Deeper than primes - Continuation

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Let us observe the length of the purple semi-circles upon infinitely many levels of semi-circles, where each level (if all the semi-circles of some given level, are included) has exactly d*(Pi/2) length (where d > 0 and has a finite length):

9053976881_5e4453c2f2_o.jpg


As shown, the right semi-circle of any arbitrary pair of smaller semi-circles (under their single larger semi-circle) is not added to the length of the left purple semi-circle, and this state of not being added to the length of the left purple semi-circle, is invariant upon infinitely many levels of semi-circles, where each level has exactly d*(Pi/2) length (if all the semi-circles of some given level, are included) (where d > 0 and has a finite length).

Conclusion: The length of the purple semi-circles < d*(Pi/2) length by the (non-included) length of permanently smaller right semi-circle, which is irreducible into 0 length, exactly because each level of the infinity many levels of smaller semi-circles (where the most right semi-circle is included (unlike the case of the purple semi-circles)) has exactly d*(Pi/2) length (length d*(Pi/2) is irreducible into length d).

By being open to the above, we can understated that 0.000...12 is a verbal_symbolic expression, which is equivalent to the invariant state of not being added to the length of the left purple semi-circle, upon infinitely many levels of semi-circles, where each level has exactly d*(Pi/2) length ((where d > 0 and has a finite length) and the length of the most right semi-circle at each level of semi-circles, is included (unlike the case of the purple semi-circles)).
 
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I'm somewhat amused that you're (apparently?) trying to disprove the existence of irrational numbers by using the irrational number pi.

If pi isn't irrational, how many digits does it have? For this example, the difference between 'infinite' pi and a truncated pi decreases extremely fast with a small number of digits - see the attached chart, where summing the first ten semicircles with a value of pi truncated after 14 (base 10) digits is only off by a trillionth - so you don't actually need a lot, but it probably wouldn't be hard to find one that needs many digits.

'Infinite' pi has been calculated to 10 trillion digits or so. Presumably your version is at least that long. The Bailey–Borwein–Plouffe formula provides a way to determine the value of any particular digit of pi (in base 16, technically). It works for any particular digit, no matter how far. Is the number of digits after the decimal in pi equal to the largest possible integer? If there are fewer digits, what would the results of the formula past that point correspond to? If there are more digits, why would the formula be unable to calculate them?


(fairly sure I didn't make any errors with the excel chart, but it's incidental anyway.)
 
uvar said:
I'm somewhat amused that you're (apparently?) trying to disprove the existence of irrational numbers ...
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Hi uvar,

I am not trying to disprove the existence of the irrational numbers.

I prove the fallacy of the Limit, as used by standard analysis, which according to it, for example, the long addition 1/2 + 1/4 + 1/8 + ... = 1 (which is equivalent to the claim that 0.12 + 0.012 + 0.0012 + ... = 1)

By using verbal_symbolic AND visual_spatial brain skills, I prove that standard analysis is wrong and the results of these long additions are < than 1 (where 1 is the value of the limit, in this case) without loss of generality.

By using verbal_symbolic AND visual_spatial brain skills, one enables to know that no collection on a given space has the power of that space (for more details please look at http://www.scribd.com/doc/98276640/Umes).
 
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doronshadmi said:
Let us observe the length of the purple semi-circles upon infinitely many levels of semi-circles, where each level (if all the semi-circles of some given level, are included) has exactly d*(Pi/2) length (where d > 0 and has a finite length):
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There is the word that's getting you into trouble. You are leaping from what seems true for all the finite cases and assuming it must be true for the infinite case. Despite your repeated attempts, Mathematics doesn't work that way.

The limit of the length of your violet curve is pi/2. The limit of the length of your blue line is 1. If you really think you can prove otherwise, then you will need to engage in a little delta-epsilon action. Bare assumptions don't do it.
 
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By using verbal_symbolic-only reasoning one simply can't understand that the length of the purple semi-circles is upon infinitely many levels of semi-circles, and there are infinitely many cases upon infinitely many levels where the most right semi-circle is not included in the length of the purple semi-circles AND it is irreducible into 0 size, exactly as explained in details in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.

Moreover, it is shown that the verbal_symbolic expression like 0.000...12 is not understood unless also visual_spatial reasoning is used in addition to verbal_symbolic reasoning.

The limit of the length of the purple semi-circles is indeed d*(pi/2) but the length of the purple semi-circles does not reach to that limit even if infinitely many purple semi-circles are involved, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.

Mathematics doesn't work properly with infinity if only verbal_symbolic reasoning is used.
 
