Deeper than primes - Continuation

[The Man]

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.

So a single location is not two locations and simply labeling that singular location with two different labels "A" and "B" does not make it two locations.

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

Yes even infinitely many different labels for a singular location can not make it anything but a singular location.

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.

"≠" doesn't prevent anything it simply represents the assertion of the inequality of, in this consideration, different 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.

No it is the continuous nature of the real number line that "prevents the state of nothing between two locations". Were the space discontinuous, as in the integers, there can be nothing between two different locations.

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).

Nope, ""≠" is an expression of" inequality. Attempting to conflate that expression with your "non-locality" drivel will not help you. Once again you have simply confused yourself by, apparently deliberately, positing the same location with different labels as two locations. Have you found that location (point) along the real number line where there is no point (location) yet?

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.

Extend it where ever you want, your purported "conclusion" is patently false as is the rest or your "extended" "non-locality" nonsense. Again please at least learn something Doron.


I'll give you one last chance Doron. You can either respond to this post properly or you can attempt to ascribe any of the above assertion to your fantasy "skillers". The former will get an appropriate response while the latter will signify that you have no intent to discuss this with anyone but yourself and your fantasies and as such you will be left indulge yourself in that regard as you see fit.

 

[doronshadmi]

The continuous nature of the real number line that "prevents the state of nothing between two locations" can't be defined by another distinct location (whether it is one more or infinitely many more locations) between any arbitrary pair of distinct locations along the real-line, simply because no matter how many distinct locations there are on infinitely many smaller scales, they are permanently distinguished by "≠" expression, which is actually the existence of at least irreducible 1-space between distinct 0-spaces, such that each 0-space is defined simultaneously by no more than one location along the real-line, and each 1-space between arbitrary distinct 0-spaces is defined simultaneously by at least two locations along the real-line.

In other words, a collection of 0-spaces can't fully cover 1-space, because an element that is defined simultaneously by at least two locations along the real-line, is irreducible into an element that is defined simultaneously by no more than one location along the real-line.

By not ignoring the non-local nature of 1-space along the real-line, one enables to understand that the power of continuum is determined by 1-space and not by collection of 0-spaces (which are naturally local, as shown above).

In order to really get it, our local (verbal_symbolic) AND non-local (visual_spatial) brain's skills must be combined into a one comprehensive framework.


For example, by "Traditional" Mathematics (which is mostly expressed by verbal_symbolic skills) 0.111... ₂ = 0.999...₁₀ = 1 where 1 is the considered mathematical object (the number itself) and 0.111...₂ or 0.999...₁₀ are some numerals (out of many representations) that represent number 1.

By using verbal_symbolic AND visual_spatial skills as follows:

one may understand that no branch of that tree actually reaches any other branch of that tree "downward" , no matter how many levels that tree has (in other words, there is no homeomorphism between 0 dimensional space (notated by "0";"1" symbols) and 1 dimensional space (notated by "_____" spatial non-composed object)).

According to this framework 0.111...₂ is a number of its own < number 1 by 0.000...1₂ where the "...1" part of that number is the irreducibility of ___ 1 dimensional space into 0 dimensional space (known as a point).

By using verbal_symbolic AND visual_spatial skills one enables to distinguish between non-local numbers like 0.111...₂ or 0.000...1₂, and local numbers like 1 or 0.

Furthermore, no collection of, for example, 0 dimensional spaces or segments on 1 dimensional space has the power of the continuum of 1 dimensional space.

By understanding the power of the continuum in terms of spatial skills, one may understand that no collection of sub-spaces of a given space (mathematical or physical) has the power of the continuum of that space, or in other words, any given collection of "hosted" sub-spaces is incomplete with respect to the "host" space.

The terms "host"\"hosted" are used here in order to clarify that the the "host" and the "hosted" are defined but not made of each other.

The non-locality of 0.111...₂ or 0.000...1₂ is "naturally vague" in terms of location, and one actually discovers/invents that the Real-line has a non-empty collection of non-local numbers between 0 dimensional space and 1 dimensional space.

