Deeper than primes

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epix said:
If you let p = 0.999... and q = 0.000...1, then also p = {0.9, 0.99, 0.999, 0.9999, 0.99999, ...}
Click to expand...

Wrong, 0.999... is not a member of set {0.9, 0.99, 0.999, 0.9999, 0.99999, ...}

0.999... = 0.9 + 0.09 + 0.009 + ... + 0.000...1 = 1
 
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It is amazing. This guy talks about infinity yet is unable to understand what it means. "0.0000....." means you repeat 0 infinite times, how can one put a "1" at the "end" of infinity?
 
doronshadmi said:
Wrong, 0.999... is not a member of set {0.9, 0.99, 0.999, 0.9999, 0.99999, ...}

0.999... = 0.9 + 0.09 + 0.009 + ... + 0.000...1 = 1
Click to expand...
:confused:
I thought that 0.999... rendered as a sum equals

0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + 0.000009 and so on without that 9 going after the bunch of 0's mysteriously changing to 1.

Well, in OM things happen different way . . .

So, Doron, what did you have for breakfast this morning? What? What clue? OM...?
Aaah, like hash browns?
No?
I was about to say that.
 
The Limit concept does not have the necessary algebraic terms, because it can't explain how a given distinct 0-dimesional space x reaches distinct 0-dimensioanl space y , such that ( there is nothing between 0(x) and 0(y) ) AND ( 0(x) ≠ 0(y) ).

Again,

Take a 1-dim element with finite size X.

Bend it and get 4 equal sides along it.

Since the size between the opposite edges is changed to the sum of only 3 sides, and since the number of the sides after the first bending is 4 sides, we have to multiply the bended 1-dim element by 1/(the number of the sides after some bending), in order to get back the finite constant size X > 0, etc ... ad infinitum ... , as shown in the diagram below.

As a result each bended 1-dim element has finite constant size X > 0, but the size between its opposite edges becomes smaller (it converges), and used to define S=2(a+b+c+d+...) .

In general, S size is unsatisfied because the bended 1-dim element has finite constant size X > 0 upon infinitely many bended levels of:

[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]

X is a constant length > 0.

Theorem: The length of X’s totally bended form ≠ The length of X’s totally non bended form.

1) Let us assume that constant X>0 is independent of the number of bends along it, by using the assumption that X is completely covered by 0-dimensioanl distinct spaces, whether it is bended or not.

2) According to (1) the length of X’s totally bended form = the length of X’s totally non bended form.

3) But the totally bended form is exactly a single 0-dimensional space and in this case X=0, which contradicts (2).

4) According to (3) we can conclude that The length of X’s totally bended form ≠ The length of X’s totally non bended form.

Q.E.D
 
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epix said:
0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + 0.000009 and so on without that 9 going after the bunch of 0's mysteriously changing to 1.
Click to expand...

mysteriously? what mysteriously? what are you talking about?


0.999...[base 10] = 0.9[base 10] + 0.09[base 10] + 0.009[base 10] + ... < 1 by 0.000...1[base 10]
 
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doronshadmi said:
The Limit concept does not have the necessary algebraic terms, because it can't explain how a given distinct 0-dimesional space x reaches distinct 0-dimensioanl space y , such that ( there is nothing between 0(x) and 0(y) ) AND ( 0(x) ≠ 0(y) ).

Again,

Take a 1-dim element with finite size X.

Bend it and get 4 equal sides along it.
Click to expand...
We've been here before. You're not even capable of describing this first step unambiguously. Why shouldn't I have a square after following those instructions? Not to mention that once you have bent your '1-dim' element, it's not 1-dimensional any more.
 
doronshadmi said:
The Limit concept does not have the necessary algebraic terms, because it can't explain how a given distinct 0-dimesional space x reaches distinct 0-dimensioanl space y , such that ( there is nothing between 0(x) and 0(y) ) AND ( 0(x) ≠ 0(y) ).
Click to expand...
That's because the concept of limits doesn't concern itself with "x reaching y"; it's the other way around.

Here is probably what you have a hard time to comprehend. Do you see any point between O and M?

O_____________________M

The line segment has only two defining points -- O and M -- so there is no other point between them. That's because you didn't chose it. Once you decide to do so, there is no way that you wouldn't be able to locate such a point.