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doronshadmi said:
The largest semicircle is the limit value that can't be reached by the value of infinitely many purple semi-circles, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.
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Proved? That would require you actually refer to the basic meaning of limit. You don't for some reason. Why is that?

For that matter, why the continued use of this red herring involving semicircles? Other than being a demonstration of your lack of understanding of the concept of a limit, or the concept of a puzzle for that matter, it lends nothing to the hand-waving in which you are now engaged.

The aggregate length of the arcs grows to its limit at exactly the same rate as the aggregate length of the diameters grows to its. So you have two similar examples of you not understanding limits. Will you be adding a third?
 
jsfisher said:
The aggregate length of the arcs grows to its limit at exactly the same rate as the aggregate length of the diameters grows to its.
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It does not change that proof that the length of infinitely many purple semi-circles < limit d*(Pi/2) or the length of infinitely many parts of the blue line < limit d.

The verbal_symbolic-only hand-waving of Karl Weierstrass does not hold water, in this case.
 
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doronshadmi said:
The limit of the length of the purple semi-circles is indeed d*(pi/2) but the length of the purple semi-circles does not reach to that limit even if infinitely many purple semi-circles are involved, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.

Mathematics doesn't work properly with infinity if only verbal_symbolic reasoning is used.
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Well, that's the "god-given property" of the limit that it can be approached but not reached. If it was possible, then 1, 2, 3, 4, ... would "reach infinity."

The proof that the addition or subtraction of rational numbers always result in a rational number is easy, so let's skip it. Now look at this Gregory-Liebniz series the way it is very often rendered:

pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - ...

Wrong. The rational series can never reach the value of irrational number pi, because the rational results of the series cannot yield an irrational number - those results are always in the form of p/q, as the series continues without bound. The reason why the result of series is written in this format without reference to a limit is that the writers cannot really imagine that the reader wouldn't be familiar with the concept of limits, and also because the presence of the ellipses gives the expression more or less a descriptive rather than an analytic one. But when algebraic formulas involved in functions are written, the limit symbolism is always included.
 
doronshadmi said:
It does not change that proof that the length of infinitely many purple semi-circles < limit d*(Pi/2) or the length of infinitely many parts of the blue line < limit d.
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If and when you actually incorporate the meaning of limit in any of your so-called proofs, maybe people will stop laughing at them. That day, though, is not today.
 
doronshadmi said:
The largest semicircle is the limit value that can't be reached by the value of infinitely many purple semi-circles, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.
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That's hardly so. The sum of the circumferences of the purple semicircles indeed converges toward pi/2, but the circumference of the largest semicircle is pi. So it cannot be the limit. Unless, you consider the limit any value that cannot be reached. :D
 
There is some person here that does not know that the length of the semi-circle is exactly d*(Pi/2), where d is the length of the diameter of the circle.
 
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Doron, true or false, and if false, why:
 

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jsfisher said:
Doron, true or false, and if false, why:
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It is false, and in order to understand why, one has to use also the visual_spatial equivalent expression of the long addition 1/2 + 1/4 + 1/8 (which is the verbal symbolic expression of the considered case).

A visual_spatial equivalent expression is (d=1):

9053976881_5e4453c2f2_o.jpg


Since there is the same rate among the purple semi-circles and the blue line, we can use the purple semi-circles in order to conclude things about the blue line.

So let's use them.

The length of the purple semi-circles is upon infinitely many levels of semi-circles, and there are infinitely many cases upon infinitely many levels, where the most right semi-circle is not included in the length of the purple semi-circles AND it is irreducible into 0 size, exactly as explained in details in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401.

In other words, the length of the purple semi-circles upon infinitely many levels of semi-circles is < d*(Pi/2) and so is the case about the blue line (it is < d=1).

The verbal_symbolic expression like 0.000...12 is not understood unless also visual_spatial reasoning is used in addition to verbal_symbolic reasoning.

The limit of the length of the purple semi-circles is d*(pi/2) but the length of the purple semi-circles does not reach to that limit even if infinitely many purple semi-circles are involved, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401, and so is the case about the blue line (1/2 + 1/4 + 1/8 + ... < limit 1).

I accept limits, but I claim that they are inaccessible to long addition like 1/2 + 1/4 + 1/8 + ... or to infinitely many purple semi-circles.
 
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doronshadmi said:
It is false, and in order to understand why, one has to use also the visual_spatial equivalent expression of the long addition 1/2 + 1/4 + 1/8 (which is the verbal symbolic expression of the considered case)....
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Yeah, yeah. We have heard this all before. Disproof by talking about something else.