By generalization, given a "host" space, no collection of "hosted" spaces has the power of the "host" space.

Even a one chance must not be given to mind's that are using only verbal-symbolic skills, in order to get this fine subject.

For example look at this response, made by verbal-symbolic-only skill(er):

"≠" doesn't prevent anything it simply represents the assertion of the inequality of, in this consideration, different locations.

"≠" prevents from different variables to actually represent only a single location along the real line, but verbal-symbolic-only skill(ers)
simply can't get that ( explained in http://www.internationalskeptics.com/forums/showpost.php?p=7948514&postcount=360 ).

Furthermore, verbal-symbolic-only skill(ers) can't get the equivalence of "≠" to 1-space (as explained above) along the real-line (their inability to deal with the real-line in terms of Cross-contexts AND Contexts-dependent framework is shown in http://www.internationalskeptics.com/forums/showpost.php?p=7887931&postcount=356 and http://www.internationalskeptics.com/forums/showpost.php?p=7895512&postcount=359 ) .

 

[The Man]

The continuous nature of the real number line that "prevents the state of nothing between two locations" can't be defined by another distinct location (whether it is one more or infinitely many more locations) between any arbitrary pair of distinct locations along the real-line, simply because no matter how many distinct locations there are on infinitely many smaller scales, they are permanently distinguished by "≠" expression, which is actually the existence of at least irreducible 1-space between distinct 0-spaces, such that each 0-space is defined simultaneously by no more than one location along the real-line, and each 1-space between arbitrary distinct 0-spaces is defined simultaneously by at least two locations along the real-line.

Probably why a linear continuum is defined as...

http://en.wikipedia.org/wiki/Linear_continuum

...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. More symbolically:

So "another distinct location" or as informally defined "between any two members there is another". Is just one aspect the "and which "lacks gaps"" is another. Both you are well aware of as they have been quoted to your before. Why do you deliberately pretend not to know what has already been explained to you and is evident in articles you have quoted yourself that expressly assert a continuum does not have what you claim it does, gaps?

In other words, a collection of 0-spaces can't fully cover 1-space, because an element that is defined simultaneously by at least two locations along the real-line, is irreducible into an element that is defined simultaneously by no more than one location along the real-line.

So have you found that location (point) along the real number line where there is no point (location) yet?

By not ignoring the non-local nature of 1-space along the real-line, one enables to understand that the power of continuum is determined by 1-space and not by collection of 0-spaces (which are naturally local, as shown above).

In order to really get it, our local (verbal_symbolic) AND non-local (visual_spatial) brain's skills must be combined into a one comprehensive framework.


For example, by "Traditional" Mathematics (which is mostly expressed by verbal_symbolic skills) 0.111... ₂ = 0.999...₁₀ = 1 where 1 is the considered mathematical object (the number itself) and 0.111...₂ or 0.999...₁₀ are some numerals (out of many representations) that represent number 1.

By using verbal_symbolic AND visual_spatial skills as follows:

[qimg]http://farm7.staticflickr.com/6142/5962015728_d2fe37cc5f_z.jpg[/qimg]

one may understand that no branch (including the right most branch that is not 1 branch) of that tree actually reaches any other branch of that tree "downward" , no matter how many levels that tree has (in other words, there is no homeomorphism between 0 dimensional space (notated by "0";"1" symbols) and 1 dimensional space (notated by "_____" spatial non-composed object)).

According to this framework 0.111...₂ is a number of its own < number 1 by 0.000...1₂ where the "...1" part of that number is the irreducibility of ___ 1 dimensional space into 0 dimensional space (known as a point).

By using verbal_symbolic AND visual_spatial skills one enables to distinguish between non-local numbers like 0.111...₂ or 0.000...1₂, and local numbers like 1 or 0.

Furthermore, no collection of, for example, 0 dimensional spaces or segments on 1 dimensional space has the power of the continuum of 1 dimensional space.

By understanding the power of the continuum in terms of spatial skills, one may understand that no collection of sub-spaces of a given space (mathematical or physical) has the power of the continuum of that space, or in other words, any given collection of "hosted" sub-spaces is incomplete with respect to the "host" space.

