Well, yes I am.

doronshadmi said:
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).
Click to expand...

No I don't. There is no gap between two distinct points on a line. There's a distance between them. All covered with, guess what, points.

doronshadmi said:
So please provide the concrete proof about your assertion that a 1-dim space is constructable by a collection of 0-dim spaces.
Click to expand...

See the definition of a line, uncountable infinity, set of real numbers to get you started.
 
doronshadmi said:
Exactly!
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If you get over your gotcha moment you will realize that by gap he meant distance. No gap where there are no points. Sorry doron, nobody agrees with you.

doronshadmi said:
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.
Click to expand...

No. A line (I assume that's what you mean by 1-dim space) is by definition made up of points and nothing but points.

doronshadmi said:
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.
Click to expand...

Define gap please, since apparently your language skills are not much better than your math skills.
 
laca said:
If you get over your gotcha moment you will realize that by gap he meant distance. No gap where there are no points. Sorry doron, nobody agrees with you.
Click to expand...

Yes, indeed. "Gap" is a word which does not have a strict mathematical definition (that I'm aware of, anyway), and is sometimes being used to mean "space between two points (which is covered in points)" and sometimes to mean "space between two points where there are no points". I think, to most people, it is clear what is meant in each case from the context. It's not usually a problem, since the latter meaning refers to something which doesn't happen, except in Doron's fevered imagination.
 
laca said:
There is no gap between two distinct points on a line. There's a distance between them. All covered with, guess what, points.
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Since "gap" and "distance > 0" is exactly the same thing in this case, that your assertion is this:

By your reasoning there is exactly 0 distance between two distinct points along a 1-dim space, because by your reasoning 1-dim space is actually a collection of 0-dim spaces, exactly because you assert that a 1-dim space is completely covered by distinct 0-dim spaces.
 
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zooterkin said:
Yes, indeed. "Gap" is a word which does not have a strict mathematical definition (that I'm aware of, anyway), and is sometimes being used to mean "space between two points (which is covered in points)" and sometimes to mean "space between two points where there are no points". I think, to most people, it is clear what is meant in each case from the context. It's not usually a problem, since the latter meaning refers to something which doesn't happen, except in Doron's fevered imagination.
Click to expand...
You are missing it again, zooterkin.

I use "Gap" as "distance > 0".

By your reasoning there is exactly 0 distance between two distinct points along a 1-dim space, because by your reasoning 1-dim space is actually a collection of 0-dim spaces, exactly because you assert that a 1-dim space is completely covered by distinct 0-dim spaces.
 
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doronshadmi said:
Since "gap" and "distance > 0" is exactly the same thing in this case, that your assertion is this:

By your reasoning there is exactly 0 distance between two distinct points along a 1-dim space, because by your reasoning 1-dim space is actually a collection of 0-dim spaces, exactly because you assert that a 1-dim space is completely covered by distinct 0-dim spaces.
Click to expand...

Doron, nobody is arguing that the distance between two distinct points is zero. That is your strawman. Stop it.

By "my" reasoning, between two distinct points there is a non-zero distance. That is what distinct means. Now, between any two distinct points on a line there are just as many points as on the line itself. Are you able to comprehend that? I think this is why 1-dim can be a "covered" by 0-dim. Not only 1-dim, by the way, any finite n-dim can too. Somebody correct me if I'm wrong.

There is really not much else that can be said.
 
laca said:
By "my" reasoning, between two distinct points there is a non-zero distance
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No laca, Since you assert that 1-dim space is completely covered by distinct 0-dim spaces, then by this assertion there is exactly 0 distance between two distinct 0-dim spaces along a 1-dim space.
 
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doronshadmi said:
No laca, Since you assert that 1-dim space is completely covered by distinct 0-dim spaces, then by this assertion there is exactly 0 distance between two distinct 0-dim spaces along a 1-dim space.
Click to expand...