Come on, Doron. Limits. They have a precise definition and meaning. Let's have some delta-epsilon action, else you have nothing.

...
The limit of the length of the purple semi-circles is d*(pi/2) but the length of the purple semi-circles does not reach to that limit even if infinitely many purple semi-circles are involved
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Well, looky here. Yet another Doron self-contradiction. It also leans towards a contradiction about the limit of the summation I posted up a post or three.

The limit is or it isn't, not both.
 
jsfisher said:
[...]

Well, looky here. Yet another Doron self-contradiction. It also leans towards a contradiction about the limit of the summation I posted up a post or three.

The limit is or it isn't, not both.
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You type faster than me.

Doron, it really looks like you contradicted yourself here. Is the limit of the series posted by jsfisher 1, or not? You say or imply both in your post. Perhaps you want to say that the limit is 1 but the sum of an infinite series cannot be defined as a limit?
 
doronshadmi said:
There is some person here that does not know that the length of the semi-circle is exactly d*(Pi/2), where d is the length of the diameter of the circle.
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LOL. Oh, you got that diameter of the largest semicircle set to 1, not 2. I was wondering what that big number 1 on the bottom of the pic was all about. I thought it might have referred to monotheism. So it's the diameter... :D
 
doronshadmi said:
The limit of the length of the purple semi-circles is d*(pi/2) but the length of the purple semi-circles does not reach to that limit even if infinitely many purple semi-circles are involved, as proved in http://www.internationalskeptics.com/forums/showpost.php?p=9298770&postcount=2401, and so is the case about the blue line (1/2 + 1/4 + 1/8 + ... < limit 1).

I accept limits, but I claim that they are inaccessible to long addition like 1/2 + 1/4 + 1/8 + ... or to infinitely many purple semi-circles.
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The point you trying to make is not overly relevant to the sum of the circumferences, because pi and 2 are constants that get factored and placed in the beginning of the formula leaving just a series to consider. So I think that

(2n - 1) / 2n = ∑(1/2n) for n = 1, 2, 3, 4, ...

makes a better argument.
 
BenjaminTR said:
You say or imply both in your post.
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No.

BenjaminTR said:
Perhaps you want to say that the limit is 1 but the sum of an infinite series cannot be defined as a limit?
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Yes.

Here is another way to look at it, by using the membership level of sets and the membership level of members of sets:

|N| > all N members (no N member = |N|).

As a result there is no such thing like the largest N member.

So the term all at the level of N members is inaccessible to |N| level.

By this approach N membership level is > all membership levels of N members.

This principle holds also by using finite values, for example:

|{}|=0 > {} member (0 = something > nothing).

|{{}}|= 1 > |{}|=0 > {} member (0 = something > nothing).

|{{{}}}|= 2 > |{{}}|= 1 > |{}|=0 > {} member (0 = something > nothing).

etc... ad infinitum.

Now let's see the difference between actual infinity, potential infinity, finite and actual finite:

|N|= actual infinity = |{...{{{}}}...}| > |...{{{}}}...| = potential infinity = (... > |{{{}}}|= finite 2 > |{{}}|= finite 1 > |{}|= finite 0) > {} member (finite 0 > nothing that is actual finite).

Let # be a placeholder for any set.

|#| > all # members (no # member = |#|).

As a result there is no such thing like the largest # member.

So the term all at the level of # members is inaccessible to |#| level.

By this approach # membership level is > all membership levels of # members.

Moreover, we do not need sets in order to show the inaccessibility into a given limit.

For example the result of the long addition 1/2 + 1/4 + 1/8 + ... is inaccessible to limit 1.

So is the case of about the equivalent long addition 0.12 + 0.012 +0.0012 + ... = 0.111...2 ,which is inaccessible to limit 1.
 
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zooterkin said:
Still hung up on Zeno's paradox, Doron?
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There is no paradox if time slows down and space is compressed during the race, as observed by SRT for bodies with mass > 0 which approach to the speed of light.
 
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It has to be stressed that in http://www.internationalskeptics.com/forums/showpost.php?p=9301162&postcount=2421 I define the level of a given set, not by the amounts of its members, but by the higher or highest level of at least one of its members (or its absence, in the case of the empty set).

In order to avoid confusion between the standard "|" and "|" notation of cardinality, and the notion of levels, let's use "]" and "[" to notate levels, as follows:

Here is another way to look at inaccessibility, by using the membership level of sets and the membership level of members of sets:

]N[ > all N members' levels.

As a result there is no such thing like the largest N member (since no member reaches the level of the set).

So the term all at the level of N members is inaccessible to ]N[ level.

