Deeper than primes - Continuation

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zooterkin said:
What units do you measure pi in? Are you familiar with what a ratio is?
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It does not matter, pi is the invariant result of circumference/diameter, where circumference and diameter sizes are > 0 AND < ∞, no matter what measurement unit is used (except that the unit is some distance on the real-line w.r.t distance 1, Where Distance can be considered as a physical concept).
 
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doronshadmi said:
It does not matter, pi is the invariant result of circumference/diameter, where circumference and diameter sizes are > 0 AND < ∞, no matter what measurement unit is used (except that the unit is some distance on the real-line w.r.t distance 1).
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Then why did you spout this nonsense?
doronshadmi said:
Pi has a physical size as distance along the real line
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zooterkin said:
This nonsense (truncated further to make it clearer):


What does 'physical' mean in Doronetics?
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Distance can be considered also as physical size (no matter what unit is used).

Pi is an invariant distance of the form circumference*unit/diameter*unit, where unit,circumference and diameter are values > 0 AND < ∞


Let us take the Wiki definition ( http://en.wikipedia.org/wiki/Dimensionless_quantity ) :

"In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension."

It is wrong since the unit can be 1mm , 1m , 1mile , 1nanometer , etc. ... ad infinitum.
 
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Originally Posted by epix
If pi is a dimensionless quantity, then what is number pi?

I guess it has to be a dimensionless number.

And what is a dimensionless number?
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The Man said:
A “quantity of dimension one”
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Actually I would avoid the circular reference and go ahead again with the definition of dimensional number; that is,
A number representing a property of a physical system, but not measured on a scale of physical units (as of time, mass, or distance). Drag coefficients and stress, for example, are measured as dimensionless numbers.
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I think I would delete the example part, coz the unit of stress is Pascal, to let the broad concept to reach the novice in the cleanest and briefest state possible, hoping that there is no SI unit for ratio, for example. ;)

Doron sometimes consults Wikipedia and the result is known. On the other hand, it's hard to ask the Wiki writers for such a high degree of anticipation. Some writers can manage to squeeze the essence out of a subject into the first couple of paragraphs, but some clearly struggle and their style requires reading through and branching to other links in order to come up with the essentials.

Now I'm not sure if my example pi miles/hour was a good idea. I guess not.
 
doronshadmi said:
It does not matter, pi is the invariant result of circumference/diameter, where circumference and diameter sizes are > 0 AND < ∞, no matter what measurement unit is used.
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If you think you have just discovered that pi is a ratio between the circumference and the diameter of a circle where 0<d<∞, then be advised that there were others before you who noticed that. If you think that someone disagrees with your statement, then you are invoking ghosts to argue with just for the sake of arguing something.

You should distinguish between accepting the correctness of formal logic and its rejection upon application. Here is an example.

INSIDE is to RAW as OUTSIDE is to NOT RAW

That is an example based on the axiom of mutual exclusivity. But its application in the particular case is in most cases the subject of rejection.

W A I T E R ! ! ! !
 
doronshadmi said:
Distance can be considered also as physical size (no matter what unit is used).
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So far so good, just remember that the same quantity of different units of length (distance), gives you different distances and “physical size”.

doronshadmi said:
Pi is an invariant distance of the form circumference*unit/diameter*unit, where unit,circumference and diameter are values > 0 AND <
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Pi is not a distance, invariant or otherwise. Pi is a quantity (technically a ratio of distances). A distance is a quantity with units of, well, distance, Pi has no such units.

doronshadmi said:
Let us take the Wiki definition ( http://en.wikipedia.org/wiki/Dimensionless_quantity ) :

"In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension."

It is wrong since the unit can be 1mm , 1m , 1mile , 1nanometer , etc. ... ad infinitum.
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That you simply want to ascribe not just some units of distance to Pi but evidently every unit of distance “ad infinitum”, doesn’t make the pervious quote wrong it just makes you wrong and blatantly so.

Above you have claimed “Pi is an invariant distance” and “unit can be 1mm , 1m , 1mile”. So let’s give that a shot.


