1=0.999... infinities and box of chocolates..Phliosophy of Math...

[Undefined]

Hi rpenner, Quantum Quack. I was just checking for typos to my last, and saw this bit. My brief observation on this 'take' on what constitutes "1" and "less than 1" may be of interest when trying to consider what an 'infinitesimal of effectiveness' QUANTUM means for your present discussion...

But just calling something "infinitesimal" is not a property that pins down a number to be specific. You have always acted like there must be one such smallest positive number, but that is not a general property of numbers less than 1. Therefore it follows if there is one such positive number smaller than any positive rational number there must be an unending progression of numbers with the same properties.

This follows because the square of a positive number is a positive number and the product of any two non-zero numbers of magnitude less than 1 is a third number with magnitude smaller than either of the first two.

That is one of the problems in the current axiomatic interpretation of "1". You have to realize that the infinitesimal QUANTUM is in EFFECT physically and logically the ORIGINAL "1" QUANTUM that starts the whole cascade of aggregation to higher level 'units' which are comprized at root BY whatever number of infinitesimal quantums make any particular aggregate number 'unit' construct.

Quickly then because I'm rushed and have to go again, once the '1' is THE infinitesimal, then that above logics used by rpenner, implying that 'infinitesimal is ASSUMED TO BE LEES THAN '1'....and then using that arbitrary designation as THE '1' concept/value to justify the argument that any product of any two infinitesimals must be less than either infinitesimal is patently circuitous 'proofing' argument....simply because of the initial assumption unwittingly 'defining' an infinitesimal quantum as less than '1' in the first place!

If one logically and physically effectively STARTS from the 'entity' of infinitesimal quantum as THE starting '1' unitary from which all higher level aggregations labeled '1' are CONSTRUCTED in the first place, then the 'product between infinitesimals CANNOT BE LESS than '1' infinitesimal quantum.

Problem solved. By starting with new axioms treating the infinitesimal as THE starting '1' unitary entity and going from there to all the higher level maths '1' entities constructed therefrom.

Sorry if that looks rushed. It was. But I think you got the drift of what a NEW and more reality-sensible 'number system' might look like soon if the axioms are changed for the better?

Gotta go. See ya round. Good luck and enjoy your discussions, guys!

 

[rpenner]

Let x and y be two positive numbers of magnitude less than 1. Therefore there exist s = 1 - x and t = 1 - y which are two positive numbers of magnitude less than 1.
Therefore all of the following products: st, sy, xt and xy are positive. But since xy + sy = (x + s) y = y, it follows xy = y - sy and so xy < y. Likewise xt + xy = x ( t + y) = x and so xy < x.

Geometrically, where lengths s + x = t + y , we can multiply (s + x)(t + y) = st + sy + xt + xy and show that 0 < x < s + x and 0 < y < t + y implies 0 < xys + x)(t + y) < x(t + y)s + x)(t + y) AND 0 < xys + x)(t + y) < (s + x)ys + x)(t + y).

$$
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[rpenner]

What would the volume of a sphere be if the diameter was 1 Plank length?
Would it change the math at all?

No, because a sphere is described in Euclidean 3-space. Because of Relativity, we already know the geometry of space-time is not Euclidean, but the approximation that space is Euclidean is a good approximation at human length scales and at human everyday relative velocities.

The Planck length is not an absolute minimum length but rather a characteristic length scale near which generalizations based on Euclidean intuitions are expected (given current understanding) to become unreliable. What this means for physics is unlikely to be learned soon as this scale is very far removed from typical experimental and observational scales. What it means to Euclidean geometry is nothing -- Euclidean geometry may have started as a model of the process of surveying fields, but its implications are fixed from its axioms, not its applications.

 

[Quantum Quack]

No, because a sphere is described in Euclidean 3-space. Because of Relativity, we already know the geometry of space-time is not Euclidean, but the approximation that space is Euclidean is a good approximation at human length scales and at human everyday relative velocities.

The Planck length is not an absolute minimum length but rather a characteristic length scale near which generalizations based on Euclidean intuitions are expected (given current understanding) to become unreliable. What this means for physics is unlikely to be learned soon as this scale is very far removed from typical experimental and observational scales. What it means to Euclidean geometry is nothing -- Euclidean geometry may have started as a model of the process of surveying fields, but its implications are fixed from its axioms, not its applications.

thank you for your erudite assessment, much appreciated.
As i said in the edit: My proposition has been utterly refuted and I thank you for your patience.

 

[Undefined]

Hi rpenner, Quantum Quack.

