Deeper than primes

[laca]

No laca, try to read for comprehension.

What are you, 8?

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

Well, yes I am.

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).

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.

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

See the definition of a line, uncountable infinity, set of real numbers to get you started.

 

[laca]

Exactly!

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.

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.

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.

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.

Define gap please, since apparently your language skills are not much better than your math skills.

 

[zooterkin]

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.

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.

 

[doronshadmi]

There is no gap between two distinct points on a line. There's a distance between them. All covered with, guess what, points.

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.

 

[doronshadmi]

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.

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.

 

[laca]

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.

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.

 

[doronshadmi]

By "my" reasoning, between two distinct points there is a non-zero distance

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.

 

[zooterkin]

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.

If there is zero distance between two points, then they are the same point, and not distinct.

 

[laca]

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.

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.

 

[doronshadmi]

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, 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)?

 

[doronshadmi]

See the definition of a line, uncountable infinity, set of real numbers to get you started.

Countable or Uncountable collocations of infinitely many objects, in both cases we deal with collections of distinct objects.

 

[laca]

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?

YES

(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)

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.

 

[doronshadmi]

If there is zero distance between two points, then they are the same point, and not distinct.

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)?

 

[laca]

Countable or Uncountable collocations of infinitely many objects, in both cases we deal with collections of distinct objects.

Are you equating countable and uncountable "collocations"? Whatever those might be, you are simply wrong.

 

[laca]

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)?

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.

 

[doronshadmi]

YES

The equivalent of that is that there is no "place" on a line where there is no point.

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

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

Well, yes I am.

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]

Are you equating countable and uncountable "collocations"? Whatever those might be, you are simply wrong.

laca, a non-empty set has distinct members, whether it is countable or uncountable.

 

[laca]

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.

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.

 

[laca]

laca, a non-empty set has distinct members, whether it is countable or uncountable.

Yes. So what? This might seem like some great idea to you but it is fairly self-evident to thinking people.

 

[doronshadmi]

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.

All we have is to combine your two "yes" answers in post http://www.internationalskeptics.com/forums/showpost.php?p=6584210&postcount=12476 .

CASE IS CLOSED.