Deeper than primes

[zooterkin]

If one specifies a location along a 1-dim element, it does not make the location a component of the 1-dim element, which stays atomic and non-local by nature.

Who says it does? It is still a point. This is one of your stumbling blocks.

 

[doronshadmi]

Who says it does? It is still a point. This is one of your stumbling blocks.

And it is still a bended atomic 1-dim element, so?

 

[zooterkin]

And it is still a bended atomic 1-dim element, so?

As soon as you have a 'bend', it is no longer 1-dimensional.

 

[doronshadmi]

Ok, so why did you include the circle? Just more noise to confuse the issue? That seems to be your hallmark trait. Scramble as much together as possible so you can bounce from the absurd to the inane without missing a beat, and always have enough remaining outlets so gibberish can save you from any criticism.

A little sense of beauty, that is all you need jsfisher (http://www.internationalskeptics.com/forums/showpost.php?p=5703507&postcount=8989):

Take a straight 1-dim with length X.

Bend it and get 4 equal sides along it.

Since the length 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 bended sides), in order to get back length X.

As a result the bended 1-dim element has length X, but the length between its opposite edges becomes smaller (it converges).

In general, this convergent series of 1/(the number of the bended sides) is resulted by 1/1+1/4+1/16+1/64+1/256+... , which has no limit exactly because length X is invariant on infinitely many convergent scales of that series.

The proof without words (http://en.wikipedia.org/wiki/Proof_without_words) was drawn by using AutoCad system, and it is accurate (each given bended 1-dim element has the same X length over infinitely many scales):

 

[zooterkin]

Ooh, shiny!

 

[jsfisher]

You are invited to provide the complete statement of the axiom, by using simple English.

I have provided a complete statement of the axiom. I'm sorry you don't understand the language in which it was expressed, but that is your problem, not mine.

Oppeness and Incompleteness are exactly the same notion about infinite collections, that can't be captured by qany limit or class.

No, terms you invent don't actually mean anything because you refuse to define them. Still, it is nice to see you admit that you also invent multiple terms all to mean the same thing. It reinforces the notion that your gibberish confuses even you.

So you can't grasp a proof withot words (http://en.wikipedia.org/wiki/Proof_without_words) as given in http://www.internationalskeptics.com/forums/showpost.php?p=5696845&postcount=8951 and http://www.internationalskeptics.com/forums/showpost.php?p=5699060&postcount=8955

You assume far too much. You also didn't comprehend the question I asked. No surprise on either count.

 

[The Man]

The Man, you simply cant rid of the use of points (localities) in your analysis.

“rid of”? So you are simply and deliberately ignoring the “points (localities)” as well as dimensions because you want to be “rid of” them?

The circle and the Koch’s fractal are both closed 1-dim elements, with not even a single point along them.

Nope, as zooterkin has pointed out to you several times already your ‘bended’ line is no longer one dimensional. You have 'bent' it at some “points (localities)” into at least one other dimension. In fact the fully iterated Koch curve (which is a fractal as noted by jsfisher) has a Hausdorff dimension of log(4)/Log(3) or about 1.2619, same as the Cantor set in 2 dimensions referred to as “Cantor dust”

ETA line: 3 such Koch curves would form the Koch snowflake or anti-snowflake.


http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

http://en.wikipedia.org/wiki/Cantor_dust#Cantor_dust

Inside each convergent circle’s length (which is a 1-dim atomic element) there is a Koch’s fractal (which is a 1-dim atomic element) which its length is invariant upon infinitely many convergent levels.

What, so now the circle is convergent? Again a circle is one dimensional because of the consistency of its radius, your ‘bended’ line has no such consistent radius (if it did it would be a circle).

As a result the convergent series of circles is infinite as long as the invariant length within each convergent circle is found, and more precisely, the increased complexity of Koch’s invariant length is found.

If this convergent series of circles’ lengths has the value of the limit (which is zero) Koch’s fractal invariant length and its infinite increased complexity is lost.

Conclusion: Koch’s fractal invariant length and its infinite increased complexity, is found as long as the value of the limit (which is 0) is not reached, and as a result we get a naturally open and incomplete convergent series.

