Deeper than primes

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jsfisher said:
As is your technique, you blame others for your failure. You again assert as fact your conclusion, yet you offer no basis for that assertion. The failure is and has always been yours.
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Wrong again jsfisher.

Here is your assertion and conclusion:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it is defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.
 
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doronshadmi said:
HatRack, support in details your argument, which asserts that the following is not formulated within a clear, precise framework of logical deduction:

Theorem: The collection of all distinct points of [0,1] can't completely cover [0,1].
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I'm going to request at this time that you word your theorem more precisely. In particular, and most importantly, give a precise definition of what you think it means for a collection of points to completely cover an interval.

This definition must be readily reducible to nothing more than familiar mathematical concepts such as sets and functions, but it need not be worded in formal logic.

For example, here is a definition that is readily reducible to a statement about sets and functions:

A set S is said to be countably infinite if there exists a bijection of N onto S, where N is the set of natural numbers.

This definition is very clear. If we can construct a bijection from N onto S, then S is countably infinite. If we can prove that no such bijection exists, then S is not countably infinite. It uses nothing more than familiar set theoretic concepts.

You are proposing a theorem that potentially upsets centuries of mathematical knowledge. Your theorem, and every definition it uses, must be readily reducible into familiar concepts so that it may be verified precisely. If that's not possible, due to a "notion" or some type of undefined concept, then you must give an appropriate axiom.

So, let's get this theorem worded precisely, using only sets and functions if possible, before proceeding.
 
HatRack said:
You are proposing a theorem that potentially upsets centuries of mathematical knowledge. Your theorem, and every definition it uses, must be readily reducible into familiar concepts so that it may be verified precisely. If that's not possible, due to a "notion" or some type of undefined concept, then you must give an appropriate axiom.
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The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.
 
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doronshadmi said:
laca, we have here jsfisher.

He is a professional mathematician, and he uses "adjacent" in the following post http://www.internationalskeptics.com/forums/showpost.php?p=6585515&postcount=12497 .

Is you wish to get an answer from a professional mathematician about "adjacent" , then just ask him.
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I'm asking you doron, because it seems like such a central point in your "argument". It's coming from the same delusion that somehow makes you think that we would need to find two distinct points with no gap between them to prove that a line is completely covered with points. I'm thinking by adjacent you mean no gap. But that's utterly ridiculous, because if there's no gap then they are one and the same point and discussing adjacency becomes moot.

This is why I'm asking you to define adjacent in the context of points.

doronshadmi said:
After all he is the one that first used "adjacent" in this thread.
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Not that I care, but I would think he was, you know, playing along, trying to communicate in your "special" language. If he wishes, he can elaborate on that.

Please not that even without defining the term, he said that no two points are adjacent. Which is correct, since the term is meaningless for points.

Let's get back to your first attempt at defining adjacent. You said adjacent means distance > 0. Do you still think this is an accurate definition or would you like to refine it?
 
doronshadmi said:
The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.
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You're hilarious. Do you even know what an axiom is? Are you even aware what is it you're trying to do? You're trying to dismiss a known and proven fact by negating it and attempting to smuggle it in as an axiom. That ain't going to happen, sorry.
 
doronshadmi said:
The axiom of the continuum:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it can't be defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.
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Awesome, we can stop right there then. I don't accept your axiom, and neither will any professional mathematician, ever.

The ZFC axioms are widely accepted because they state things that are relatively simple and obvious. Your axiom makes an assertion about what infinity can and cannot be, something that should be a theorem which follows from much simpler axioms.

But, the true problem with your axiom is that it makes ZFC inconsistent. The negation of your axiom can in fact be derived from ZFC. That is because ZFC implies the existence of the real number line (1-dimensional space), which is in fact an uncountably infinite collection of distinct points (0-dimensional spaces).

Well, that wraps it up. Unless you annihilate one or more ZFC axioms, you have been disproven.
 
doronshadmi said:
Wrong again jsfisher.

Here is your assertion and conclusion:

If 0-dimensional space exists, then given any dimensional space greater than 0-dimensional space, it is defined as a collection of all distinct and uncountably infinite 0-dimensional spaces.
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Nope. Again you misrepresent, misunderstand, and fantasize. That is not my assertion nor my conclusion.
 
laca said:
he said that no two points are adjacent. Which is correct, since the term is meaningless for points.
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Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?
 
doronshadmi said:
Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?
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Because the term cannot apply to points, as I said earlier. We know what adjacent means, in general. Because points don't occupy any space themselves, if there is no distance between them (the normal definition of adjacent), then they are the same point. Therefore, adjacent does not apply as a meaningful term for points.
 
