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19th January 2019, 02:22 PM  #3281 
Penultimate Amazing
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Wholeness is not necessarily Comleteness
Wholeness is not necessarily Completeness, as seen in http://www.internationalskeptics.com...postcount=2798 exactly because infinitely many things are infinitely weaker that actual infinity (as seen in http://www.internationalskeptics.com...postcount=3095).
In order to deal with such notions, philosophy and mathematics are inseparable of each other (http://www.internationalskeptics.com...postcount=3280). 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

29th January 2019, 04:36 AM  #3282 
Penultimate Amazing
Join Date: Mar 2008
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Please look at the following diagram:
It was known as "2X=X√2 paradox" (This is an old "problem" that was known at least to Leibniz and probably to the Greeks). Actually, this is not a paradox at all since no integer is an irrational number, and a straightforward way to show it, is by X=1, that is, 2>√2. By observing the top of the attached diagram, one finds the convergent series a+b+c+d+... 1) Please pay attention that this series is rigorously defined by the intersections of the black straight lines (which go through the peaks of the zigzag (black, red, green, magenta, blue, cyan) lines with constant length 2X) with each side of the square. 2) It means that the mathematical fact that 2X>X√2, is inseparable of the mathematical fact that 2X>2(a+b+c+d+...). Let X (one side of the square) = 1 In that case (a+b+c+d+...) is actually (1/2+1/4+1/8+1/16...). By (2) 2(1)>2(1/2+1/4+1/8+1/16...), which can be reduced into 1>1/2+1/4+1/8+1/16... It has to be stressed that no partial sums like a, a+b, a+b+c, ... are involved in this argument, but not less than the series a+b+c+d+... If one does not agree with the argument above, one has to prove (according to the considered diagram) that series a+b+c+d+... is not defined by the zigzag lines (where, again, no partial sums like a, a+b, a+b+c, ... are involved in such proof). Moreover, if one proves it, one also demonstrates why visualization is insufficient for rigorous mathematical results. I am fully aware that what is called "not a summation in the usual sense" means a+b+c+d+... ≤ X, where the semantics (meaning) of ≤ (in the considered case) is "not greater than" X, or "at most" X. Since series a+b+c+d+... is strictly defined by all the zigzag lines such that 2X is strictly > X√2, series a+b+c+d+... can't be but strictly < X. So I still do not see how ≤ is relevant to the diagram above. 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

29th January 2019, 07:21 AM  #3283 
Penultimate Amazing
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Posts: 12,883

I wish to stress that, for example:
S = 1/2+1/4+1/8+1/16... 2S = 1+1/2+1/4+1/8+1/16... 2S  S = 1  S is not a proof of the considered case because: 1) By omitting S from 2S there is no guarantee that the omitted value (= 1/2+1/4+1/8+1/16...) is equal to the nonomitted value (= 1). 2) The separability between 2>√2 and 1>1/2+1/4+1/8+1/16... has not been proven. 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

29th January 2019, 07:30 PM  #3284 
Graduate Poster
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error

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I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

31st January 2019, 08:30 AM  #3285 
Penultimate Amazing
Join Date: Mar 2008
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The standard notion of set (according to Prof. Melvin Randall Holmes):
Quote:
By logically going beyond the notion of collection {} is tautology and {} is contradiction, such that any given collection is ~contradiction AND ~tautology. As about cardinality: {} = 0 {} = ∞ = the cardinality of actual infinity {...} = any cardinality > 0 AND < ∞ Some examples: {{}} = 1 {{}} = ∞ {{}} = 0 {{1,2}} = 1 {{1,2}} = 2 {1,2} = 2 {{1,{},2}} = 0 {{1,{},2}} = ∞ Nested cardinality examples: {{1,{},2}} = (((0)3)1)∞ {{1,{},2}} = (((0)1)1)∞ {{1,{},2}} = (((0)1,1)1)∞ {{1,{},2}} = (((0)1,1,1)1)∞ {} = (0)∞ etc. ...  As can be seen, the standard notion of collection is a very limited mathematical framework. 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