Again a set does not have to have its limit (or limits) as a member (or members) of that set. Again if you simply make “open and incomplete” synonymous with “infinite” then your are simply claming “as a result we get a naturally ” infinite “convergent series“.

So did you simply miss that fact or are you deliberately ignoring it?

Again your simple and deliberate ignorance combined with your morbid fantasies do not constitute facts (other than you being deliberate ignorant and enjoying your morbid fantasy).

“So what” clearly summarizes your “deep” understanding of this subject.

Again we see that you simply mentioning “DNA” and “the mass of a shrinked star” “clearly summarizes your “deep” understanding of” those subjects. Doron you simply mentioning those topics does not suddenly imbue either you or your OM with any understanding of them (“deep” or otherwise).

Again we see how your standard notion that is based on stretched points as the basis of lines, prevents from you to understand the real complexity, which is the result of Non-locality\Locality Linkage.

Who ever said anything about “stretched points”, other then you?

The terms of the series are exactly the reason of why the given series has no limit and no sum (it is inherently incomplete).

What? So now the series does not even have a limit? Again you simply making “incomplete” sysnonomus with “infinite” means all you are claiming is that the series “is inherently” infinite.

Simply wrong, and your insistence to claim otherwise does not change the fact that it is a wrong argument of jsfisher, on this subject.

Your simple and apparently deliberate ignorance of what constitutes a fractal does not infact make them fractals

As already explained to you before, any infinite collection is inherently incomplete, exactly because it is not limited to any class.

Um “incomplete” would be a class of collections (even when you do simply make it mean infinite), again since you make “incomplete” synonymous with “infinite” you are again simply stating that “any infinite collection is inherently” infinite.

This “proof” was wrong 2,300 years ago, it is wrong now, and it will stay wrong for the next coming 2,300 years.

In other words, it is a time independent false.

Again your simple and deliberate ignorance as well as your preference for your morbid fantasy do not refute that proof.

Closed under classes, isn't it The Man?

No, it is closed under an operation of succession as clearly stated and affirmed by your own assertion.

Any knowledge (obsolete or novel, it does not matter) uses also "baby staps" in order to help people to grasp it, so?

Until you stop deliberately choosing to be just a baby about things. Any chance of you growing up any time soon Doron?

Yes, also by Head/Hammer scholars.

It was Head/Hummer that I recommended, if you recall.

 

[doronshadmi]

Nope, as zooterkin has pointed out to you several times already your ‘bended’ line is no longer one dimensional.

Bended line has no impact on its length, and this is exactly the reason of why infinitely many bended levels are inherently incomplete (the 0 length at the limit is not reached, otherwise we get the false equation (X>0)=0).

Standard Math can't deal with this anomaly by using Limits.

Again a circle is one dimensional because of the consistency of its radius, your ‘bended’ line has no such consistent radius

1) The invariant length of Koch's fractal is measured under 1-dim (only the length is considered).

2) The length of the 1-dim element , which clearly converges at each additional bended level, is not influenced by the infinitely many additional bended levels.

3) So we get an anomaly of standard analysis, where at the base of 1/1+1/4+1/16+1/64+1/256+… convergent series of bended levels there is an invariant length > 0, which is resulted by an incomplete number of bended levels, because it logically can't reach the limit point, which its length is exactly 0.

4) By including the limit point as a part of the convergent series of bended levels, we actually claim that the invariant length > 0 is also equal to the length of the limit point, which is exactly 0, which is a contradiction notated as (X>0) = 0.

5) Conclusion:

X>0

X+X+X+X+… stands at the basis of a convergent series like 1/1+1/4+1/16+1/64+1/256+… and as a result this series is incomplete exactly because (X>0) = 0 is false.

Again a set does not have to have its limit (or limits) as a member (or members) of that set.

The limit (which has length 0) or any bended location along the 1-dim element, do not contribute anything the invariant X>0 length of the 1-dim element.

The bended locations along X invariant length contributes only the convergence of X upon infinitely many bended levels, which are incomplete, exactly because X length is invariant under infinitely many convergent bended levels ((X>0) = 0 is not a part of the reasoning).