HatRack said:
But, the true problem with your axiom is that it makes ZFC inconsistent.
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No HatRack, this axiom is strictly not ZFC, because ZFC is closed under the concept of Collection.

Any one that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.
 
zooterkin said:
Because the term cannot apply to points, as I said earlier. We know what adjacent means, in general. Because points don't occupy any space themselves, if there is no distance between them (the normal definition of adjacent), then they are the same point. Therefore, adjacent does not apply as a meaningful term for points.
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You are wrong, zooterkin.

"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.
 
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jsfisher said:
Nope. Again you misrepresent, misunderstand, and fantasize. That is not my assertion nor my conclusion.
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Please answer only by yes or no:

k=1 to ∞

Does any given k-dimensional space is distinct from the collection of all distinct and uncountably infinite 0-dimensional spaces?
 
HatRack said:
I don't accept your axiom, and neither will any professional mathematician, ever.
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Since when you are the voice of any professional mathematician?

Again,

Any one (professional mathematician or not) that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.
 
doronshadmi said:
No HatRack, this axiom is strictly not ZFC, because ZFC is closed under the concept of Collection.

Any one that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.
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Oh, so you choose to annihilate ALL of the ZFC axioms then. Don't you hate it when the pesky mathematics that has been developed over the centuries by all those cranks such as Newton and Cauchy and Weierstrass gets in the way of your grand new theory? Well, to each his own. Good luck deducing some useful theorems from your theory and its lone axiom.

Hey, Doron really did come up with a good idea. Next time I run into a mathematical theorem that is above my understanding, I'll just introduce an axiom which asserts its falsity. :D
 
doronshadmi said:
Since when you are the voice of any professional mathematician?

Again,

Any one (professional mathematician or not) that gets infinity only in terms of the concept of Collection, can't comprehend infinity in terms of this axiom.
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The fact that you've been going on about this nonsense for how many years now and you've yet to produce a peer-reviewed paper that has gained acceptance in the mathematical community (let alone an Internet forum) is a pretty good indication that my assertion is true.
 
doronshadmi said:
You are wrong, zooterkin.
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No, I'm not. See my signature. ;)

"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.
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Well, yes, if two points are adjacent (distance=0), then we are, in fact, talking about just one point; but I don't think you've quite grasped that yet.
 
doronshadmi said:
Laca, how exactly jsfisher's proposition "no two points are adjacent" is a correct proposition, if the term "adjacent" (which is a part of jsfisher's proposition) is (as you claim) "meaningless for points"?
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Well, think about it real hard... I know you can do it! Oh, wait...
 
doronshadmi said:
"Not adjacent" (distance>0) is related to more than one 0-dimensional space along 1-dimensional space.

"Adjacent" (distance=0) is related to exactly one 0-dimensional space along 1-dimensional space.
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In that case, adjacent is equivalent to "the same". And you just made a fool out of yourself by arguing essentially for "no two distinct points are the same". Congrats. Quite an achievement.
 
HatRack said:
The fact that you've been going on about this nonsense for how many years now and you've yet to produce a peer-reviewed paper that has gained acceptance in the mathematical community (let alone an Internet forum) is a pretty good indication that my assertion is true.
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Your assertion is true as long as the mathematical community gets infinity only in terms of the concept of Collection (it can't comprehend infinity in terms of this axiom).
 
doronshadmi said:
Please answer only by yes or no:

k=1 to ∞

Does any given k-dimensional space is distinct from the collection of all distinct and uncountably infinite 0-dimensional spaces?
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You're asking whether one space is distinct from a collection of spaces. I'm almost sure you're not as stupid as that question leads us to believe.
 
laca said:
In that case, adjacent is equivalent to "the same". And you just made a fool out of yourself by arguing essentially for "no two distinct points are the same". Congrats. Quite an achievement.
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No, it makes you fool because by your assertion a 1-dimensional space is a collection of all distinct and uncountably infinite 0-dimensional spaces.

Exactly like jsfisher, you can't comprehend the term "distinct" if it is related to a collection of more than a single 0-dimensional space.
 