The terms "host"\"hosted" are used here in order to clarify that the the "host" and the "hosted" are defined but not made of each other.

The non-locality of 0.111...₂ or 0.000...1₂ is "naturally vague" in terms of location, and one actually discovers/invents that the Real-line has a non-empty collection of non-local numbers between 0 dimensional space and 1 dimensional space.

By generalization, given a "host" space, no collection of "hosted" spaces has the power of the "host" space.

All this blathering still gets you back to the same problem a continuum doesn't have gaps so it is by definition completely covered by points. A discrete space has gaps but you evidently just don't want to talk about a discrete space and simply insist on pretending that a continuous space can be, well, discrete.

 

[doronshadmi]

Verbal_symbolic-only skill(ers) do not understand that no collection has the power of the continuum.

The continuum doesn't have gaps exactly because it is not a collection. In other words, the real-line (which is not less than 1-space AND 0-spaces, or in other words, it is not only 1-space (which has the power of the continuum)) does not have the power of the continuum.

For example, the collection of all 0-spaces along 1-space does not have the power of the continuum of 1-space, exactly because such collection exists because there is always 1-space "room" between any arbitrary pair of 0-spaces along it, which is actually equivalent to "≠" expression between them.

"≠" expression enables the existence of different 0-spaces along a given 1-space, in the first place.

In that case we can ask to show even a single case, where there is no "≠" expression between 0-spaces along a given 1-space.

Once again, an element that is defined simultaneously by at least two locations along the real-line, is irreducible into an element that is defined simultaneously by no more than one location along the real-line, or in other words, the real-line is not less than 1-space AND collection of 0-spaces along it (where the term "space" is mathematical or physical), and such association (known as the real-line) does not have the power of the continuum (as 1-space has).

No Wikipedia quotes of this subject (that are currently based on Verbal_symbolic-only skills) can change this simple fact.

 

[The Man]

Verbal_symbolic-only skill(ers) do not understand that no collection has the power of the continuum.

Once again if you have such problems with your fantasy "skill(ers)" then have better fantasies.

The continuum doesn't have gaps exactly because it is not a collection. In other words, the real-line (which is not less than 1-space AND 0-spaces, or in other words, it is not only 1-space (which has the power of the continuum)) does not have the power of the continuum.

No Doron a continuum does not have gaps explicitly because a lack of such gaps is part of the definition of a continuum.

For example, the collection of all 0-spaces along 1-space does not have the power of the continuum of 1-space, exactly because such collection exists because there is always 1-space "room" between any arbitrary pair of 0-spaces along it, which is actually equivalent to "≠" expression between them.

Again what part of the collection is not part of the collection such that there is "room" in the collection for, well, some part of the collection?

"≠" expression enables the existence of different 0-spaces along a given 1-space, in the first place.

Nope, a simple assertion of inequality still seems to confuse you.

In that case we can ask to show even a single case, where there is no "≠" expression between 0-spaces along a given 1-space.

Well you can certainly ask, but I doubt most here have any need to. Once again you seem to be deliberately conflating that two different points are not equal (by that different definition) with the symbol representing inequality ("≠") as somehow 'enabling' that difference.

Once again, an element that is defined simultaneously by at least two locations along the real-line, is irreducible into an element that is defined simultaneously by no more than one location along the real-line, or in other words, the real-line is not less than 1-space AND collection of 0-spaces along it (where the term "space" is mathematical or physical), and such association (known as the real-line) does not have the power of the continuum (as 1-space has).

No Wikipedia quotes of this subject (that are currently based on Verbal_symbolic-only skills) can change this simple fact.

For your own edification "two locations" in time are not simultaneous. Do please at least learn something Doron.

A one dimensional element is not reducible to a non-dimensional element but it can be represented as a collection of one dimensional elements, non-dimensional elements or some combinations thereof. Thank you for clearly demonstrating the advantages of mathematics over your non-local fantasies.

 

[zooterkin]

Verbal_symbolic-only skill(ers) do not understand that no collection has the power of the continuum.