2nd February 2019, 11:52 PM  #3286 
Penultimate Amazing
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By going beyond the notion of collection (which is a composed thing) the noncomposed is defined by noncomposed opposite extremes, which are NOthing and YESthing, where the cardinalities (the magnitudes) of them are {} = (0)∞
So the cardinality of any give collection is > 0 AND < ∞, which means that no collection is accessible to that has cardinality 0 (NOthing) or cardinality ∞ (YESthing). By being aware of the composed and the noncomposed, one enables to understand why a collection with endless members is not actual infinity, simply because it is inaccessible to YESthing (that has cardinality ∞). 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

20th March 2019, 07:08 PM  #3287 
Philosopher
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Doronshadmi, look into palindromic numbers and their sums. The key lies there. Trust me.

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Fill the seats of justice with good men; not so absolute in goodness as to forget what human frailty is.  Thomas Jefferson What region of the earth is not filled with our calamities?  Virgil 

22nd March 2019, 03:10 PM  #3288 
Penultimate Amazing
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__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

22nd March 2019, 04:02 PM  #3289 
I say nay!
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I am 100% confident all psychics and mediums are frauds. 

23rd March 2019, 08:43 AM  #3290 
Philosopher
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Location: Detroit
Posts: 5,431

I capitalize but not symbolic

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Fill the seats of justice with good men; not so absolute in goodness as to forget what human frailty is.  Thomas Jefferson What region of the earth is not filled with our calamities?  Virgil 

23rd March 2019, 02:39 PM  #3291 
Penultimate Amazing
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__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

27th March 2019, 07:26 AM  #3292 
Penultimate Amazing
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__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

27th March 2019, 05:55 PM  #3293 
Graduate Poster
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Doron, why are you replying to a post that was made over 6 months ago? Why are you not replying to my question?

__________________
I'm an "intellectual giant, with access to wilkipedia [sic]" "I believe in some ways; communicating with afterlife is easier than communicating with me." Tim4848 who said he would no longer post here, twice in fact, but he did. 

27th March 2019, 07:43 PM  #3294 
Philosopher
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Posts: 5,431

Doron, you are clearly afraid of the implications of palindromic number summation. You do not surprise me in this.
But if you can overcome your timidity and resume silence, much can still be achieved. 
__________________
Fill the seats of justice with good men; not so absolute in goodness as to forget what human frailty is.  Thomas Jefferson What region of the earth is not filled with our calamities?  Virgil 

29th March 2019, 06:51 PM  #3295  
ETcorngods survivor
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A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!"  Monketey Ghost 

29th March 2019, 09:49 PM  #3296 
Philosopher
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Location: Mesa, AZ
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"At the Supreme Court level where we work, 90 percent of any decision is emotional. The rational part of us supplies the reasons for supporting our predilections." Justice William O. Douglas "Humans aren't rational creatures but rationalizing creatures." Author Unknown 