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doronshadmi said:
Your assertion is true as long as the mathematical community gets infinity only in terms of the concept of Collection (it can't comprehend infinity in terms of this axiom).
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I'm sure you've gotten this before. But, the problem is: no one knows what the hell you're talking about.
 
laca said:
You're asking whether one space is distinct from a collection of spaces. I'm almost sure you're not as stupid as that question leads us to believe.
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This question is stupid as long as one comprehends infinity only in terms of collection.
 
zooterkin said:
No, I'm not. See my signature. ;)



Well, yes, if two points are adjacent (distance=0), then we are, in fact, talking about just one point; but I don't think you've quite grasped that yet.
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These is no such a thing like "two points are adjacent", because distance 0 is exactly one distinct point.

If there are more points, then and only then "Not adjacent" (distance>0) holds between any collection of distinct points, even if this collection is the collection of all distinct and uncountably infinite 0-dimensional spaces.
 
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doronshadmi said:
No, it makes you fool because by your assertion a 1-dimensional space is a collection of all distinct and uncountably infinite 0-dimensional spaces.
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How does that make me a fool? It's a proven fact in the commonly accepted axiomatic system. Before you ask, let me tell you why it's commonly accepted: it's useful, consistent and makes sense. None of which is applicable to your gibberish.

doronshadmi said:
Exactly like jsfisher, you can't comprehend the term "distinct" if it is related to a collection of more than a single 0-dimensional space.
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That's, to put it bluntly, a lie. It's not true, because you have been informed that we actually do comprehend what it means to say two points are distinct. You repeating it makes it a lie. Stop lying. Doesn't look well on your resumé.
 
HatRack said:
Hey, Doron really did come up with a good idea. Next time I run into a mathematical theorem that is above my understanding, I'll just introduce an axiom which asserts its falsity. :D
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You can't do that, because first you have to understand the tautology of this mathematical theorem.
 
laca said:
How does that make me a fool? It's a proven fact in the commonly accepted axiomatic system. Before you ask, let me tell you why it's commonly accepted: it's useful, consistent and makes sense. None of which is applicable to your gibberish.
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Wrong, it enables to use actual infinity, which is non-local by nature, and no collection has its non-local property.

One of the main future areas is Quantum Computers, which use Non-locality\Locality Bridging as their calculation methods.

You remind me a person who claims that candles are applicable light sources, and no further research (even if it is only theoretical at its first stages, exactly as OM is at this stage) is need in order to defined deeper principles for light sources.

Limited thinkers like you can be found all along the history of our civilization.
 
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doronshadmi said:
Wrong, it enables to use actual infinity, which is non-local by nature, and no collection has its non-local property.

One of the main future areas is Quantum Computers, which use Non-locality\Locality Bridging as their calculation methods.
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You get 10 points on the wacko scale for bringing in quantum.

doronshadmi said:
You remind me a person who claims that candles are applicable light sources, and no further research (even if it is only theoretical at its first stages, exactly as OM is at this stage) is need in order to defined deeper principles for light sources.
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Nowhere have I said that. Actually, using your analogy, it's like we got candles and you are trying to say that candles don't work. We see their light all the time, but you just can't accept that candles work. See, that is a valid analogy. Next time try not to come off as slanderous and do attempt an analogy that has some truth in it.

doronshadmi said:
Limited thinkers like you can be found all along the history of our civilization.
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I'm sure. And loonies like you can be found in any institution for the mentally ill.
 
laca said:
Nowhere have I said that. Actually, using your analogy, it's like we got candles and you are trying to say that candles don't work.
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Wrong, I see that it works, but I have a novel idea about the concept of source of light, that your candle-only view can't comprehend.

laca said:
Nowhere have I said that.
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It is an analogy, poor laca.

laca said:
I'm sure. And loonies like you can be found in any institution for the mentally ill.
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Wrong again, candle-only thinkers like you are usually tended to put any one, which sees beyond their candle-only view, in institution for the mentally ill.

This is a typical way of thinking of dogmatic persons, who wish to eliminate anything that they can't comprehend.
 
