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Old 19th January 2019, 02:22 PM   #3281
doronshadmi
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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).
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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.

Last edited by doronshadmi; 19th January 2019 at 02:46 PM.
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Old 29th January 2019, 04:36 AM   #3282
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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 zig-zag (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 zig-zag 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 zig-zag 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.
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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.

Last edited by doronshadmi; 29th January 2019 at 06:23 AM.
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Old 29th January 2019, 07:21 AM   #3283
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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 non-omitted value (= 1).

2) The separability between 2>√2 and 1>1/2+1/4+1/8+1/16... has not been proven.
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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.

Last edited by doronshadmi; 29th January 2019 at 07:44 AM.
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Old 29th January 2019, 07:30 PM   #3284
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Old 31st January 2019, 08:30 AM   #3285
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The standard notion of set (according to Prof. Melvin Randall Holmes):
Quote:
A set is a collection determined by its elements. Finite sets are often written {a, b, c} (for example), by listing their elements. Order does not matter and repeated items do not change the intended meaning.

The elements of the sets are not parts of the set. The set is not made by conglomerating its elements together. This is a common misunderstanding.

To see this it is enough to play with the notation. {x} is not the same object as x: if a set were made up of its elements as parts, this would not make sense. If you don’t believe this, look at {{2, 3}}: this is a set with one element, while its sole element is a set with two elements, so they are different.

Another way of seeing it is to notice that a relation of part to whole should be transitive. If a is part of b and b is part of c, then a is part of c. But notice that 2 ∈ {2, 3} and {2, 3} ∈ {{2, 3}}, but 2 is not a member of {{2, 3}}


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.
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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.

Last edited by doronshadmi; 31st January 2019 at 08:50 AM.
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Old 2nd February 2019, 11:52 PM   #3286
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By going beyond the notion of collection (which is a composed thing) the non-composed is defined by non-composed 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 non-composed, one enables to understand why a collection with endless members is not actual infinity, simply because it is inaccessible to YESthing (that has cardinality ).
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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.

Last edited by doronshadmi; 3rd February 2019 at 12:11 AM.
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Old 15th March 2019, 01:42 AM   #3287
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Math Over Matrix (MOM) vs Math Under Matrix (MUM)

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 notion of 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).

Math Over Matrix or MOM axiomatic framework is given in http://www.internationalskeptics.com...postcount=3095 , http://www.internationalskeptics.com...postcount=3108 and http://www.internationalskeptics.com...postcount=3110 and it is done by using visual_spatial AND verbal_symbolic reasoning.

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

Math Under Matrix or MUM is done by using only verbal_symbolic reasoning.

By MUM, for example, 0.999...[base 10] = 1, and a person named Chirs Seib ( please look at the discussions in https://www.youtube.com/watch?v=SDtFBSjNmm0&t=3s ) provided a generalization for any base > 1, as follows:

First we observe that the base value is always greater by 1 from the digits that are used in the numbers of the form B.BBB... (where B is any digit > 0), for example:

If B = 1, then the base value is 2.
If B = 2, then the base value is 3.
If B = 3, then the base value is 4.
...
If B = 9, then the base value is 10.
etc. ...

Here is @Chris Seib general form for any base > 1:

(B+1) * 0.BBB... = B.BBB...
B * 0.BBB... + 0.BBB...= B + 0.BBB...
B * 0.BBB... = B
0.BBB... = 1
------------------

Let B = 9 (without loss of generality) and in that case the base value is 9+1=10

In that case we have the following steps:
10 * 0.999... = 9.999...
(9+1) * 0.999... = 9.999...
9 * 0.999... + 0.999...= 9 + 0.999...
9 * 0.999... = 9
0.999... = 1

Let's do it in "slow motion" (as suggested by @logic892173):

10 * 0.999... = 9.999...
(9+1)*0.999... = 9.999...
9 * 0.999... + 1*0.999... = 9.999...
9 * 0.999... + 0.999... = 9.999... (as @logic892173 wrote "the distributive property say that (a+b)*c = a*c + b*c for any real numbers a,b,c")
9 * 0.999... + 0.999...= 9 + 0.999...
9 * 0.999... = 9
(9*0.999...)/9 = 9/9
1*0.999... = 9/9
0.999... = 1
Alternatively
(9*0.999...)/0.999... = 9/0.999...
9*1 = 9/0.999...
9 = 9/0.999...
9/9 = (9/0.999...)/9
1= (9/0.999...)/9

So the general form as given by @Chris Seib holds perfectly (or so it seems at first glance as long as one is unaware of MOM and MUM difference).