So The Man, after all 1+1+1+… is relevant under a convergent series like 1/1+1/4+1/16+1/64+1/256+…

Who ever said anything about “stretched points”, other then you?

You are right about that. You used the term "dragged point" in order to define a line.

In that case please show us how a dragged 0-dim, which is local by nature, can define a 1-dim, which is non-local by nature.

What? So now the series does not even have a limit?

Good morning The Man, your limit notion is only a part of your false dream.

Um “incomplete” would be a class of collections

No.

Incompleteness is exactly the notion that is not bounded by the Class notion.

No, it is closed under an operation of succession

Succession it a Trojan horse of any closed notion, exactly as the invariant X>0 is a trojan horse of the convergent series 1/1+1/4+1/16+1/64+1/256+… which is naturally opened (the inconsistency of (X>0)=0 is not a part of the consistent reasoning of the openness of any infinite collection).

Until you stop deliberately choosing to be just a baby about things. Any chance of you growing up any time soon Doron?

Any chance that (X>0)=0 will not be a part of your reasoning, The Man?

Doron you simply mentioning those topics does not suddenly imbue either you or your OM with any understanding of them (“deep” or otherwise).

The "sudden effect" of these topics is the result of your obsolete Limit-oriented knowledge of these topics.

 

[doronshadmi]

I have provided a complete statement of the axiom. I'm sorry you don't understand the language in which it was expressed,

Nonsense.

Real universal notions are not limited to any particular language or form of expression.

So this time please translate the universal notion to plain English and do not hide behind your particular expression of it.

No, terms you invent don't actually mean anything because you refuse to define them.

Wake up jsfisher it is already done at http://www.internationalskeptics.com/forums/showpost.php?p=5704824&postcount=9004 (both by a diagram and a string of symbols) but you can't get it because your reasoning is closed under the expression of a sting of linear representation of symbols (which is nothing but a particular expression of a notion, and if a notion is limited to some particular representation, its universality is lost).

Still, it is nice to see you admit that you also invent multiple terms all to mean the same thing.

This is the whole idea of a universal notion, it is not limited to any particular expression of it.

Again your linear-only step-by-step local-only reasoning rising its limited head.

 

[doronshadmi]

What, so now the circle is convergent? Again a circle is one dimensional because of the consistency of its radius, your ‘bended’ line has no such consistent radius (if it did it would be a circle).

Again generalization problems The Man?

The general thing here is Length.

By providing the proof without words (http://en.wikipedia.org/wiki/Proof_without_words) below, I clearly expose an anomaly at the Limit-oriented reasoning, where a convergent length is linked with an invariant length, over infinitely many convergent scale levels, and as a result the convergent series of that convergent length can't have 0 as a part of that series.

As a result we get infinitely many and incomplete convergent scale levels, that approaches but can't get length 0.

In fact the fully iterated Koch curve (which is a fractal as noted by jsfisher) has a Hausdorff dimension of log(4)/Log(3) or about 1.2619,

1) 1.2619… is not an accurate value, so you have no accurate sum of Koch's fractal dimension, and your Limit-oriented approach fails also in this case.

2) 1.2619… is some (and non-accurate) result of a particular logs's base, so also in this case 1.2619… does not provide the dimension of Koch's fractal.

 

[doronshadmi]

Again a set does not have to have its limit (or limits) as a member (or members) of that set.

EDIT:

So they are least upper bounds or greatest lower bounds, which are known also as the limits of a given set, which are not necessearly members of a given set.

It does not change the fact that the limit element is not the final element of an infinite collection, whether it is a member of a given set or not.

We are dealing here with a limit of a convergent series of added length, and I provided a proof without words in http://www.internationalskeptics.com/forums/showpost.php?p=5704824&postcount=9004 which proves that an infinite convergent series has no limit.

This proof is an anomaly of standard math of this subject, and therefore cannot be comprehended by using the standard knowledge of this subject.

 

[The Man]

Bended line has no impact on its length, and this is exactly the reason of why infinitely many bended levels are inherently incomplete (the 0 length at the limit is not reached, otherwise we get the false equation (X>0)=0).

The point wasn’t about length, but dimensions. Are you simply or deliberately unable to distinguish length from dimensions?