3.14 Milimeter
3.14 Meter
3.14 Mile

So much for your “invariant distance”, which part (if not all) of your self-contradicting claims is wrong Doron?
 
epix said:
Actually I would avoid the circular reference and go ahead again with the definition of dimensional number; that is,

I think I would delete the example part, coz the unit of stress is Pascal, to let the broad concept to reach the novice in the cleanest and briefest state possible, hoping that there is no SI unit for ratio, for example. ;)

Doron sometimes consults Wikipedia and the result is known. On the other hand, it's hard to ask the Wiki writers for such a high degree of anticipation. Some writers can manage to squeeze the essence out of a subject into the first couple of paragraphs, but some clearly struggle and their style requires reading through and branching to other links in order to come up with the essentials.

Now I'm not sure if my example pi miles/hour was a good idea. I guess not.
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Well, miles/light hour might have been a better example.
 
Pi is an invariant distance along the real line, which is determined by the ratio between two distances, which is not impacted by any unit measurement (as long as unit > 0 AND < ∞) exactly as shown by the expression circumference*unit/diameter*unit (where also abs(circumference) and abs(diameter) > 0 AND < ∞).

3.1410 < pi, and the distance of pi along the real line is not changed by the measurement unit.

It does not mean that the result of pi*unit is invariant as pi.

On the contrary, the result of pi*unit is based on the invariant distance of pi along the real line and the variant property of the unit, such that we get infinitely many distances that are derived form pi as an invariant distance along the real line, for example:

... pi*1nanometer < pi*1mm < pi*1cm < pi*1m < pi*1km ... etc. ... ad infinitum.

By understanding invariant values like pi (which are > 0 AND < ∞) one easily understands why distances that are derived from ∞ curvature do not belong to the class of objects that are based on pi as an invariant distance along the real line.

Furthermore, no amount of elements that are derived from ∞ curvature can fully cover the invariant distance of pi along the real line.

Let's do a further step and define the absolute unit measurement as the distance of elements that are derived from ∞ curvature.

By using Traditional Math I wish to see how some amount of such units define an invariant distance like pi along the real line (be aware of the fact that pi*0unit = 0 distance, or in other words, the invariant distance of pi along the real line is eliminated by 0unit, which is the absolute unit measurement that is derived from ∞ curvature.
 
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doronshadmi said:
By using Traditional Math I wish to see how some amount of such units define an invariant distance like pi along the real line (be aware of the fact that pi*0unit = 0 distance, or in other words, the invariant distance of pi along the real line is eliminated by 0unit, which is the absolute unit measurement that is derived from ∞ curvature.
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"Traditional math" can only answer questions who are structurally sound, so you can wish indefinitely.

You should brush up on the difference between global absence and local absence. If fruit called apples doesn't exist, then apples won't appear in the basket; if they do exist, it doesn't mean that they will be placed in some particular basket.

You've been clearly struggling with something that comes to most folks naturally. So even if traditional math makes some attempt to repair your question so it could be considered, you won't be able to see the change that has been made. If you fail to understand that length, volume, area, pressure, energy, charge and so on have a unit of measurement, but ratio doesn't
then

pi = circumference/diameter = "dimensionless quantity"

You misapplied the term in your scheme, coz the initial subject never called for bringing dimensionless quantities into the picture. The subject was your previous claim that 1-dim object, such as circle, cannot be reduced to a 0-dim object, which is point. Now you vehemently argue that when radius=0, circle becomes point. Since you don't have the type of writing instrument that math uses all the time, you won't be able to prove either case through comprehensive presentation apart from the usual verbal declarations studded with phantasmagorical syntax.

You used once the implicit equation for unit circle; that is

y2 + x2 = 1

but you never understood its meaning, otherwise you would be able to trace the case when r is reduced to 0 and prove the result.

How do we call the folks who take incessantly an issue with a subject without being able to understand it?
(doron s. with a single replacement.)

Since the devil has so much on his mind, then you keep going . . .
 
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A person that does not understand the concept of class, says something like:

"If fruit called apples doesn't exist, then apples won't appear in the basket; if they do exist, it doesn't mean that they will be placed in some particular basket."

A person that does understand the concept of class naturally knows that it is an association between the global (invariant and cross-contexts) and the local (the variant and context-dependent).

Pi is an invariant distance along the real-line that is used as cross-contexts property among infinitely many context-dependent properties of curvature's degrees, and as a result we get the class of all circles.

By using only verbal_symbolic skills, one by mistake defines X2 + Y2 = R2 as a circle even if R = 0 or ∞.