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http://en.wikipedia.org/wiki/Absolute_Infinite
The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God.[1] He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.
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Cantor is quoted as saying:
“ The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type. "[2]
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[1] # ^ §3.2, Ignacio Jané (May 1995). "The role of the absolute infinite in Cantor's conception of set". Erkenntnis 42 (3): 375–402. doi:10.1007/BF01129011. ( http://www.springerlink.com/content/m70382468x5h5205/ )

[2] # ^ Quoted in Mind Tools: The Five Levels of Mathematical Reality, Rudy Rucker, Boston: Houghton Mifflin, 1987; ISBN 0395383153.

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http://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html
Cantor distinguished three levels of existences: 1) in the mind of God (the Intellectus Divinum); 2) in the mind of man (in abstracto); and, 3) in the physical universe (in concreto.) Cantor believed that Absolute Infinity exists only in the mind of God. But he argued that God instilled the concept of number, both finite and transfinite, into the mind of man. Cantor frequently appealed to their existence as eternal ideas in the mind of God as the basis for the existence of the transfinites in the mind of man.18 I will pursue the implications of this appeal for our understanding of contingent rationality under Epistemology below. Cantor adamantly defended the existence of the transfinites in abstracto, even arguing that God had put them into man's mind to reflect his own perfection.19 Cantor advanced infinite series representations of irrationals to claim that their existence was equivalent to that of the transfinites.20
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doronshadmi said:
Wrong, I see that it works, but I have a novel idea about the concept of source of light, that your candle-only view can't comprehend.
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There is no candle-only view. It's just that we won't take your BS and you don't like it.

doronshadmi said:
It is an analogy, poor laca.
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I should have known that I need to be more explicit when talking to special people. Nowhere have I said that there is no need for developments in mathematics. It's just that you don't even come close to understanding the concepts involved, let alone develop your own theory. You're in way over your head, whether you realize it or not.

doronshadmi said:
Wrong again, candle-only thinkers like you are usually tended to put any one, which sees beyond their candle-only view, in institution for the mentally ill.

This is a typical way of thinking of dogmatic persons, who wish to eliminate anything that they can't comprehend.
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No, doron. Mentally ill people get put in institutions for mentally ill people. There's no dogma here except your own. It has been shown to you why you're wrong and yet you keep spouting the same old tired nonsense. That's dogmatic.
 
laca said:
It has been shown to you why you're wrong
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No, it has been shown that persons like you gets infinity only in terms of collections.

As a result you can't comprehend the non-local property of 1-dimensional space, which is infinity beyond the concept of collection.

laca said:
You're in way over your head, whether you realize it or not.
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Actually the size of your head is exactly 0-dimensional space and from this size you can't comprehend 1-dimensional space.
 
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doronshadmi said:
No, it has been shown that persons like you gets infinity only in terms of collections.

As a result you can't comprehend the non-local property of 1-dimensional space, which is infinity beyond the concept of collection.



Actually the size of your head is exactly 0-dimensional space and from this size you can't comprehend 1-dimensional space.
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Yeah, yeah, let's cut to the chase. Do you have anything resembling reason to add as to why points cannot completely cover a line? As things stand, you got squat.
 
doronshadmi said:
Please answer only by yes or no:

k=1 to ∞

Does any given k-dimensional space is distinct from the collection of all distinct and uncountably infinite 0-dimensional spaces?
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Gibberish cannot be answered with either a yes or no response. Did you mean to ask if for any integer N>0 is N-space distinguishable from a collection of uncountably many points? Did you have a particular collection of points in mind?
 
doronshadmi said:
Actually the size of your head is exactly 0-dimensional space and from this size you can't comprehend 1-dimensional space.
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You might want to review your membership agreement, Doron. It would be a shame for you to get suspended from the forums, or worse, for silly rule violations.
 
laca said:
Do you have anything resembling reason to add as to why points cannot completely cover a line?
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Yes.

Look how simply is understood by this little game:

For this game we need two points, a line and a plane.

Let the plan be the screen itself.

Now we locate two arbitrary points on the screen and call them point A and point B, as follows:

.A B.

Point A can be simultaneously at most at one location.

Point B can be simultaneously at most at one location.

Now add the line to the game, as follows:

A B

The line can be simultaneously at least at location A AND at location B.

Being simultaneously at least at location A AND at location B is possible in this case, because the line has a non-local property.

Being simultaneously at most at one location is the best that we can get from a point, because a point has local property.

It does not matter how infinitely many scale levels are defined (they can be also uncountable infinitely many), the non-local property of the line is not vanished, and so is the local property of the points.