30th March 2019, 01:59 AM  #3297 
Penultimate Amazing
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Please very carefully observe Prof. Edward Frenkel's YT video on vectors https://www.youtube.com/watch?v=PFkZGpN4wmM .
The essence of his notion is given from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video, where he introduces the idea which I call "Math Over Matrix", which is actually our ability to deduce also beyond the notion of collections (the mug is not a collection of its projections). I find that his idea of the difference between "the thing in itself" and its representations (he is an expert in Representation Theory) mathematically airs its view by the following two axioms: (1) The Axiom Of NonComplexity: There exists, at least, object _ (1dim object), such that it is not a collection of shorter or shortest objects. (a shortest object is at least . (0dim object)). (2) The Axiom Of Markers: Given a collection of shorter or shortest objects, they define values with respect to, at least, object _ Without (1) there is no, at least, object _ , and without (2) no value can be defined with respect to, at least, object _ So a useful mathematical framework, in this case, is based on, at least, (1) AND (2). Please pay attention that (1) and (2) are deduced by using both visual_spatial AND verbal_symbolic reasoning, which actually enables to distinguish between "the thing in itself" (the noncomposed existence, given by visual_spatial) and a collection (the composed, given by verbal_symbolic reasoning) of representations that are related to it but they are not the same as the "the thing in itself" (the mug ("the thing in itself") is not a collection of its projections (it is not its representations)). Please pay attention that _ or . is noncomposed, where this common property actually enables _ . deduction by the same framework. But in order to do useful deduction, _ or . are also distinguishable of each other such that at least _ (1dim object) is irreducible into at least . (0dim object) AND at least . (0dim object) is not expansible into at least _ (1dim object). Now let's deduce spatially AND symbolically according to (1) and (2) by using what is known as positional numeral system (if only verbal_symbolic reasoning is used). Since the considered mathematical framework is based on at least visual_spatial AND verbal_symbolic reasoning, one actually uses positional number system. Let's observe, for example 0.111...[base 2] and 0.222...[base 3] (which are numbers of their own that are not equal to 1, if visual_spatial AND verbal_symbolic reasoning is used), by the following visual_spatial AND verbal_symbolic diagram: Since at least _ (1dim object) is irreducible into at least . (0dim object), for example, 0.222...[base 3] < 1 by at least 0.000...1[base 3], then the ...1 in 0.000...1[base 3] is exactly the irreducibly of at least _ (1dim object) into at least . (0dim object). By using visual_spatial AND verbal_symbolic reasoning, the notion of invariant proportion is used among potential infinitely many scales (where potential infinity is simply the result of at least _ (1dim object) that is irreducible into at least . (0dim object)) as follows: 0.222...[base 3] < 1 by at least 0.000...1[base 3], but this is only the least case under [base 3]. Now, please (by using visual_spatial AND verbal_symbolic reasoning) very carefully pay attention to the potential infinitely many stairs in 0.222...[base 3] along the diagram above. By doing so we get the following: 0.222...[base 3] < 1 by at least 0.000...1[base 3] but 0.22...[base 3] < 1 by at least 0.000...2[base 3] where the invariant proportion among 0.000...2[base 3] (___) and 0.000...1[base 3] (_) is kept along the diagram, exactly because _ or ___ (1dim form) are irreducible into . (0dim form). ___ is 3 times longer than _ by the invariant proportion along the potential infinitely many scales of the [base 3] positional number system (of the example above). [base 3] positional number system is used here for any [base > 1] positional number system, without loss of generality ( https://en.wikipedia.org/wiki/Withou..._of_generality ). 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

30th March 2019, 09:25 AM  #3298 
Penultimate Amazing
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In the previous post I wrote:
By more careful observation, I have realized that ...x[base y] form has to be involved with the power value of some base value along the potential infinitely many scales of a given positional number system (without being confused with the base^{power} at the left side of the radix point), so the right representation of ...n[base y] form (as done above) has to be as follows: Let b be the base value, which is represented by any natural number > 1 Let p be the power value of b, which is represented by any natural number ≥ 0 The general form is ...b^{p} If (for example) b = 3, then a better representation of an invariant proportion along potential infinitely many scales (of this case) is done as follows: 0.000...1[base 3] is actually ...3^{0} 0.000...2[base 3] is actually ...3^{1} etc. (I knew that I used 3 as a part of [base 3], but it is done as a "short cut" in order to air my view more simply at first glance) Now let's rewrite the quote by using the new symbolic representation (including some highlighted corrections): 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

31st March 2019, 10:53 AM  #3299 
Penultimate Amazing
Join Date: Mar 2008
Posts: 12,883


__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

31st March 2019, 11:45 AM  #3300 
Penultimate Amazing
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Posts: 12,883

Invariant proportion
As for the notion of invariant proportion, it is easily understood by pi (if defined by circle's circumference\diameter ratio), which is irreducible into . AND not extensible into straight _
This invariant proportion is found among a collection of many circles, but since a collection is a composed thing, it has at most potential infinitely many circles, exactly because actual infinity is the noncomposed property of at least 1dim thing. By ACTUALLY USING visual_spatial AND verbal_symbolic reasoning, one gets the following hierarchy of existence: 1) Actual infinity (al least the noncomposed _______ thing) 2) Finite (the finitely composed that is finitely weaker than the noncomposed _______ thing and therefore can be added up to a given value (for example 1, that is defined at the edge of 0______1) 3)Potential infinity (the infinitely composed that is infinitely weaker than the noncomposed _______ thing and therefore can't be added up to a given value (for example 1, that is defined at the edge of 0______1) 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