By using also visual_spatial skills one naturally knows that no circle exists if X2 + Y2 = 0 or X2 + Y2 = ∞.

The verbal_symbolic-only skill(er) can't comprehend this simple fact, he has no struggle with the mistake that X2 + Y2 = 0 or X2 + Y2 = ∞ are not circles.

"structurally sound" holds only if not less than verbal_symbolic ("sound") AND visual_spatial ("structure") skills are used.

This is defiantly not the case of Traditional Math, which gets only the verbal_symbolic aspect of an expression like X2 + Y2 = R2
Because of these limited used skills, for example, 0.999...10 = 1
 
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doronshadmi said:
This is defiantly not the case of Traditional Math, which gets only the verbal_symbolic aspect of an expression like X2 + Y2 = R2
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Really? Unlike Doronetics, traditional math notices that X2 + Y2 = R2 is identical to the Pythagorean Theorem, which is easy to visualize.

pic-pythagorasT.gif


pythag_squares.gif


Traditional math can also see what what Doronetics can't - something that is not drawn in the first picture but easily can be.

CircleEquation.GIF


If you convert the implicit equation to its "explicit" or function form, and plot it, where r=3 with the domain
-3 ≤ x ≤ 3,
the function will draw a nice semicircle. Then you can use -f(x) to draw the other semicircle. Everyone would be able to see it - done step by step.

circle.gif



Only you wouldn't be able to see it, coz of the confusion in the Fifth Law of Motion: We stir from right to left, right?

By using only verbal_symbolic skills, one by mistake defines X^2 + Y^2 = R^2 as a circle even if R = 0 or ∞.
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Just set x=0, r=0 and see what the function returns in term of a point drawn on the y axis. You can't set x=∞, coz the tradititional math regards the expression as not compatible with the concept of infinity.
 
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As I said, whether the verbal_symbolic-only expression is X2 + Y2 = R2 or a2 + b2 = c2, if R or c = 0 OR ∞, then the results are not in the class of results that are derived from R or c > 0 AND < ∞ (where in this case the class is the collection of all circles, which exists only if pi exists).

It is so simple but Traditional Math that is based only on verbal_symbolic skills, can't comprehend it exactly because it does not distinguish between "R or c = 0 OR ∞" and "R or c > 0 AND < ∞" results of X2 + Y2 = R2 or a2 + b2 = c2 expressions (the visual_spatial skills, which are essential for this distinction, are not used by Traditional Math).

The verbal_symbolic-only expressions are worshiped by verbal_symbolic-only skill(ers) up to ignorance.
 
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doronshadmi said:
As I said, whether the verbal_symbolic-only expression is X2 + Y2 = R2 or a2 + b2 = c2, if R or c = 0 OR ∞, then the results are not in the class of results that are derived from R or c > 0 AND < ∞ (where in this case the class is the collection of all circles, which exists only if pi exists).

It is so simple but Traditional Math that is based only on verbal_symbolic skills, can't comprehend it exactly because it does not distinguish between "R or c = 0 OR ∞" and "R or c > 0 AND < ∞" results of X2 + Y2 = R2 or a2 + b2 = c2 expressions (the visual_spatial skills, which are essential for this distinction, are not used by Traditional Math).

The verbal_symbolic-only expressions are worshiped by verbal_symbolic-only skill(ers) up to ignorance.
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You've been spewing one of the deepest dogma ever publicly available. Even when faced with hard evidence that the "verbal_symbolic" rendition is actually drawing tool that Doronetics doesn't have and never will, you just keep repeating your claims. But that's the way things are highlighted. A rational approach to your gibberish is to ignore it, but things are not that simple - something is running in the background that needs to become visible. That something does have meaning. Care for an example?

Total packed states are considered as total exactly because they can't be packed further (or more).
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Those "packed states" actually refer to arrangements, such as

IJPAM

which is the acronym for "International Journal of Pure and Applied Mathematics" which was inserted into the works through seriously flawed statement made by Dr. Gosh. But when

Non-total packed states are considered as non-total exactly because they can be packed further (or more).
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shows up, it means that IJPAM can be "unzipped" to form a short sentence, as it was the case with

I AM P.J.