Some functions are not defined for certain x, like f(x) = log 0, so there is no corresponding point y, but there is a corresponding point for any x > 0 and that includes a value that you notate as 0.000...1

Here are some y-values for f(x) = log(10) x

0.1 = -1
0.01 = -2
0.001 = -3
0.0001 = -4

You can see that as x approaches zero, y approaches negative infinity, or

[lim x → 0] log(10) x = -∞

x never reaches zero or any other limit number, coz that would collapse the concept of infinity: if the train stops, the name of the town is "Finity."

I very much doubt that the following link would change your mind, but I post it anyway.
http://ltcconline.net/greenl/courses/115/functionGraphLimit/limits.htm
 
doronshadmi said:
mysteriously? what mysteriously? what are you talking about?


0.999...[base 10] = 0.9[base 10] + 0.09[base 10] + 0.009[base 10] + ... < 1 by 0.000...1[base 10]
Click to expand...
The Summerian tablets are an exercise in eloquency when compared with this sorrowful scribble.

In your previous rendition, 0.000...1 was one of the addends
0.999... = 0.9 + 0.09 + 0.009 + ... + 0.000...1 = 1
Click to expand...
and now it is not the part of the expansion and sits at the end of the whole expression separated from the expansion by < and preceded by by. Did you mean to invoke the situation where 0.000...1 is leaving saying bye-bye to the rest of the expression?
 
epix said:
Once you decide to do so, there is no way that you wouldn't be able to locate such a point.
Click to expand...
Once you decide to do so, there is no way that you wouldn't be able to avoid a line between some arbitrary pair of points, where ...1 of the expression 0.000…1 is such a line.

epix said:
x never reaches zero or any other limit number, coz that would collapse the concept of infinity: if the train stops, the name of the town is "Finity."
Click to expand...
This is exactly OM's claim, which does not agree with Traditional Math, which claims that, for example 1/2+1/4+1/8+… = to limit 1.

Now we see that you simply wish to be against OM by principle, no matter if you agree or does not agree with what it claims.
 
epix said:
The Summerian tablets are an exercise in eloquency when compared with this sorrowful scribble.

In your previous rendition, 0.000...1 was one of the addends

and now it is not the part of the expansion and sits at the end of the whole expression separated from the expansion by < and preceded by by. Did you mean to invoke the situation where 0.000...1 is leaving saying bye-bye to the rest of the expression?
Click to expand...

Do you understand that 0.9, 0.09, 0.009., ... etc. are 0() spaces, where ...1 of 0.000...1 expression is 1() space ?
 
epix said:
That's not a theorem due to the triviality of the proposition: (4X)/3 ≠ X.
Click to expand...

Please look at

[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]

The length of X > 0 is invariant as long as the given element is not totally bended (where each bend is an 0-dimensional space).

Traditional Math claims that since 1-dimasional space is totally covered by 0-dimensional spaces, then constant X>0 is defined
even if 1-dimensioan space is totally bended.

OM shows the impossibility of such a claim, which is also related to the impossibility that, for example, 1/2+1/4+1/8+... = to limit 1.
 
doronshadmi said:
Please look at

[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]

The length of X > 0 is invariant as long as the given element is not totally bended (where each bend is an 0-dimensional space).

Traditional Math claims that since 1-dimasional space is totally covered by 0-dimensional spaces, then constant X>0 is defined
even if 1-dimensioan space is totally bended.
Click to expand...
Traditional math doesn't claim anything like that, coz the Koch curve is not a 1-dimensional space. Traditional math "claims" and can prove that the fractal dimension of Koch's curve is d = (log 4)/(log3).

Evidence of what traditional math claims and what it doesn't:
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
 
epix said:
Traditional math doesn't claim anything like that, coz the Koch curve is not a 1-dimensional space. Traditional math "claims" and can prove that the fractal dimension of Koch's curve is d = (log 4)/(log3).

Evidence of what traditional math claims and what it doesn't:
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
Click to expand...

It has infinite length.

http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
Click to expand...

It seems that you are unaware of Traditional Math's claim about the equality of |R|, whether we deal with finite of infinite length.