It's odd that you keep going on about the supposed limitations of others when you are the one who seems to be unable to grasp some fairly basic maths concepts. What do you think a continuum is?

 

[doronshadmi]

What do you think a continuum is?

The answer is in http://www.internationalskeptics.com/forums/showpost.php?p=7952882&postcount=364.

 

[doronshadmi]

Two locations in time are comparable simultaneously by memory, and yet the location in the future is potential, so Time can't be defined only in terms of collection of localities.

Step-by-step minds are tuned to get things only in terms of collections, and as a result they can't get the true nature of the continuum, which is not reducible to collections.

 

[zooterkin]

The answer is in http://www.internationalskeptics.com/forums/showpost.php?p=7952882&postcount=364.

So, you don't understand what a continuum actually is. Got it.

 

[doronshadmi]

So, you don't understand what a continuum actually is. Got it.

No zooterkin, you do not understand that the continuum is not a collection.

 

[zooterkin]

No zooterkin, you do not understand that the continuum is not a collection.

Who said it was?

 

[doronshadmi]

Who said it was?

Anyone who claims that there is a collection of 0-spaces with no gaps between them.

Such mind can't get:

http://www.internationalskeptics.com/forums/showpost.php?p=7878241&postcount=354

http://www.internationalskeptics.com/forums/showpost.php?p=7952072&postcount=362

http://www.internationalskeptics.com/forums/showpost.php?p=7952882&postcount=364

http://www.internationalskeptics.com/forums/showpost.php?p=7953665&postcount=368

All such mind can do is to quote Wiki and other similar sources that get the continuum in terms of collection.

Here is some nonsense given by local-only mind:

"A one dimensional element is not reducible to a non-dimensional element but it can be represented as a collection of one dimensional elements, non-dimensional elements or some combinations thereof."

And after this nonsense (which does not distinguish between 1-space and a collection of 0-spaces on it (where such collection of many 0-spaces, actually does not exist without 1-space as its "host" space)) the local-only mind writes this extra nonsense:

"Thank you for clearly demonstrating the advantages of mathematics over your non-local fantasies."

And here is another nonsense given by this local-only mind:

"Once again you seem to be deliberately conflating that two different points are not equal (by that different definition) with the symbol representing inequality ("≠") as somehow 'enabling' that difference."

This is funny, by this mind "≠" expression among collection of unique values (for example R members) does not naturally related to their inequality w.r.t each other (this local-only mind does no get that "≠" is not transformable to "=" among R members without eliminating their unique values w.r.t each other, which is resulted by no more than a single 0-space).

Actually, this is another example of the inability of this local-only mind to get, for example, http://www.internationalskeptics.com/forums/showpost.php?p=7948514&postcount=360.

 

[doronshadmi]

Here is another nonsense of this local-only mind:

"Were the space discontinuous, as in the integers, there can be nothing between two different locations."

Nothing between value A and value B is expressed as "A = B", which is defiantly not two things (locations, quantities, sizes or whatever).

This is another example of the inability of this local-only mind to get, for example, http://www.internationalskeptics.com/forums/showpost.php?p=7948514&postcount=360.

Here is another nonsense:

"Once again you have simply confused yourself by, apparently deliberately, positing the same location with different labels as two locations."

A = B is exactly the same thing, so the phrase above simply exposes the inability of this local-only mind to get what I wrote in http://www.internationalskeptics.com/forums/showpost.php?p=7948514&postcount=360.

 

[zooterkin]

Here is another nonsense of this local-only mind:

"Were the space discontinuous, as in the integers, there can be nothing between two different locations."

Nothing between value A and value B is expressed as "A = B", which is defiantly not two things (locations, quantities, sizes or whatever).

I see your confusion. It doesn't mean that at all. For example, talking of integers, there is no value that is an integer between 3 and 4. That doesn't mean that 3=4.

Here is another nonsense:

"Once again you have simply confused yourself by, apparently deliberately, positing the same location with different labels as two locations."