By this approach N membership level is > all membership levels of N members.

This principle holds also by using finite values, for example:

]{}[ = 0 > {} member (0 = something > nothing).

]{{}}[ = 1 > ]{}[ = 0 > {} member (0 = something > nothing).

]{{{}}}[ = 2 > ]{{}}[ = 1 > ]{}[ = 0 > {} member (0 = something > nothing).

etc... ad infinitum.

Now let's see the difference between actual infinity, potential infinity, finite and actual finite:

]N[ = actual infinity = ]{...{{{}}}...}[ > ]...{{{}}}...[ = potential infinity = (... > ]{{{}}}[ = finite 2 > ]{{}}[ = finite 1 > ]{}[ = finite 0) > {} member (finite 0 > nothing, which is actual finite).

Let # be a placeholder for any set.

]#[ > all # members' levels.

As a result there is no such thing like the largest # member (since no member reaches the level of the set).

So the term all at the level of # members is inaccessible to ]#[ level.

By this approach # membership level is > all membership levels of # members.

Moreover, we do not need sets in order to show the inaccessibility into a given limit.

For example the result of the long addition 1/2 + 1/4 + 1/8 + ... is inaccessible to limit 1.

So is the case of about the equivalent long addition 0.12 + 0.012 +0.0012 + ... = 0.111...2 ,which is inaccessible to limit 1.
 
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No matter how many pages of inane gibberish you produce, doron, reality does not agree with you. Do you have anything that resembles a result? Anything at all to show for these decades you spent on doronetics? So far its use seems limited to the production of high-quality gibberish. The market is pretty saturated though, so forgive me for not being impressed in the least.
 
jsfisher said:
Well, looky here. Yet another Doron self-contradiction. It also leans towards a contradiction about the limit of the summation I posted up a post or three.

The limit is or it isn't, not both.
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There is a limit, and it is inaccessible to a given convergent infinite series, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=9300229&postcount=2415.

Very simple.

jsfisher said:
Let's have some delta-epsilon action, else you have nothing.
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Let's have also some visual_spatial action, else you have nothing.

Here is the rigorous verbal_symbolic AND visual_spatial proof of the inaccessibility to limit d*(Pi/2) or to limit d:

9076101160_b50490da84_o.jpg


The details are at http://www.internationalskeptics.com/forums/showpost.php?p=9300229&postcount=2415.
 
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Please look at http://www.internationalskeptics.com/forums/showpost.php?p=9301513&postcount=2425.


The claim that the length of the purple semi-circles = d*(Pi/2) or the claim that the length of the blue line = d, is equivalent to the claim that the infinitely many levels of semi-circles with constant length d*(Pi/2), are reducible into length d.

But it can't be since there are infinitely many levels of semi-circles with constant length d*(Pi/2).

Since Standard Analysis excludes infinitesimals, is can't capture that there are infinitely many levels of semi-circles with constant length d*(Pi/2).

By including infinitesimals, an expression like 0.000...12 is simply the value > 0 that prevents from infinitely many levels of semi-circles with constant length d*(Pi/2) to be reduced into length d.
 
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zooterkin said:
Nonsense.
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Only if you exclude infinitesimals.

Also look at http://www.internationalskeptics.com/forums/showpost.php?p=9301293&postcount=2422.

That has mass > 0 can't reach the speed of light, exactly as the constant length d*(Pi/2) upon infinitely many levels of semi-circles, can't reach length d.

If you push the system to reach the limit (length d, in this case), you get finitely many levels of constant length d*(Pi/2), and then quantum jump into length d.
 
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doronshadmi said:
If you push the system to reach the limit (length d, in this case), you get finitely many levels of constant length d*(Pi/2), and then quantum jump into length d.
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Are you saying this just to emphasize your total lack of understanding of limits, or do you just like spouting nonsense?
 
jsfisher said:
Are you saying this just to emphasize your total lack of understanding of limits, or do you just like spouting nonsense?
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Are you saying this just to emphasize your total lack of understanding of infinitesimals, or do you just like spouting nonsense?

More about infinitesimals can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=9204465&postcount=2330 and http://www.internationalskeptics.com/forums/showpost.php?p=9301700&postcount=2427, and it is well known that Standard Analysis arbitrarily excludes them.

What is the reasoning behind this arbitrary exclusion?
 
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doronshadmi said:
Are you saying this just to emphasize your total lack of understanding of infinitesimals, or do you just like spouting nonsense?
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Neither.

More about infinitesimals can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=9204465&postcount=2330 and http://www.internationalskeptics.com/forums/showpost.php?p=9301700&postcount=2427, and it is well known that Standard Analysis arbitrarily excludes them.