31st March 2019, 12:59 PM  #3301 
Penultimate Amazing
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Posts: 12,883

Dear Apathia please be aware that my framework is not only over matrix, as observed in the posts following your "Thank you!" post.
If you carefully observe them, you may find how I am doing my best in order to define nonfixed numbers in therms of invariant proportion, which is a matrix fantasy that is not used by traditional mathematicians. By this matrix fantasy positional numeral system is defined as positional number system that enables deduction about nonfixed numbers, which are currently not accepted by traditional mathematicians, exactly because they define actual infinity in terms of the composed (in terms of collections). As you know, more details are already given in http://www.internationalskeptics.com...postcount=3095 , but now (in order to not be closed under subjectobject duality) I suggested to exchange "object" by "thing". More details are also given in http://www.internationalskeptics.com...postcount=3285 and http://www.internationalskeptics.com...postcount=3286 (in this post the term "thing" is used ( NOthing can also be visually represented by . and YESthing can also be visually represented by __ , in addition to the symbolic representation {}, where the outest "{" and "}" are the same as __ and the void between {} is the same as . )). 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

3rd April 2019, 03:55 AM  #3302 
Penultimate Amazing
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Posts: 12,883

Here is a quote taken from https://plato.stanford.edu/entries/cusanus/#MetOnt about Nicolas of Cusa:
Quote:
The noncomposed in our awareness exercise is defined as an endless noncomposed straight 1dimensional thing (...____...) and as a point (0dimensional thing (.) . By length observation ...____... is the biggest length and . is the smallest length. By curvature observation ...____... is the smallest curvature and . is the biggest curvature.  The most symmetrical thing which not ...____... AND not . , is called a circle. There are endlessly many circles that share a common property known as PI, which is the invariant proportion between circle's circumference and circle's diameter. So by this awareness exercise (which is done both visually AND verbally) no circle is ...____... or . (where ...____... or . are the extreme states that share the noncomposed as their common property), whether it is observed in terms of length or in terms of curvature. Yet ...____... and . are reachable from a given circle or a collection of given circles, by finitely many steps. So finite collections (which are finitely composed things) are strong enough in order to actually reach ...____... or . noncomposed extremes. Therefore finite collections are defined as finitely weaker than the noncomposed (which is the common property of ...____... and .). On the contrary, infinite collections (which are infinitely composed things) are not strong enough in order to actually reach ...____... or . noncomposed extremes. Therefore infinite collections are defined as infinitely weaker than the noncomposed (which is the common property of ...____... and .). Generally, more you composed less you accessible to the noncomposed (this observation is logically supported by visual_spatial AND verbal symbolic deduction, as given by https://drive.google.com/file/d/0B3h...ew?usp=sharing which rigorously demonstrates how endless upwards growth of the bivalent tree (which is the most basic logical structure) is developed in the opposite direction w.r.t the noncomposed (called Unity in this diagram)).  By generalization of the awareness exercise the noncomposed (which is the common property of ...____... and .) is defined as actual infinity, where the composed is finitely weaker OR infinitely weaker than actual infinity. The infinitely weaker than actual infinity is defined as potential infinity. By this generalization, actual infinity (which is the noncomposed) can't be measured or be used as a measurement of the composed, whether it is finitely OR infinitely composed. So no collection (which is a composed thing ( circles' multiplicity is the result of not being ...____... AND not being . )) is mensurable in terms of actual infinity (which is the noncomposed that is the common property of ...____... and .).  Another aspect of this awareness exercise, is the ability to observe circles "objectively" (from the outside) from ...____... to the collection of circles, or "subjectively" (from the inside) from . to the collection of circles.  What is given above can't be known by using verbal_symboliconly observation, and unfortunately the treatment of modern mathematics (which is also known as traditional mathematics) of infinity, is done only in the domain of the composed, and because of this reason actual infinity is understood and defined in terms of the composed (in terms of collections).  This Fundamental Conceptual Mistake simply prevents one of being aware of actual infinity (which is essentially noncomposed).  By being aware of the noncomposed without being lost within its finite or infinitely many expressions, life phenomena is actually fulfilled. 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