Once the P.J. initials are decoded with all the implications, you are going to be a free man. That means your gibberish will become a priority subject to be squarely ignored, as something that we shouldn't bring back from the Dark Ages. Actually you may feel like talking to yourself for a while, but I think all the between-the-lines stuff has been recovered. You are not the only one, Doron; there've been other "messengers", like Richard Dawkins, for example.
 
When recess is over and you come back in from the playground with the other kindergarteners, let me know.
 
No matter what concept is used, there can't be more than one expression of it without the association among its host state and its hosted state (mathematical or physical) such that being host and hosted means that its expressions are defined but not made of each other.

For example, a curve with 0 curvature's degree is the host state (has the power of continuum) that gathers infinitely many distinct hosted curves with curvature's degree > 0, such that the gathered hosted curves exist as a collection only if they do not have the power of continuum of the host state.
 
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doronshadmi said:
Pi is an invariant distance along the real line, which is determined by the ratio between two distances, which is not impacted by any unit measurement (as long as unit > 0 AND < ∞) exactly as shown by the expression circumference*unit/diameter*unit (where also abs(circumference) and abs(diameter) > 0 AND < ∞).

3.1410 < pi, and the distance of pi along the real line is not changed by the measurement unit.
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Again Pi is a quantity not a distance. That any quantity can be represented as a location along the real number line does not make it a distance. It is units of distance that make some quantity of those units a distance, Pi has no such units. You are evidently and perhaps deliberately confusing a specific location along the real number line with some invariant distance. The location of some quantity along the real number line is not changed by different units of distance, but its distance from any other locations is. So your claim that "Pi is an invariant distance along the real line of " is still demonstrably false. You keep demonstrating that you simply do not know and apparently do not want to know basic geometry. Learn something please Doron.

doronshadmi said:
It does not mean that the result of pi*unit is invariant as pi.
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The ratio of a circles circumference to its diameter is only invariant in Euclidean geometry. Learn something please Doron.

doronshadmi said:
On the contrary, the result of pi*unit is based on the invariant distance of pi along the real line and the variant property of the unit, such that we get infinitely many distances that are derived form pi as an invariant distance along the real line, for example:

... pi*1nanometer < pi*1mm < pi*1cm < pi*1m < pi*1km ... etc. ... ad infinitum.
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Wait so now your claiming your "invariant distance" is "infinitely many distances"? Can you ever just agree with even yourself Doron?

doronshadmi said:
By understanding invariant values like pi (which are > 0 AND < ∞) one easily understands why distances that are derived from ∞ curvature do not belong to the class of objects that are based on pi as an invariant distance along the real line.
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By understanding geometry one easily understands that Pi is not a distance and that the ratio of a circles circumference to its diameter is an invariant quantity only in Euclidean geometry. Learn something please Doron.


doronshadmi said:
Furthermore, no amount of elements that are derived from ∞ curvature can fully cover the invariant distance of pi along the real line.

Let's do a further step and define the absolute unit measurement as the distance of elements that are derived from ∞ curvature.

By using Traditional Math I wish to see how some amount of such units define an invariant distance like pi along the real line (be aware of the fact that pi*0unit = 0 distance, or in other words, the invariant distance of pi along the real line is eliminated by 0unit, which is the absolute unit measurement that is derived from ∞ curvature.
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Again Pi has no "such units", it is not a distance and the ratio of a circles circumference to its diameter is only an invariant quantity in Euclidean geometry. That you simply expect "Traditional Math" to ascribe some "such units" at your whim and refer to it as "the absolute unit measurement that is derived from ∞ curvature." simply demonstrates that you don't understand, units, geometry or Pi. You continue to simply string words together that you clearly do not understand into nonsensical statements that evidently just have some ill defined meaning only to you.

Your failure is complete Doron. Please learn something.
 
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doronshadmi said:
Please look how a verbal_symbolic-only skiller can't get his mistake as shown in http://www.internationalskeptics.com/forums/showpost.php?p=7869061&postcount=341
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You take issues with the "traditional math" using expressions, such as "x = ∞". Even when reminded that such an expression is misleading and that math doesn't use it, you just simply use it again. How far your ignorance can go? ignorance = ∞? If you learn and start to adjust and be more correct, then your gibberish will lose the purpose of why it is out and kicking. That's why you simply ignore and repeat the same thing all over again.