I admit, I am doing a lot of mistakes during my work, and the last one was my nonsensical treatment about (5*2)=10 and 9*0.999...=9 (as I did in https://www.youtube.com/watch?v=SDtFBSjNmm0&t=3s).

So let's do it very carefully in order to avoid (as much as possible) nonsensical treatments about 0.999...

First, here is again the relevant part of my argument to @Mathologer about 0.999... :

"@Mathologer at 12:16 you subtract M=9+9/10+9/100+9/1000+... from 10M=90+9+9/10+9/100+9/1000+... and you are left with 9M=90. But by doing so you simply eliminate M that is the result of endless added values that all of them > 0 (which means that there is always some value > 0 that permanently prevents us from actually reaching (to be equal) a given limit value. So by your last step ((9M=90)/9=1M=10) the result M=10 has nothing to do with the omitted "tail" (as you call it in 12:16) M=9+9/10+9/100+9/1000+... (they are not the same mathematical object, even if they have the same name (called in this case "M")). In other words, your "slow motion" analysis does not prove that 9+9/10+9/100+9/1000+... = 10."

Now, let's carefully observe 9 * 0.999... = 9 as follows:

It has to be stressed that every number ≠ 0 that is divided by itself is equal 1, and so is the case about 0.999.../0.999... = 1

So, by this division (9 * 0.999...)/0.999... = 9/0.999... the property of 0.999... (at the left side of the equation) as an endless addition of numbers that every one of them > 0, simply "gets off stage", as seen in the left side of the equation, as follows:

(9 * 0.999...)/0.999... = 9*(0.999../0.999...) = 9*1 (the property of 0.999... (at the left side of the equation) as an endless addition of numbers that every one of them > 0, simply "gets off stage")

The result of this division is equivalent to the result of subtract M=9+9/10+9/100+9/1000+... from 10M=90+9+9/10+9/100+9/1000+... in
@Mathologer's argument (in both cases the property of 0.999... as an endless addition of numbers that every one of them > 0, simply "gets off stage").

So by taking 9 * 0.999... = 9 and dividing both sides by 0.999..., we get 9*1 = 9/0.999...

And by 9/9 = (9/0.999...)/9 we get 1 = 1/0.999... exactly because at the previous step we did 0.999../0.999..., which actually eliminates the property of 0.999... as an endless addition of numbers that every one of them > 0.

Also please pay attention that by dividing both sides of 9 = 9/0.999... by 9 ( as follows: 9/9 = (9/0.999...)/9 ) we get 1 = 1/0.999...

Please very carefully pay attention that 0.999... is not impacted by this division (the division is done only on 9 in both sides of the equation) but since it is compared to a number that is divided by itself (where the result can't be but 1) we have lost the ability to distinguish between a fixed number like 1 and a non-fixed number like 0.999... as an endless addition of numbers that every one of them > 0, which is fundamentally different than 1 (if it is defined by using also visual_spatial reasoning, which is something that traditional mathematicians don't do about arithmetic).

Let's return once again to 9 * 0.999... = 9 and divide its both sides by 9 as follows:

(9*0.999...)/9 = 9/9
9/9*0.999... = 9/9 (every number ≠ 0 that is divided by itself is equal 1)
and we get 0.999... = 1

Also in this case please very carefully pay attention that 0.999... is not impacted by this division (the division is done only on 9 in both sides of the equation) but since it is compared to a number that is divided by itself (where the result can't be but 1), we have lost the ability to distinguish between a fixed number like 1 and a non-fixed number like 0.999... as an endless addition of numbers that every one of them > 0, which is fundamentally different than 1 (if it is defined by using also visual_spatial reasoning, which is something that traditional mathematicians don't do about arithmetic).

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

Both 0.999...[base 10] = 0.9+0.09+0.009+0.0009+... or 2(a+b+c+d+...) (please observe ) are known as forms of convergent series, so the discussion generally argues about such forms, and this is exactly how fruitful mathematical discussion is done, by going beyond some particular form (0.999...[base 10] in this case).