This claim is based on the reasoning which claims that 1-dimensional space is totally covered by 0-dim spaces, and according to this reasoning, it does not matter if the considered form has infinitely of finitely bends along some considered length (finite or not) > 0.

In other words, you have no understanding of Traditional Math's claim about the equality of |R| of any arbitrary given length > 0, whether it is bended or not.

According to Traditional Math totally bended length X and totally not bended length X have cardinality |R|, but I prove in http://www.internationalskeptics.com/forums/showpost.php?p=6514886&postcount=12204 that this is a false claim.
 
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epix said:
Traditional math "claims" and can prove that the fractal dimension of Koch's curve is d = (log 4)/(log3)
Click to expand...
We are not talking here about fractal dimensions, but about the difference between totally bended and totally not bended (or not totally bended) forms.

Traditional Math simply can't understand that if the Koch's fractal (or any other arbitrary bended form) is totally bended, then its length = 0.

Instead, Traditional Math claims that a totally bended form (and Koch's fractal is just an example here) has an infinite length, and this mistake is clearly seen and written in http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html, which leads to the mistake that the number of points along a totally bended form, has cardinality (Cardinality in terms of Traditional Math) > 1.

In other words epix, your replies have nothing to do with my posts, on this subject.
 
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In general, any converges or diverges value, which its size is defined upon infinitely many scale levels, is measured by non-strict numbers of the form 0.000…x, and it has a non-strict value, as can be seen by the following example of Koch's fractal:

Fractal_koch.png
.
http://upload.wikimedia.org/wikipedia/commons/8/89/Fractal_koch.png
 
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The Man said:
No Doron you are simply and entirely wrong as usual. Once again a space is not independent of its sub-spaces.
Click to expand...

The Man you have no case because you do not distinguish between the complex and the non-complex.

You are talking about the dependency under the complex results among independent spaces.

Let me give you an example:

The existence of 1-dim and 0-dim spaces are independent of each other, such that if one of them is missing, the other one still exists.

Now let's observe a complex like line-segment, which is the result of the linkage among 1() and 0().

Even if the length of 1() space under the complex, called line-segment, depends on the values of 0() spaces, it does not mean the 1() or 0() existence depend on each other.
 
doronshadmi said:
This claim is based on the reasoning which claims that 1-dimensional space is totally covered by 0-dim spaces, and according to this reasoning, it does not matter if the considered form has infinitely of finitely bends along some considered length (finite or not) > 0.
Click to expand...
I was right in my suspicion that you don't have the vaguest idea about the concept of points. If there is a location on 1-dim object, such as the initiator line of the Koch curve, that a point cannot define, then you can't "cut and bend" the line infinitely. That's because for that activity you need to locate the vertex and the inflection points. Any modification or a construction of objects with d>0 requires the presence of defining points, and you are totally oblivious to that fact. You only see that after one million iterations, the Koch curve still comprise combination of line segments and points and therefore there would be always a point-line-point segment, e.g. there would be always a space between two points "not covered by a point." But after another couple of million iterations you find that this particular line segment is littered with vertex points as the bending continues. Go back and look at the example once again. Do you see any other point apart from the endpoints?

O_____________________M

No?
The conclusion is then that this line segment is not entirely covered by points. That also means that this line segment can't become the generator of the Koch curve, coz you can't locate the necessary inflection and vertex points to modify the line. The reason why you can't locate the points is the absence of any points between O and M. That in turn means that the finite length of the O_M line cannot be divided by number 3, coz the result of such a division looks like

O_______|_______|_______M

and since there are no points between O and M, there is no way to define and locate the points through (M - O)/3. The general consequence is that Organic Mathematics discovered numbers greater than zero that are indivisible. (Applause.)

BUT! The concept of the limit says that there is always a space between two points on a curve, otherwise there would be no derivative of the function that draws such a curve, and without us being able to compute derivatives, the science time would stop in the 16th century. The length of such a pointless segment approaches zero -- zero is the limit. So in this respect, it's true that "line cannot be entirely covered by points." On the other hand, any line length can be further divided. That's a contradiction that the concept of infinity solves. You can locate any point on a curve and at the same time compute the derivative of the function that draws the curve. The result in the approximate format would be "infinitely precise," but not "exact."

PI is to EXACT as 3.1415... is to INFINITELY PRECISE
 
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