Since you bring up length anyway, the Koch curve has an infinite length giving the Koch snowflake an infinite perimeter, but it still encompasses a finite area.

Standard Math can't deal with this anomaly by using Limits.

Try actually learning and using what you call “Standard Math”, at least then you might understand the difference between length and dimension.

1) The invariant length of Koch's fractal is measured under 1-dim (only the length is considered).

That “length” still involves more than one dimension so the appropriate term would be perimeter.

2) The length of the 1-dim element , which clearly converges at each additional bended level, is not influenced by the infinitely many additional bended levels.

In the actual construction of a Koch snowflake the perimeter increases by 1/3 with each iteration so the value of it’s perimeter is divergent not convergent.

Now in limiting that perimeter to a finite value (1 in this case) you wind up with an interesting problem. In that you have to scale down the construction (of an actual Koch snowflake) by 1/3 with each iteration to maintain your finite perimeter requirement. So you are simply scaling down your inscribing circle by 1/3 each time and thus end up with a convergent series with a common ration of 1/3 that starts at a value of 2*p*{[Tan(60)*1/6]-[Tan(30)*1/6]} or about 1.209 (the circumference of your first circle). Making the difference between the 3 times series and your original as 6*p*{[Tan(60)*1/6]-[Tan(30)*1/6]} (3 times circumference of your first circle) and thus the sum of your series as 3*p*{[Tan(60)*1/6]-[Tan(30)*1/6]} or about 1.814 (3/2 times the circumference of your first circle).

So your inclusion of your perimeter limited pseudo-Koch snowflake is entirely superfluous, as jsfisher has already noted.

3) So we get an anomaly of standard analysis, where at the base of 1/1+1/4+1/16+1/64+1/256+… convergent series of bended levels there is an invariant length > 0, which is resulted by an incomplete number of bended levels, because it logically can't reach the limit point, which its length is exactly 0.

The “anomaly of standard analysis” simply remains yours, based once again on your simple or deliberate misunderstanding and misrepresentation of what you call “Standard Math”.

4) By including the limit point as a part of the convergent series of bended levels, we actually claim that the invariant length > 0 is also equal to the length of the limit point, which is exactly 0, which is a contradiction notated as (X>0) = 0.

5) Conclusion:

X>0

Again the limit of a set need not be a member of that set and the perimeter in the construction of an actual Koch snowflake is divergent not convergent. The fractal has an infinite perimeter, but encompasses a finite area.

X+X+X+X+… stands at the basis of a convergent series like 1/1+1/4+1/16+1/64+1/256+… and as a result this series is incomplete exactly because (X>0) = 0 is false.

“1/1+1/4+1/16+1/64+1/256+…” is not “X+X+X+X+…” it is however X+X/4+X/16+X/64+X/256.…. Again “X+X+X+X+…”, having a common ratio equal to 1, would be divergent not convergent. Once again your assertions are easily demonstrated to be simply false.

The limit (which has length 0) or any bended location along the 1-dim element, do not contribute anything the invariant X>0 length of the 1-dim element.

A limit contributes exactly that, a limitation. Exactly as your perimeter value always equals one is just your own limitation in this instance.

The bended locations along X invariant length contributes only the convergence of X upon infinitely many bended levels, which are incomplete, exactly because X length is invariant under infinitely many convergent bended levels ((X>0) = 0 is not a part of the reasoning).

If “X” is “invariant” then a series of infinitely many “X”s is not convergent.

So The Man, after all 1+1+1+… is relevant under a convergent series like 1/1+1/4+1/16+1/64+1/256+…

No Doron even after all your nonsensical gibberish “1+1+1+…” is still only divergent and “1/1+1/4+1/16+1/64+1/256+…” only convergent.

You are right about that. You used the term "dragged point" in order to define a line.

In that case please show us how a dragged 0-dim, which is local by nature, can define a 1-dim, which is non-local by nature.

It was already given to you before Doron.

http://en.wikipedia.org/wiki/Dimension

Inductive dimension
Main article: Inductive dimension
An inductive definition of dimension can be created as follows. Consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an n+1-dimensional object by dragging an n dimensional object in a new direction.
The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that (n + 1)-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.

http://en.wikipedia.org/wiki/Inductive_dimension

Inductive dimension
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.
The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.