What is the reasoning behind this arbitrary exclusion?
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First, you simply inserting a word, infinitesimals, clumsily into a post doesn't mean "[m]ore about infinitesimals can be seen in" the post. Second, there is nothing arbitrary about it; infinitesimals are flawed by contradiction. Now, we all know contradiction and you are long time friends, but it is not usually a healthy characteristic in Mathematics.


How's that delta-epsilon demonstration coming along? Surely, you can support your limitless claims with something that actually uses the meaning of limits at its base.
 
jsfisher said:
Neither.
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Both.


jsfisher said:
infinitesimals are flawed by contradiction.
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According to your Limit(ed) reasoning.

Since infinitesimals are derived from the association among different levels of spaces (for example: the association among the composed level of lower dimensional spaces with the non-composed level of a given higher dimensional space) they are the distinct signature of consistency.

On the contrary, contradiction is possible only if things are at the same level, and indeed Limits are derived form a mathematical universe that has one level, which is the composed level of collections.
 
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doronshadmi said:
Both.



According to your Limit(ed) reasoning.

Since infinitesimals are derived from the association among different levels of spaces (for example: the association among the composed level of lower dimensional spaces with the non-composed level of a given higher dimensional space) they are the distinct signature of consistency.

On the contrary, contradiction is possible only if things are at the same level, and indeed Limits are derived form a mathematical universe that has one level, which is the composed level of collections.
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*yawn*
When you have something to show, either an actual result made possible by the mysterious Doronetics or disprove something in Mathematics with something that includes all the required elements of a proof, then I might be interested.
 
doronshadmi said:
That has mass > 0 can't reach the speed of light, exactly as the constant length d*(Pi/2) upon infinitely many levels of semi-circles, can't reach length d.
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That's an analogy that compares real and imaginary and therefore it may not be true. Think about it a bit. If a series is assumed convergent but at one point that series is capable of reaching its limit, then there is nothing that would prevent the series becoming divergent, which is a contradiction to the initial assumption.

Any series can be converted into a sequence through its cumulative sums. In the case of

Ser. = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

the sequence of partial sums is

Seq. = 1/2, 3/4, 7/8, 15/16, 31/32, ...

As the sequence progresses without bound, there will be always a |unit| difference between the numerator and the denominator. In order to compute the limit, a unit is added to the numerator of one of the terms, because the terms are made of naturals and the minimum difference between two naturals is |unit|. That means

(n - 1 + 1)/n = n/n = 1

and the limit of sequence Seq. is 1, and so must be the limit od series Ser.

You are trying to outflank an opponent who doesn't exist.
 
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jsfisher said:
*yawn*
When you have something to show, either an actual result made possible by the mysterious Doronetics or disprove something in Mathematics with something that includes all the required elements of a proof, then I might be interested.
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In the following diagram

[qimg]http://farm6.staticflickr.com/5445/9076101160_b50490da84_o.jpg[/qimg]

The length of the semi-circles is the constant value d*(Pi/2) upon infinite amount of levels, exactly because a 2-dim space (the area of each semi-circle in this diagram) is irreducible into 1-dim (the limit line with length d) (there is no homomorphism between n dimensional space and n-1 dimensional space, where n = 1 → ∞ (where → means "approaches by infinitely many steps")).

So is the case about the size of the added areas under infinitely many levels of semi-circles (starting form the second level), where this size is calculated by the long addition (1/2 of the area under the first semi-circle) + (1/4 of the area under the first semi-circle) + (1/8 of the area under the first semi-circle) + (1/16 of the area under the first semi-circle) + (1/32 of the area under the first semi-circle) + ... < (1/1 area under the first semi-circle).

As long at the notion of levels is not used by a given reasoning, this reasoning can't avoid contradiction.

Standard Analysis and ZF(C) are two examples of single-level reasoning.
 
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doronshadmi said:
Standard Analysis and ZF(C) are two examples of single-level reasoning.
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OK. Let's play a game, doron. I show you some stuff that those awful, lame single-level reasoning frameworks have brought us and you show me some stuff that your kewl multi-level doronetic reasoning achieved. Deal?
 
jsfisher said:
*yawn*
When you have something to show, either an actual result made possible by the mysterious Doronetics or disprove something in Mathematics with something that includes all the required elements of a proof, then I might be interested.
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Lol. You are for an infinite wait, because Doron will not present his case without involving Doronetics.
If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited.
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Actually that's not Doron but Simplicius seeing glimpses of the continuum hypothesis. Well, maybe not.
 
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