If there is zero distance between two points, then they are the same point, and not distinct.
 
doronshadmi said:
No laca, Since you assert that 1-dim space is completely covered by distinct 0-dim spaces, then by this assertion there is exactly 0 distance between two distinct 0-dim spaces along a 1-dim space.
Click to expand...

No doron. I asked you to stop with the strawman. Two distinct points cannot have 0 distance between them, as this would contradict them being distinct in the first place.
 
laca said:
No doron. I asked you to stop with the strawman. Two distinct points cannot have 0 distance between them, as this would contradict them being distinct in the first place.
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Laca, please answer only by yes or no to the following question:

Do you claim that a 1-dimensional space is completely covered by distinct 0-dim spaces (be careful before you reply, because if you claim that 1-dimensional space is completely covered by distinct 0-dimensional spaces, then you actually assert that given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them)?
 
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laca said:
See the definition of a line, uncountable infinity, set of real numbers to get you started.
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Countable or Uncountable collocations of infinitely many objects, in both cases we deal with collections of distinct objects.
 
doronshadmi said:
Laca, please answer only by yes or no to the following question:

Do you claim that a 1-dimensional space is completely covered by distinct 0-dim spaces?
Click to expand...

YES

doronshadmi said:
(be careful before you reply, because if you claim that 1-dimensional space is completely covered by distinct 0-dimensional spaces, then you actually assert that given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them)
Click to expand...

NO

Stop with the strawman already. Nowhere have I, or anyone else for that matter, asserted anything even remotely like that. It does not follow from the premises. It is self-contradictory. If I claim that a line can be completely covered by points, I claim exactly that. The equivalent of that is that there is no "place" on a line where there is no point. This is what I'm asserting. It is not equivalent to what you are saying. If you feel it is, feel free to prove so. Until you do, the two positions are not equivalent and I would like to ask you for the umpteenth time to show some shred of human decency and stop misrepresenting our position.
 
zooterkin said:
If there is zero distance between two points, then they are the same point, and not distinct.
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zooterkin, please answer only by yes or no to the following question:

Do you claim that a 1-dimensional space is completely covered by distinct 0-dim spaces (be careful before you reply, because if you claim that 1-dimensional space is completely covered by distinct 0-dimensional spaces, then you actually assert that given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them)?
 
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doronshadmi said:
Countable or Uncountable collocations of infinitely many objects, in both cases we deal with collections of distinct objects.
Click to expand...

Are you equating countable and uncountable "collocations"? Whatever those might be, you are simply wrong.
 
doronshadmi said:
zooterki, please answer only by yes or no to the following question:

Do you claim that a 1-dimensional space is completely covered by distinct 0-dim spaces (be careful before you reply, because if you claim that 1-dimensional space is completely covered by distinct 0-dimensional spaces, then you actually assert that given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them)?
Click to expand...

Yes, doron, we already know you are capable of parroting the same nonsense ad infinitum. You can stop. Several posters have confirmed that yes, that is the claim. The claim is not equivalent to the nonsensical gibberish you keep demanding proof of.
 
laca said:
YES
Click to expand...

laca said:
The equivalent of that is that there is no "place" on a line where there is no point.
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Since all places along 1-dim space are covered by distinct 0-dimensional space, and since you agree with the following:
laca said:
doronshadmi said:
You actually assert that 1-dim space is actually a collection of 0-dim spaces.
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Well, yes I am.
Click to expand...

Your YES has one and only one result which is:

Given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them.

CASE CLOSED.
 
doronshadmi said:
Since all places along 1-dim space are covered by distinct 0-dimensional space, and since you agree with the following:


Your YES has one and only one result which is:

Given two 0-dimensional spaces along 1-dimensional space, they are both distinct AND have 0 distance between them.
Click to expand...

Please show how that is equivalent with my assertion. Unless you can do so, you have no right to equate the two positions. I'm going to ask you once more to stop the strawmanning. Thank you.
 
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