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

Let's observe 2(a+b+c+d+...) it terms of MOM, which is a deduction that is done by by using visual_spatial AND verbal_symbolic reasoning, as follows:

During your observation of https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg please be focused on the following:

1) There are aleph0 bent 1-dim orange forms that all of them have length 1.

2) These aleph0 bent 1-dim orange forms construct 2(a+b+c+d+...) along the straight 1-dim orange form, which has length 1.

3) 2(a+b+c+d+...) = the length of the straight 1-dim orange form iff there is at least one bent 1-dim orange form (which is one of the aleph0 constructors of 2(a+b+c+d+...) along the straight 1-dim orange form), that its length = 0.

4) Since (3) does not hold (aleph0 bent 1-dim orange forms are inaccessible to the point at the bottom of the diagram (length=0)) 2(a+b+c+d+...) < the length of the straight 1-dim orange form.

5) By (4) 2(a+b+c+d+...) does not have a sum exactly because of this inaccessibility.

Again, please observe https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg and carefully follow all of these 5 steps during the observation (by all I mean that do not stop before all of the 5 steps are done During the observation).

Please let me know what you have discovered.

Thank you.


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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.

Last edited by doronshadmi; 15th March 2019 at 01:59 AM.
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Old 15th March 2019, 01:56 PM   #3288
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What is 0.000...1 by visual_spatial AND verbal_symbolic reasoning?

By deduce visually AND symbolically one understand that the ...1 of 0.000...1 simply indicates that any given infinite collection (and in this case the infinitely many 0.9+0.09+0.009+...) is infinitely weaker than actual infinity, which is non-composed by nature (addressed as non-composed ________).

So by deduce visually AND symbolically it is easily understood that 0.000...1 symbolically represents the endlessly smaller that can't be the smallest, where again being endlessly smaller is not any particular smaller value, but it is the general inability to be the smallest (there is always _ at the "right side" of _________ that can't be reduced into a point (can't become the smallest, and this is exactly what is given visually AND symbolically in https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg)).

Those who deduce only symbolically are closed under the matrix of collections and therefore wrongly deduce actual infinity in terms of collections, and as seen in "0.(9) + 0.(0)1 = 0.(9)1" symbolic gibberish , they are unaware of being under the symbolic-only matrix.
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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.

Last edited by doronshadmi; 15th March 2019 at 02:16 PM.
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Old 15th March 2019, 03:52 PM   #3289
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Originally Posted by doronshadmi View Post
... we have lost the ability to distinguish between a fixed number like 1 and a non-fixed number like 0.999...

I certainly have lost that ability. What is a non-fixed number? Does it move around? Is it greater today than it was yesterday? And why do you think it's a number? A variable could easily pull off all that.
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Old 15th March 2019, 08:53 PM   #3290
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Please define 1-dim.
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Old 16th March 2019, 08:51 AM   #3291
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Originally Posted by Hevneren View Post
I certainly have lost that ability. What is a non-fixed number? Does it move around? Is it greater today than it was yesterday? And why do you think it's a number? A variable could easily pull off all that.
A non fixed number is a value that does not have a sum.

For example 0.999...[base 10] is a non-fixed number, where 1 is a fixed number.

0.999...[base 10] < 1 exactly by 0.000...1[base 10] (where "exactly" is not necessarily "fixed").

More about 0.000...1[base 10] (which is also a non-fixed number) is already given in http://www.internationalskeptics.com...postcount=3288 .
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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.

Last edited by doronshadmi; 16th March 2019 at 08:54 AM.
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Old 16th March 2019, 09:11 AM   #3292
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Didn't you just show 0.999... = 1 ?
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Old 16th March 2019, 09:13 PM   #3293
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Originally Posted by Little 10 Toes View Post
Please define 1-dim.
That has length > 0 AND width = 0.
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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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Old 16th March 2019, 09:14 PM   #3294
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Originally Posted by zooterkin View Post
Didn't you just show 0.999... = 1 ?
No.
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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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Old Yesterday, 01:26 AM   #3295
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login892173: The symbol "<" represent an order relation. It is usually defined in the real numbers but if that sum has no value then it is not a real number.

0.999...[base 10] < 1 by 0.000...1[base 10], where ...1 at 0.000...1 indicates the inaccessibility of potential infinity (infinity that is based on the notion of collections) to actual infinity (infinity that is beyond the notion of collections, and therefore it is a non-composed whole) exactly as axiomatically given in http://www.internationalskeptics.com...postcount=3095 , http://www.internationalskeptics.com...postcount=3108 and http://www.internationalskeptics.com...postcount=3110.