If you come across a wooden stick 3.14 feet long, then its length coincides with a well-known constant. That's all. It depends on circumstances that would relate meaningfully 3.14 to pi. But if you come across 3.141... in the environment that is basically free of any curves, then the likelyhood that such an incidence is purely coincidental is 1/n where n → ∞. Then it's time to learn more about pi.
 
Verbal_symbolic-only skill(ers) get quantity representation only in terms of location along the real line, for example:

"Again Pi is a quantity not a distance. That any quantity can be represented as a location along the real number line does not make it a distance."

An accurate distance is the absolute result between any two locations along the real line, for example:

Accurate distance 1 = abs(0-1) = abs(2-3) = abs(24.1-25.1) etc. ad infinitum.

Accurate distance 0 = abs(0-0) = abs(2-2) = abs(24.1-24.1) etc. ad infinitum.

Accurate distance pi = abs(0-pi) = (2pi-3pi) = abs(24.1pi-25.1pi) etc. ad infinitum.

An accurate distance > 0 is located simultaneously at any arbitrary two different locations along the real line.

An accurate distance = 0 is located simultaneously at one location along the real line.

Verbal_symbolic-only skill(ers) get the real-line only as a collection of accurate 0 distances.

No wonder that they can't get pi or 1 as accurate distances > 0 along the real line.

Furthermore, they can't get non-accurate distances like 0.999...10 along the real line.

Once again Distance is limited only to Metric space by minds that get the real-line only as a collection of accurate 0 distances.

Distance is not limited only to Metric space by minds that get the real-line not only as a collection of accurate 0 distances.

For example, the distance between emotions is actually the measurement of distinction between them.
 
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It is good to see Doron continues to be unconstrained by actual meaning of words and phrases. If you are going to ignore reality, his approach certainly makes it more convenient.
 
Verbal_symbolic-only skill(ers) can't be aware of the objective and non-personal state of mind, which is the only way to capture the subjective personal aspect of reality.

As a result they try to force naturally subjective aspects of reality as if they are objective.

One of the results of this forcing is the mechanic reductionist definition of the real-line as a collection of accurate 0 distances, where order has no significance.
 
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Please look at this verbal_symbolic-only notion:

"Wait so now your claiming your "invariant distance" is "infinitely many distances"?"

In this case the verbal_symbolic-only skill(er) can't distinguish between an invariant distance like pi and the variant distance, which is the result of pi*unit.

Also be aware of the fact that verbal_symbolic-only skill(er) is blind to http://www.internationalskeptics.com/forums/showpost.php?p=7851200&postcount=306 .
 
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jsfisher said:
It is good to see Doron continues to be unconstrained by actual meaning of words and phrases. If you are going to ignore reality, his approach certainly makes it more convenient.
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Well certainly amusing to say the least. Unfortunately, it is an old shtick that Doron somehow thinks he invented.
 
doronshadmi said:
Verbal_symbolic-only skill(ers) can't be aware of the objective and non-personal state of mind, which is the only way to capture the subjective personal aspect of reality.

As a result they try to force naturally subjective aspects of reality as if they are objective.

One of the results of this forcing is the mechanic reductionist definition of the real-line as a collection of accurate 0 distances, where order has no significance.
Click to expand...


Once again if you find your fantasy “Verbal_symbolic-only skill(ers)” so troubling then you should have better fantasies.

“where order has no significance”? That must be some funky real number line fantasy you’ve got going there.
 
This nonsense was taken from http://en.wikipedia.org/wiki/Linear_continuum :

Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound.
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It must be stressed that there is always a gap for more elements between the least upper bound and all members of the non-empty subsets, simply because no amount of 0-spaces actually reaches (has the identity of) a least upper bound, whether the least upper bound is a member of a given subset, or not.

No axiom can construct 1-space by any amount of 0-spaces, or in other words, no amount of 0-spaces has the power of continuum of 1-space.

Traditional Math gets the power of continuum in terms of collection, and this is exactly its fundamental failure, no matter what verbal_symbolic-only expressions are used (whether they are called, axioms, definitions, or whatever).

Once again, verbal_symbolic-only skill(ers) are blind to http://www.internationalskeptics.com/forums/showpost.php?p=7851200&postcount=306.
 
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doronshadmi said:
This nonsense was taken from http://en.wikipedia.org/wiki/Linear_continuum :


It must be stressed that there is always a gap for more elements between the least upper bound and all members of the non-empty subsets, simply because no amount of 0-spaces actually reaches (has the identity of) a least upper bound, whether the least upper bound is a member of a given subset, or not.