Just another way of considering or defining the dimensions of an object or space in what you like to call “Standard Math”.

Good morning The Man, your limit notion is only a part of your false dream.

Good morning Doron, oh wait, sorry, you still haven’t woken up from your morbid dream of saving our civilization with your OM fantasies. Well, maybe tomorrow then?

No.

Incompleteness is exactly the notion that is not bounded by the Class notion.

Unbounded sets are a class of sets just as bounded sets are.

http://en.wikipedia.org/wiki/Unbounded_set

Succession it a Trojan horse of any closed notion, exactly as the invariant X>0 is a trojan horse of the convergent series 1/1+1/4+1/16+1/64+1/256+… which is naturally opened (the inconsistency of (X>0)=0 is not a part of the consistent reasoning of the openness of any infinite collection).

No the inclusion in the set of the successor of any element in that set is specifically what closes the set under an operation of succession on any element of that set. If you think it is a “Trojan horse” that attacks your notions that’s just because you brought it in yourself.

Any chance that (X>0)=0 will not be a part of your reasoning, The Man?

It never has been Doron, it is simply part of your deliberately failed “reasoning” that you like to ascribe to others.

The "sudden effect" of these topics is the result of your obsolete Limit-oriented knowledge of these topics.

“sudden effect”? So you do think that simply mentioning some topic out of hand has a "sudden effect" that you hope will confuse people into thinking you actually know something about that topic?

Again generalization problems The Man?

Again Doron “generalization” does not mean simply claiming whatever you think suits your notions.

The general thing here is Length.

Again the appropriate term would be perimeter since more than one dimension is involved.

By providing the proof without words (http://en.wikipedia.org/wiki/Proof_without_words) below, I clearly expose an anomaly at the Limit-oriented reasoning, where a convergent length is linked with an invariant length, over infinitely many convergent scale levels, and as a result the convergent series of that convergent length can't have 0 as a part of that series.



As a result we get infinitely many and incomplete convergent scale levels, that approaches but can't get length 0.

[qimg]http://farm5.static.flickr.com/4070/4417179545_d4e9c86236_o.jpg[/qimg]​

Again a limit need not be a member of the set. For some reason you seem to think that the set not including its limit or limits as members somehow makes it incomplete. Of course since you simply make “incomplete” synonymous with “infinite” your assertion of “As a result we get infinitely many and incomplete convergent scale levels” simply means “As a result we get infinitely many and” infinite “convergent scale levels”.

1) 1.2619… is not an accurate value, so you have no accurate sum of Koch's fractal dimension, and your Limit-oriented approach fails also in this case.

“sum of Koch's fractal dimension”? Who said anything about a “sum of Koch's fractal dimension”, other than you. So now you are just going to deliberately confuse the Hausdorff dimension of a fractal with the sum of a convergent series? Yes “1.2619… is not an accurate value” that is why I said “or about 1.2619” the exact value is log(4)/Log(3) also as clearly given before.

2) 1.2619… is some (and non-accurate) result of a particular logs's base, so also in this case 1.2619… does not provide the dimension of Koch's fractal.

It provides the Hausdorff dimension of that fractal and others. As jsfisher has asked before….

do you have any understanding whatsoever of the impact fractals have had on the meaning of length and dimension?

What you call “standard math” is not the dogmatic adherence to previous understanding that you like to pretend and profess it to be.

 

[zooterkin]

Now in limiting that perimeter to a finite value (1 in this case) you wind up with an interesting problem. ...

I think this should be noted as a red-letter day. Doron has added something interesting to the thread.

 

[The Man]

EDIT:

So they are least upper bounds or greatest lower bounds, which are known also as the limits of a given set, which are not necessearly members of a given set.

Not all sets are bounded.

It does not change the fact that the limit element is not the final element of an infinite collection, whether it is a member of a given set or not.

If it is a member (and least upper bound) then it is the finail element of that collection.

We are dealing here with a limit of a convergent series of added length, and I provided a proof without words in http://www.internationalskeptics.com/forums/showpost.php?p=5704824&postcount=9004 which proves that an infinite convergent series has no limit.