Moreover, the notion of existence that goes beyond the notion of collections (what I define as actual infinity) is very simply addressed in Prof. Edward Frenkel's video as observed from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

login892173: In traditional Math infinite sums are defined to have a value and in this case 0.(9) can be defined to be the value of the infinite sum of the terms 9/10^n for all natural numbers n. Since your interpretation of your drawings gives you a different definition, the answer you arrive with your drawings is irrelevant because it represent different objects.

What you call traditional Math is a Math Under Matrix framework (MUM) exactly because it is fundamentally values things by the notion of collections.

I use Math Over Matrix framework (MOM) that fundamentally values things by also going beyond the notion of collections.

MOM enables to use potential infinity (the composed existence of infinite collections) and actual infinity (the non-composed existence beyond finite or infinite collections) in one framework.

That is also over the matrix, is aware of the matrix.

That is under the matrix, is unaware of the matrix.

In order to be also over the matrix, one uses visual_spatial AND verbal_symbolic reasoning.

More details are given in http://www.internationalskeptics.com...postcount=3287 and http://www.internationalskeptics.com...postcount=3291.
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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.

Last edited by doronshadmi; Yesterday at 03:13 AM.
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Old Yesterday, 03:33 AM   #3296
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The actual limit is beyond any infinite collection.

Chris Seib: The limit of the "sequence" of the bent orange lines is the vertex of the triangle and is a point. It is NOT a member of the sequence.

You are right, the limit of the composed (where the composed in https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg is a collection of infinitely many bent 1-dim orange forms with length=1 that construct 2(a+b+c+d+...)) is the non-composed that is beyond the composed (actual infinity is inaccessible to every infinite collection).
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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.

Last edited by doronshadmi; Yesterday at 03:48 AM.
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Old Yesterday, 06:34 AM   #3297
Little 10 Toes
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Originally Posted by doronshadmi View Post
That has length > 0 AND width = 0.
Any reason why you can't say one dimensional?

As soon as you "bend" a line segment , you now have a two dimensional object.
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Old Yesterday, 01:31 PM   #3298
doronshadmi
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Originally Posted by Little 10 Toes View Post
As soon as you "bend" a line segment , you now have a two dimensional object.
Wrong, you have a fractal dimensional object, which is ≈ 1.2618

But in https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg all we care is the length of the orange forms, whether they are straight or bent.
__________________
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.

Last edited by doronshadmi; Yesterday at 01:34 PM.
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Old Yesterday, 01:55 PM   #3299
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The symbolic illusion of partial sums argument

It is argued that 0.999... is not a term of the sequence (0.9, 0.99, 0.999, ...)

By carefully observe (0.9, 0.99, 0.999, ...) one easily discovers that

Code:
        1
        ↓         2
0.9 = 0.9         ↓      3
0.99 = 0.9 ( + 0.09)     ↓
0.999 = 0.9+0.09 ( + 0.009)
...
is actually the same as

Code:
  1         2         3   ...
  ↓         ↓         ↓
0.9  +   0.09  +  0.009 + ...
or in other words, 0.999... is actually embedded in (0.9, 0.99, 0.999, ...) |N| terms.

So the partial sums argument (0.9, 0.99, 0.999, ...) of |N| terms provides exactly the same result as given by 0.999... = 0.9 + 0.09 + 0.009 + ...

By rejecting this simple fact (by arguing that 0.999... is not a term of the sequence (0.9, 0.99, 0.999, ...)),
one actually claims that 0.9 + 0.09 + 0.009 + ... < 0.9 + 0.09 + 0.009 + ...
__________________
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.

Last edited by doronshadmi; Yesterday at 02:28 PM.
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Old Yesterday, 09:53 PM   #3300
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Originally Posted by doronshadmi View Post
Wrong, you have a fractal dimensional object, which is ≈ 1.2618

But in https://farm5.staticflickr.com/4015/...f5b36c0f_o.jpg all we care is the length of the orange forms, whether they are straight or bent.
Like most of your posts, this does not make any sense.
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Old Yesterday, 10:01 PM   #3301
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Originally Posted by doronshadmi View Post
It is argued that 0.999... is not a term of the sequence (0.9, 0.99, 0.999, ...)
Really? Who is arguing this idea?
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