‏¹ cayley tree
http://mathworld.wolfram.com/CayleyTree.html

bad_logic wrote:

All it says is, no matter how close you get to L, the sequence gets closer.
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Again, there are two different points of view in this case, which are:

1) The sequence point of view is based on the ratio between x and 1 (notated as x/1) where x is any element in the sequence, and we do not need any hypothetic limit in order to define the exact place of any x along the sequence.

It is also can be done by |x-L| = epsilon, where epsilon > |L-L| = 0.

In any case, since we do not need the limit, then nothing becomes closer to any hypothetic limit.

2) If we try to compare any x to |L-L| (which is the "gap" between the limit to itself) we actually use the expression x/0, and from the point of view of the limit itself, we do not have any information about the explored sequence.

I am going to add my previous posts to this post, and this time please do not ignore them, thank you:

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Originally Posted by Wade-d
Since "scale levels" has not been rigorously defined, this is a non-answer.
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It is rigorously defined, and all we have to do is to understand x/1 expression.

x/1 is the ratio between each R member to number 1, and by this ratio it is clearly understood that .999... is no more than a non-finite long addition of the rational numbers 0.9 + 0.09 + 0.009 + 0.0009 ... where each one of them exists in a different scale level (which in this particular case is based on base 10 fractal)
and the sum of this long addition is not equivalent to 1.

Therefore .99999... < 1

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bed_logic wrote:

there are no triangles in the koch fractal's boundary. Although, to be fair, we shouldn't be talking about triangles. After the first iteration, no iteration has any "triangles" on it
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You are wrong about that, for example:

http://www.andymeneely.com/Prog/Fractals/PracticalFr...

0.99999... = 1 if and only if these sharp patterns can form a smooth sharpless boundary.

Since this is not the case, 0.99999... < 1

As you can see, I did not use any triangle here, but you, with out any good reason, continue to talk about it, so please overcome this "problem".

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Any limit of some non-finite sequence in the common Math, is always based on a gap > 0 between any element of the sequence and the hypothetic limit.

The gap of this non-finite sequence is the absolute value |x-L| where L is the limit value itself and x is any arbitrary value of the sequence, that cannot be L itself, if we are talking about a non-finite sequence.

In other words, there is a clear dichotomy between the non-finite sequence and any hypothetic limit of it, simply because no non-finite collection (ordered or not) can be limited by any shape or form, otherwise it is not a non-finite collection, but a finite collection.

Again we define smaller or greater elements in some sequence, because we take one of the gaps which is > 0, then we call it 1, then we compare the entire gaps to it, and only then we can determinate if some element is greater or smaller than 1.

It cannot be done if we try to do that by using the "gap" between L to itself |L-L|, which is always 0.

If we want to understand the behavior of some sequence, all we have to do is to use the expression x/1.

If the absolute result is always < 1 we say that the sequence's elements become smaller, otherwise they do not change or become bigger, and we do not need the epsilon-delta 1-1 mapping technique , or any L as the limit of a non-finite sequence.

As I wrote, we can understand the behavior of the sequence, only if we compare any one of its members to some value, which is not 0, for example x/1.

If we try to compare any x to |L-L| (which is the "gap" between the limit to itself) we actually use the expression x/0, and from the point of view of the limit itself, we do not have any information about the explored sequence.

A part of my work:

Let us say that we have a sports car (where the name of the back wheels is "epsilon" and the name of the front wheels is "delta") and our mission is to cross the zero point of X,Y-axis with both "delta" and "epsilon" wheels.

http://www.geocities.com/complementarytheory/SportsC...

We are seated in the car and trying to reach point zero.

We realize that no matter how fast we drive, we are not getting any closer to the zero point, and the reason is: the faster we drive, the smaller we become (as can be seen in the picture above) and we have here a Lorenz-like transformation.

Shortly speaking, our mission cannot be completed.

In the same manner set R is an incomplete collection.

Actually we reach point zero, if and only if we don’t have a car anymore but a single point, which is a phase transition that cuts the infinitely many smaller states, and we don’t have an incomplete collection over infinitely many scales, but a finite collection of many scales.

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bad_logic wrote:

If I can't follow your logic, it's not being rigorous.
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No, it says that you did not grasp yet the deep meaning of Godel's work about the incompetence of the axiomatic framework itself.

Here it is again:

Gödel used mathematical rigorous methods in order to research the language of Mathematics itself, and he found an interesting connection between consistency and completeness concepts.

In other words, no consistent framework is necessarily complete, and no complete framework is necessarily consistent.

He showed that in any, so called, consistent framework, there can be theorems which are based on some axiomatic system, but they cannot be solved within the framework of this axiomatic system.

Strictly speaking, we have no choice but to add new axioms to the current axiomatic system, in order to deal with these theorems, but then we discover that this addition is an “endless story”.

In other words, no axiomatic framework can be proven as both consistent and complete, within the framework of the axiomatic method.

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As I said bad_logic, you did not grasp yet that the Cantorean conceptual mathematical world does not exist anymore after Gödel's work, but the current community of professional mathematicians does not understand it (yet) and continue to use formal technical grammatical rules without any deep insights behind them (in the case of the Cantorean non-finite).

Since a non-finite collection of axioms cannot be both consistent and complete, we have no choice but conclude that a non-finite collection cannot be completed, in order to save its consistency (or in other words, its "mathematical existence").

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bad_logic wrote:

It doesn't say a "non-zero length segment becomes a 0-length segment".
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In this case |x-L| = epsilon > 0, where epsilon is the unclosed interval between x and L.

In other words, no infinitely many elements can eliminate this epsilon > 0.

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bad_logic wrote:

Godel. Axioms. Completeness. Consistancy. Ho-fucking-Hum. It doesn't say mathematics is wrong.
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No, it says that Mathematics is an open framework that deeply can be changed by paradigm-shifts, when new fundamental insights are created/discovered.

In this case my insight about the non-finite is richer, simpler and much more interesting then the Cantorean transfinite universe.

Because you did not read carefully all of http://www.iidb.org/vbb/showpost.php?p=2388287&p... you do not realize it yet, so please this time read all of it, thank you.

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bad_logic wrote:

Don't tell me I don't understand Godel. Don't tell me the cantorean infinity is wrong. I don't believe you, Doron. I have no reason to.
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Do not believe to my posts, all you need is a real , courage in order to be able to see things from a deeper point of view about the fundamental concepts of the language of Mathematics and its logical reasoning.

Until this very moment we did not find the strength within you to do this paradigm-shift in your mind, that will let you to read and, for the first time, understand:

http://www.geocities.com/complementarytheory/TAP.pdf

http://www.geocities.com/complementarytheory/My-firs...

And also http://www.geocities.com/complementarytheory/No-Naiv...

where at the end of it (pages 9 to 19) you can find the logical reasoning which I use in my framework.

I call this logical reasoning Complementary Logic, and please read the above link if you want to understand it.

If you don't understand it after you read it, then please ask me questions about it, thank you.

x=1/(n+1)

|x-0|>0

n>0
1>0

1/n>0

n>=1

n/n>=1/n
1>=1/n

x=1/n

|x-0|>0