4th April 2019, 01:17 AM  #3303 
Penultimate Amazing
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By http://www.internationalskeptics.com...postcount=3302 it is concluded that the whole (the noncomposed) is not the sum of its expressions, which means that the noncomposed is THE EXISTENCE (also called BEING, BRAHMAN, EINSOF, etc.) that is not limited by its collection of (finite or potentially infinite) expressions.
Life phenomenon that is aware of THE EXISTENCE during its collection of (finite or potentially infinite) expressions, has TRUE free will known also as Samādhi (also called Unity Consciousness). But as you notice, no representation (visual_spatial, verbal_symbolic etc. or their associations) is THE EXISTENCE (The thing in itself). 
__________________
That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

14th April 2019, 07:17 AM  #3304 
Penultimate Amazing
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Zeno's Achilles\Tortoise Race
First, one has to define what are the minimal conditions that enable a race, in the first place.
Both in Mathematics and Physics there are primitive notions which are themselves undefined, and they are used as the building blocks of deduction or induction in order to avoid endless regression (An endless regression prevents a strict solution of some problem). In that case we may ask: Is a solution is always a strict solution, or there are also solutions that are not strict (which means that endless regression (or endless progression) is also a possible solution)? In order to demonstrate these notions, let's carefully observe, for example, point and line as primitive notions under Hilbert's axiom system of Euclidean Geometry: "1. For every two points A and B there exists a line a that goes through A and goes through B ." If we understand primitive notions in terms of atoms, then atoms are actually noncomposed things (they are not defined by other building blocks since they are themselves building blocks). In that case a line is not a collection of points (it is a noncomposed thing exactly as a point is a noncomposed thing) which means that the term "goes through" does not define a point as a building block of a line, and no collection of points is a line. Now, by using these observations, let's carefully observe the minimal conditions that enable a race, in the first place. If only a line is considered, we can't define points along the line that define at least two competitors. If only a point is considered, then nor competitors neither a race path are definable. So, the minimal conditions that enable a race is a line and at least two points along it, such that a point and a line are noncomposed (they are not building blocks of each other and yet they can associate through their common property of being noncomposed). The existence of a line as a noncomposed thing (as a buildingblock) is stronger then any collection of distinct points along it. As a result, in order to actually reach a given point from another given point, it is done by a single noncomposed line segment, which simultaneously connects the given pair of distinct points with each other. Such connections, are always done by finitely many steps. Therefore, a finite collection of points is defined as finitely weaker than the existence of a line (which is a noncomposed thing). Moreover, by this observation, the existence of a line as a noncomposed thing, defines it as actual infinity, which is reachable by a collection of finitely many steps (a finite collection is finitely weaker than actual infinity). In that case a collection of infinitely many points along a given noncomposed line is defined as infinitely weaker than actual infinity, and therefore its cardinality can't be defined by strict value (as done in case of finite cardinality), which means that infinite collections are no more than potential infinity that can't reach a given point by infinitely many steps. Let's use the observation of clapping hands. By this observation, one of your hands actually reaching your other hand only by finitely many steps, since finitely many steps are finitely weaker than actual infinity (which is the noncomposed gap between your hands). Since potential infinity is infinitely weaker than the noncomposed gap between your hands, no infinitely many steps of your hand actually reaching your other hand (since potential infinity is inaccessible to actual infinity, by definition).  