No you didn’t, you simply imposed your own limit of the perimeter value for your superfluous pseudo Koch snowflake must be one and that still didn’t remove the limit of 0 for your scaled down inscribing circles or the sum of that series, which I have given you in my previous post.

This proof is an anomaly of standard math of this subject, and therefore cannot be comprehended by using the standard knowledge of this subject.

The “anomaly” is just in your brain Doron that has you preferring your own misinterpretation and misrepresentations and therefore you simply can not comprehend “the standard knowledge of this subject”.

 

[The Man]

I think this should be noted as a red-letter day. Doron has added something interesting to the thread.

Its about time.

 

[Manopolus]

Proof by intimidation

As in, this whole thread seems to be a feeble attempt at it.

 

[The Man]

Does it count if the only one Doron actually intimidates into believing his notions or “proof” is himself?

 

[jsfisher]

Nonsense.

Real universal notions are not limited to any particular language or form of expression.

So this time please translate the universal notion to plain English and do not hide behind your particular expression of it.

Since "real universal notions are not limited to any particular language or form of expression" you should have no trouble dealing with the form I provided.

In other words, you deal with the translation.

Was there something you found lacking in my formulation of the Axiom of Infinity? If so, let's discuss that rather than your lack of fluency in formal languages.

Wake up jsfisher it is already done at http://www.internationalskeptics.com/forums/showpost.php?p=5704824&postcount=9004 (both by a diagram and a string of symbols) but you can't get it because your reasoning is closed under the expression of a sting of linear representation of symbols (which is nothing but a particular expression of a notion, and if a notion is limited to some particular representation, its universality is lost).

No, it wasn't already done back in that or any URL. You don't define your terms, ever.

This is the whole idea of a universal notion, it is not limited to any particular expression of it.

It can be a universal notion only if it is, um, what's the word, oh yeah, universal. Yours are not. They are just nonsense. What to prove me wrong? Then define something.

 

[jsfisher]

By providing the proof without words....

Setting aside for a moment your bogus attempt at using AutoCAD as an automated theorem generator, what is it you think you actually proved? Can you give a concise statement of your theorem?

 

[doronshadmi]

Since you bring up length anyway, the Koch curve has an infinite length giving the Koch snowflake an infinite perimeter, but it still encompasses a finite area.

No, it has an incomplete area based on infinite interpolation that has no accurate sum.

You still do not get that any infinite collection is inherently incomplete, whether it is expressed as infinite interpolation or infinite extrapolation.

In the actual construction of a Koch snowflake the perimeter increases by 1/3 with each iteration so the value of it’s perimeter is divergent not convergent.

Now in limiting that perimeter to a finite value (1 in this case) you wind up with an interesting problem. In that you have to scale down the construction (of an actual Koch snowflake) by 1/3 with each iteration to maintain your finite perimeter requirement. So you are simply scaling down your inscribing circle by 1/3 each time and thus end up with a convergent series with a common ration of 1/3 that starts at a value of 2**{[Tan(60)*1/6]-[Tan(30)*1/6]} or about 1.209 (the circumference of your first circle). Making the difference between the 3 times series and your original as 6**{[Tan(60)*1/6]-[Tan(30)*1/6]} (3 times circumference of your first circle) and thus the sum of your series as 3**{[Tan(60)*1/6]-[Tan(30)*1/6]} or about 1.814 (3/2 times the circumference of your first circle).

So your inclusion of your perimeter limited pseudo-Koch snowflake is entirely superfluous, as jsfisher has already noted.

All you did is to show that an infinite interpolation has no accurate sum (in your language it is about 1.814…), which is exactly the same result of my infinite converges series, which has an invariant length of Koch's fractal > 0 that can't reach the limit value, which is exactly 0.

And I also showed it without using any circle, but simply show how the convergent length between the opposite edges of the Koch's fractal can't be 0, as long as these edges are in common with the invariant length of the Koch's fractal, upon infinitely many bended levels, here it is again (no circles are used):

You simply have no real notion of the inherent incompleteness of infinite collection (and in this case, infinite collection of bended levels).