No axiom can construct 1-space by any amount of 0-spaces, or in other words, no amount of 0-spaces has the power of continuum of 1-space.

Traditional Math gets the power of continuum in terms of collection, and this is exactly its fundamental failure, no matter what verbal_symbolic-only expressions are used (whether they are called, axioms, definitions, or whatever).
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It must be stressed that there is a least one element of the set between any two members of that set which is why the set "lacks gaps" and is “densely ordered” as noted. Of course we have been over this before. Please indentify the location of your “gap”, Doron, in the set where there can be no elements of the set. Once again you are deliberately confusing what is expressly “another” element of the set in a set “which "lacks gaps"” as some “gap for more elements”. Do please tell us what element(s) of the set are not element(s) of the set such that you can have “more elements” of that set for your gap in a set that explicitly, well, "lacks gaps"? One might conclude that you were simply mistaken if it were not for the explicit statements you quoted that expressly contradict your assertion of such a “gap for more elements”. So your misrepresentation is as deliberate as it is blatantly obvious.


doronshadmi said:
Once again, verbal_symbolic-only skill(ers) are blind to http://www.internationalskeptics.com/forums/showpost.php?p=7851200&postcount=306.
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Once again if your fantasy “verbal_symbolic-only skill(ers)” trouble you so then have better fantasies.
 
Even among verbal_symbolic-only skill(ers) (which get everything (including the power of continuum) only in terms of Collection) Q set has gaps even if "there is at least one element of the set between any two members of that set".

By using also visual_spatial skills, it is easily understood that no collection of 0-spaces (and R members are equivalent to 0-spaces, where 0-spaces are logically, verbally or spatially can't be but exact locations) has the power of continuum of 1-space.

So expressions like "Please identify the location of your “gap”" can't get a gap, because a gap is non-local by definition, which is equivalent at least to 1-space that is found simultaneously at AND not-at any given 0-space.

It is not about being or not being a member of a collection of elements, which are equivalent to 0-space.

It is about the inability of any amount of 0-spaces to be 1-space, simply because no amount of 0-spaces (which are local by definition), has the power of continuum of 1-space (which is non-local by definition).

The fact that there is no homeomorphism between a collection of 0-spaces and 1-space (whether "space" is understood in terms of mathematical abstraction or physical realm) is extended, without loss of generality, to any pair of different spaces, such that the lower level of a given space is hosted w.r.t the higher level of a given space, and the higher level of a given space is host w.r.t to all lower spaces.

The closed "Singing box" of hosted-only spaces, can't get the host space and its non-locality w.r.t any collection of lower spaces.
 
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If we analyze 0 and ∞ in terms of Length or Curvature we find that 0 and ∞ are context-dependent, for example:

0 is the smallest Length, known as a point, where ∞ is the largest Length, known as an endless straight line.

∞ is the largest Curvature's degree, known as a point, where 0 is the smallest Curvature's degree, known as an endless straight line.

But there is also a cross-contexts view of 0 and ∞, such that 0 is a hosted space w.r.t ∞ space and ∞ space is a host space w.r.t 0 space, no matter if Curvature or Length are considered.
 
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Let's take two arbitrary locations on the real-line.

If there is nothing between them, than the, so-called, two locations are actually the same location or in other words, nothing between location A and location B is actually A = B, which is unconditionally resulted by a single location.

The same result (being a single location) is found even if ...=A=B=C=... expression is infinitely long.

This is not the case among finite "A≠B" or infinite "...≠A≠B≠C≠..." expressions, which actually enable the existence of more than a single location exactly because "≠" expression prevents the state of nothing (notated by "=" expression) among locations.

That is prevents the state of nothing between two locations can't be another location, because another location is always located as some end case of any given pair, no matter what scale level is considered.

Conclusion: "≠" is an expression of non-locality (known at least as 1-space), such that no amount of the form "...≠A≠B≠C≠..." completely covers a 1-space (know as a line).

Without a loss of generality, this conclusion is extended to all spaces, such that the lower spaces are local w.r.t the higher spaces and the higher spaces are non-local w.r.t the lower spaces, where the higher spaces are represented by at least "≠" expression among the lower spaces.
 
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