Zeno's Achilles\Tortoise Race It is argued that Zeno's Achilles\Tortoise Race is not a paradox in real life because we can summarize nonfinite values (where each value > 0) that are added to some initial value. By doing that we are able to get an accurate value, which is different from the initial value. For example: 1 is the initial value and 1+1/2+1/4+1/8+…= 2, where 2 is an accurate value that is different from the initial value 1. Actually the whole idea of Limits is somehow motivated by the desire to solve Zeno's Paradox. Let us investigate two different cases of Achilles\Tortoise Race: Case A: Achilles wins against the Tortoise OR Achilles and the Tortoise are on the same position, and the Race stops. Case B: Achilles does not win against the Tortoise, and the Race continues (actually forever). Case A: Distance = Speed * Time The next position of Achilles and the Tortoise along the Race = previous position + Distance. Case A exists only if neither Speed nor Time are changed during the Race. Let us show it by using an algorithm (no particular programming language is used here): Position X1 = 0 Position X2 = 10 Achilles Speed = Aspeed = 10 Tortoise Speed = Tspeed = 1 Time = 1 Do Loop K from 1 to forever Achilles position = position X1 + distance ( = Aspeed * Time) Tortoise position = position X2 + distance (= Tspeed * Time) Position X1 = Achilles position Position X2 = Tortoise position If X1 ≥ X2 then STOP Next Loop K Loop K=1: Achilles position = 0 + (10 * 1) = 10 Tortoise position = 10 + (1 * 1) = 11 Position X1 = Achilles position = 10 Position X2 = Tortoise position = 11 If X1 ≥ X2 then STOP (The Race continues after Loop K=1, since Tortoise position > Achilles position) Next Loop K=1 Loop K=2: Achilles position = 10 + (10 * 1) = 20 Tortoise position = 11 + (1 * 1) = 12 Position X1 = Achilles position = 20 Position X2 = Tortoise position = 12 If X1 ≥ X2 then STOP (The Race stops after Loop K=2, since Tortoise position < Achilles position) Next Loop K=2 Case A is finitely weaker than actual infinitely.  Case B: Distance = Speed * Time In case B , Time is changed during the Race. Let us show it by using an algorithm (no particular programming language is used here): Position X1 = 0 Position X2 = 10 Achilles Speed = Aspeed = 10 Tortoise Speed = Tspeed = 1 Time = 1 Do Loop K from 1 to forever Achilles position = position X1 + distance ( = Aspeed * Time) Tortoise position = position X2 + distance (= Tspeed * Time) Position X1 = Achilles position Position X2 = Tortoise position If X1 ≥ X2 then STOP Time = Time / Aspeed (Achilles Speed = Aspeed = 10) Next Loop K Loop K=1: Achilles position = 0 + (10 * 1) = 10 Tortoise position = 10 + (1 * 1) = 11 Position X1 = Achilles position = 10 Position X2 = Tortoise position = 11 If X1 ≥ X2 then STOP Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.1 (The Race continues after Loop K=1) Next Loop K=1 Loop K=2: Achilles position = 10 + (10 * 0.1) = 11 Tortoise position = 11 + (1 * 0.1) = 11.1 Position X1 = Achilles position = 11 Position X2 = Tortoise position = 11.1 If X1 ≥ X2 then STOP Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.01 (The Race continues after Loop K=2) Next Loop K=2 Loop K=3: Achilles position = 11 + (10 * 0.01) = 11.1 Tortoise position = 11.1 + (1 * 0.01) = 11.11 Position X1 = Achilles position = 11.1 Position X2 = Tortoise position = 11.11 If X1 ≥ X2 then STOP Time = Time / Aspeed (Achilles Speed = Aspeed = 10) = 0.001 (The Race continues after Loop K=3) Next Loop K=3 ... Loop K n+1 The Race continues forever because any given position of Achilles and the Tortoise that comes next is the result of previous positions + Distances values, where each Distance value > 0 (Achilles position < Tortoise position is an invariant state). Case B is infinitely weaker than actual infinitely. For more details, please observe http://www.internationalskeptics.com...postcount=3302 .  Traditional mathematicians take such loops as a sequence of partial sums. For example, they are claim that 0.999... is not a term of the following sequence of partial sums: (0.9, 0.99, 0.999, ...) Let's check this claim. The result of Code:
1 ↓ 2 0.9 = 0.9 ↓ 3 0.99 = 0.9 ( + 0.09) ↓ 0.999 = 0.9+0.09 ( + 0.009) ... Code:
1 2 3 ... ↓ ↓ ↓ 0.9 + 0.09 + 0.009 + ... What we care is about the result of the series of infinitely many 0.9+0.09+0.009+... added finite numbers (where every one of them > 0) which are embedded in the infinitely many terms of the sequence (0.9, 0.99, 0.999, ...).  Also we are aware that any infinite collection is infinitely weaker than actual infinity (which is noncomposed, as observed in this post). 
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That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. 

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