A finite value has an accurate sum, an infinite value does not have an accurate sum, but you simply can't get it because you do not understand the real nature of infinite interpolation\extrapolation.

“1/1+1/4+1/16+1/64+1/256+…” is not “X+X+X+X+…” it is however X+X/4+X/16+X/64+X/256.…. Again “X+X+X+X+…”, having a common ratio equal to 1, would be divergent not convergent. Once again your assertions are easily demonstrated to be simply false.

If “X” is “invariant” then a series of infinitely many “X”s is not convergent.

Again, the convergent length between the opposite edges of the Koch's fractal can't be 0 (where 0 is the value of the limit), as long as these edges are in common with the invariant length of the Koch's fractal, upon infinitely many bended levels, as shown above, but you prefer to ignore this proof without words, because it does not fit your Limit-oriented notion, which is obsolete.

A limit contributes exactly that, a limitation. Exactly as your perimeter value always equals one is just your own limitation in this instance.

Exactly because the Koch's fractal has an invariant length > 0 , there are infinitely many bended levels that can't reach the value of the limit, which is exactly 0.

The limitation is entirely the result of your Limit-oriented approach, that prevents the notion of infinite interpolation\extrapolation, which is inherently non-limited.

No Doron even after all your nonsensical gibberish “1+1+1+…” is still only divergent and “1/1+1/4+1/16+1/64+1/256+…” only convergent.

You do not get that in my model there is no dichotomy between invariant AND convergent length, as clearly shown and explained above, but you can't get this anomaly because you are using a model that is based on dichotomy.

It was already given to you before Doron.

Just another way of considering or defining the dimensions of an object or space in what you like to call “Standard Math”.

Both Wiki sources are not based on the qualitative difference between the local and non-local atomic aspects that send at the basis of any complex like Set (for example).

Again an obsolete knowledge is used by you The Man, which is irrelevant to OM's novel notions, of this subject.

Unbounded sets are a class of sets just as bounded sets are.

No, the incompleteness of a collection is not closed under Class, where a set is a particular case of a collection that is closed under Class.

No the inclusion in the set of the successor of any element in that set is specifically what closes the set under an operation of succession on any element of that set. If you think it is a “Trojan horse” that attacks your notions that’s just because you brought it in yourself.

Since any infinite collection (where a set is a particular case of a collection that is closed under Class) is inherently incomplete (it has a “Trojan horse”), it does not have "closed gates".

You simply ignore the inherent incompleteness of any collection, whether it is closed under Classes or not.

“sudden effect”? So you do think that simply mentioning some topic out of hand has a "sudden effect" that you hope will confuse people into thinking you actually know something about that topic?

Again, the "“sudden effect” is the result of your obsolete Limit-oriented notion of this subject.

Again the appropriate term would be perimeter since more than one dimension is involved.

No, it is about an anomaly of the standard model, which shows how a length can be both invariant AND variant (converges, in this case) in a one model.

“sum of Koch's fractal dimension”? Who said anything about a “sum of Koch's fractal dimension”, other than you. So now you are just going to deliberately confuse the Hausdorff dimension of a fractal with the sum of a convergent series? Yes “1.2619… is not an accurate value” that is why I said “or about 1.2619” the exact value is log(4)/Log(3) also as clearly given before.

This is another example of your inability to understand the generalization of this subject.

If it is a member (and least upper bound) then it is the finail element of that collection.

Any infinite collection is inherently incomplete, or in other words, any given element of that collection is not its final element, and order has no impact on that fact.

No you didn’t, you simply imposed your own limit of the perimeter value for your superfluous pseudo Koch snowflake must be one and that still didn’t remove the limit of 0 for your scaled down inscribing circles or the sum of that series, which I have given you in my previous post.

On the contrary, your Limit-oriented notation, can't get the model where both invariant AND variant (converges, in this case) length are found.

Your built-in dichotomy of your notions simply can't get this anomaly of Standard Math model of this subject.

The “anomaly” is just in your brain Doron that has you preferring your own misinterpretation and misrepresentations and therefore you simply can not comprehend “the standard knowledge of this subject”.

A typical reply of a person that uses built-in dichotomy and Limit-oriented notions of this subject.

How boring.​