If you understand this little game, then you immediately and unconditionally understand that a line can't completely be covered by points even if the collection of points is complete (no point is missing).

By understanding this beautiful fact, you also immediately and unconditionally understand that a line has a degree of infinity which is strictly greater than any given collection of points along it.

The same holds among any collection of given dimensions, such that:

The infinity of a line is strictly greater than the infinity of collection of points.

The infinity of a plan is strictly greater than the infinity of collections of lines and points.

Etc …

Fullness (that has no successor), which is the opposite of Emptiness (that has no predecessor), has the greatest infinity, which is grater than any collection of dimensional spaces even if this collection has uncountable infinitely many dimensions.
 
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doronshadmi said:
Look how simply is understood by this little game:

For this game we need two points, a line and a plane.

Let the plan plane be the screen itself.
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FIFY. "The screen"? Did you mean my computer monitor screen? It's is not really a plane, but I suppose we can accept it as a metaphor.

Now we locate two arbitrary points on the screen and call them point A and point B, as follows:

.A B.
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Did anyone besides Doron not recognize what important word he omitted?

Point A can be simultaneously at most at one location.

Point B can be simultaneously at most at one location.
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Oddly phrased, but ok.

Now add the line to the game, as follows:

A B
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Here's where that missing word becomes important. There is no guarantee a unique line as been determined at this point in the game.

The line can be simultaneously at least at location A AND at location B.
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No, it can't. This is where Doron seriously abuses both language and mathematical concept. Part of the line (exactly one point, in fact) coincides with point A, and part of the line (again, exactly one point) coincides with point B.

Language and concept abuse devolves into the following gibberish:

Being simultaneously at least at location A AND at location B is possible in this case, because the line has a non-local property.

Being simultaneously at most at one location is the best that we can get from a point, because a point has local property.

It does not matter how infinitely many scale level are used (they can be also uncountable infinitely many), the non-local property of the line is not vanished, and so is the local property of the points....
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...and so on.
 
jsfisher said:
Here's where that missing word becomes important. There is no guarantee a unique line as been determined at this point in the game.
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All we need is one line.

jsfisher said:
No, it can't. This is where Doron seriously abuses both language and mathematical concept. Part of the line (exactly one point, in fact) coincides with point A, and part of the line (again, exactly one point) coincides with point B.
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Ho, yes it can.

This is where jsfisher seriously abuses both language and mathematical development, becuse his notion is closed under the concept of Collection.

jsfisher your mental limitations can't prevent the development of the mathematical science beyond the concept of Collection.

The concept of Collection is still there but it is not the only player in the game anymore.
 
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doronshadmi said:
jsfisher said:
Here's where that missing word becomes important. There is no guarantee a unique line as been determined at this point in the game.
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All we need is one line.
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Be that as it may, this is just another example of Doron not being able to express himself with any clarity or precision.

jsfisher said:
No, it can't. This is where Doron seriously abuses both language and mathematical concept. Part of the line (exactly one point, in fact) coincides with point A, and part of the line (again, exactly one point) coincides with point B.
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Ho, yes it can.
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The word you are struggling to find on your keyboard is "oh". Ho is slang for prostitute. Please use the correct word.

This is where jsfisher seriously abuses both language and mathematical development, becuse his notion is closed under the concept of Collection.
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Again you blame me for your failure. You have no basis for this bald accusation, and it isn't true. It may be convenient for you to think otherwise, but you are simply lying to yourself.

Really, Doron. Stop lying about me to cover your own blunders and cognitive limitations. If doronetics has any merit, focus on that. Your continual lying and inability to stay on topic only underscores that it doesn't.

jsfisher your mental limitations can't prevent the development of the mathematical science beyond the concept of Collection.

The concept of Collection is still there but it is not the only player in the game anymore.
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As I have said right along. Knock yourself out! Develop a new mathematical foundation. Mathematics has plenty of room for new ideas.

Heck, you even have your first axiom for Doronetics. Admittedly, it is not a very constructive axiom, and it lacks some foundation, but its yours.

On the other hand, stop accusing the parts of Mathematics you can't understand as being wrong. They aren't. They certainly aren't wrong just because you'd like something else. Build your something else; try to avoid the obvious contradictions and inconsistencies that plague all of your work to date; then maybe we can explore its utility.

So far, Doron, all you do harp about how everyone else is wrong and how everyone else just doesn't get it. You really need a mirror.
 
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