A = B is exactly the same thing, so the phrase above simply exposes the inability of this local-only mind to get what I wrote in http://www.internationalskeptics.com/forums/showpost.php?p=7948514&postcount=360.

If A and B are points, and A=B, then A and B are two labels for the same thing.

 

[doronshadmi]

I see your confusion. It doesn't mean that at all. For example, talking of integers, there is no value that is an integer between 3 and 4. That doesn't mean that 3=4.

I see your confusion. "there is no value that is an integer between 3 and 4" doesn't mean that there is nothing between 3 and 4, and in this case "there is a difference between 3 and 4" (notated by ≠) ,such that "3=4" is a false expression.

Your understanding of Nothing is not comprehensive.

If A and B are points, and A=B, then A and B are two labels for the same thing.

If A=B, than the variables are two labels for the same thing, whether it is a point, a line, or any possible (abstract or not) thing.

------------------

Let us talk about points. You are invited to show two points, such that there is nothing (understood also as no-gap) between them.

If you do that, then I agree that 1-space is completely covered by 0-spaces.

 

[jsfisher]

Let us talk about points. You are invited to show two points, such that there is nothing between them.

Bundled into your confusion are all sorts of assumptions about those points. In your muddle, you fail to recognize all these hidden assumptions, and so you leap to very wrong conclusions. Ironic, too, you continuing to miss hidden assumptions, but we've come to expect no less of you.

For the case at hand, if I define the integers as my universe of points and apply their natural ordering, then point 3 and point 4 are two distinct points in this universe. There is nothing, however, between point 3 and point 4.

 

[The Man]

Well Dorn, I did caution you about misattributing quotes to your fantasies, so enjoy your self-indulgent fantasies as that is evidently all you want.

 

[Little 10 Toes]

I see your confusion. "there is no value that is an integer between 3 and 4" doesn't mean that there is nothing between 3 and 4, and in this case "there is a difference between 3 and 4" (notated by ≠) ,such that "3=4" is a false expression.

I see your confusion. You don't know what an integer^WP is. If I have a number line made only of integers, and you ask me place 2.5 on it, I will say you can't because there is nothing between 3 and 4 that is an integer. There is no integer between 3 and 4. 3 and 4 are different.

Your understanding of Nothing is not comprehensive.

Your understanding of integers is not comprehensive.

If A=B, than the variables are two labels for the same thing, whether it is a point, a line, or any possible (abstract or not) thing.

So then you agree that A=B. Thank you.

Let us talk about points. You are invited to show two points, such that there is nothing (understood also as no-gap) between them.

If you do that, then I agree that 1-space is completely covered by 0-spaces.

Okay then. I have a number line made up of integers. What is between 3 and 4?

 

[doronshadmi]

Okay then. I have a number line made up of integers. What is between 3 and 4?

The difference between 3 and 4 is notated as 3≠4 or 3__4, where ___ is the line and 3 and 4 are two different values along it.

For more details, please look at page 3 in http://www.scribd.com/doc/17039028/OMDP .

Some local-only mind may say: "if I define the integers as my universe of points and apply their natural ordering, then point 3 and point 4 are two distinct points in this universe. There is nothing, however, between point 3 and point 4."

Filtering out elements does not mean that they do not exist, for example ...__-4__-3__-2__-1__0__1__2__3__4__... is an example of the form of points if only the integers are considered along the real-line, which is, again, not less than Non-locality (at least 1-space) AND a collection of localities (0-spaces), where the term "space" can be mathematical or physical.

What you see above is not limited to Geometry, but it is Formalism that is based of verbal_symbolic AND visual_spatial skills.

----------

Once again, you are invited to show two points, such that there is nothing (understood also as no-gap) between them.

If you do that, then I agree that 1-space is completely covered by 0-spaces.

 

[jsfisher]

The difference between 3 and 4 is notated as 3≠4 or 3__4, where ___ is the line and 3 and 4 are two different values along it.

That's just supreme silliness. The difference between 3 and 4 is notated as |3 - 4|.

However, difference between does not have the same meaning as between, so you have again demonstrated your weakness in reading comprehension. Please try again.