Let x and y be two positive numbers of magnitude less than 1. Therefore there exist s = 1 - x and t = 1 - y which are two positive numbers of magnitude less than 1.
Therefore all of the following products: st, sy, xt and xy are positive. But since xy + sy = (x + s) y = y, it follows xy = y - sy and so xy < y. Likewise xt + xy = x ( t + y) = x and so xy < x.

Geometrically, where lengths s + x = t + y , we can multiply (s + x)(t + y) = st + sy + xt + xy and show that 0 < x < s + x and 0 < y < t + y implies 0 < xys + x)(t + y) < x(t + y)s + x)(t + y) AND 0 < xys + x)(t + y) < (s + x)ys + x)(t + y).

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Under the current axiomatic system of maths, sure. But I just pointed out that a new axiomatic system where an infinitesimal quantum of effectiveness is EFFECTIVELY and LOGICALLY the original "1", and nothing is smaller than that original "1" quantum as a 'number' representing a FUNDAMENTAL 'unity' state.

Under that new axiomatic maths, th "1" you are currently using would be a COMPOSITE UNITY state/number constructed up of whatever number of fundamental infinitesimal "1" quanta makes up whatever UNIT ENTITY under study/consideration in any particular context where the 'composite unitary' may be a 'camel' (see arfa branes' and my posts above using that example/context) or whatever.

That is why pure abstraction leads to loss of contextual information which is crucial to prevent maths from falling into the unfortunate 'logic/asR current axiomatic treatment/construct, not in the new axiomatic improved treatment/construct alluded to above.

Did you note and understand what I said in my post #1121, and its implications for the logics/arguments/proofs you used in your post #1118? The 'product' of the original '1' quantum of effectiveness is NOT less than '1' such quantum '1' in my new axiomatic maths construct; and the 'fractional maths' exercises are totally contextual to clearly specify that the '1' in your current maths is a COMPOSITE '1', made up of whatever number of infinitesimal quantum '1's that make up that particular 'composite unity' in the context you are working in.

The 'fraction numbers' of a composite unity '1' represent each of the individual infinitesimal quantums going up to make up that composite '1' represent...and most importantly, there are not an 'inifity of them, but a complex string of infinitesimal quantum 'numbers' that together give the overall '1' composition unit number in any one context.

No, because a sphere is described in Euclidean 3-space. Because of Relativity, we already know the geometry of space-time is not Euclidean, but the approximation that space is Euclidean is a good approximation at human length scales and at human everyday relative velocities.

The Planck length is not an absolute minimum length but rather a characteristic length scale near which generalizations based on Euclidean intuitions are expected (given current understanding) to become unreliable. What this means for physics is unlikely to be learned soon as this scale is very far removed from typical experimental and observational scales. What it means to Euclidean geometry is nothing -- Euclidean geometry may have started as a model of the process of surveying fields, but its implications are fixed from its axioms, not its applications.

You should be aware that Shaun Carroll et al have measured the angles related to the WMAP data 'features', and they found that space is FLAT and EUCLIDEAN and goes on INFINITELY beyond the observable universal extent, because their measurements determined that the 'triangles' they employed in their measurement construct add up to ONLY 180 degrees, to a precision to three decimal places.

You should also be aware that SPACE in reality is Euclidean; while the NON-Euclidian 'mathematics/geometry' concepts/abstractions are introduced into the 'analysis' when the motional dynamics of the energy features arising, interacting, subsiding in the ENERGY-SPACE (not space-time, since that is abstract and misleading mathematical 'artifact') are observed and modeled in an abstract NON-Euclidian 'space-motion' (ie, space-time currently called) construct which effectively is dealing with the DYNAMICS of ENERGY features arising, changing and subsiding as part of the energy-space substrate of the universal phenomena.

So, basically, energy-space is flat and Euclidian to infinity at the largest scales; while the NON-Euclidian context arises only in relation to FEATURES within that energy-space, not energy-space itself in its ground state globally.

So, please note from the above: Nature has no NON-Euclidian 'space', only NON-Euclidian 'surfaces/densities/gradients etc' of localised configurations/accumulations etc of energy-space 'features' arising from that flat Euclidian substrate of infinitely extending energy-space in its ground state.

That's as far as I am at liberty to explain of my ToE perspective on these matters. So I'll leave you all to it. Enjoy your explorations/discussions into all the yet to be teased out subtleties in both the mathematics and the physics contextual realities!

Good luck, and see y'all round, rpenner, QQ, arfa, everyone!

 

[arfa brane]

Under the current axiomatic system of maths, sure. But I just pointed out that a new axiomatic system where an infinitesimal quantum of effectiveness is EFFECTIVELY and LOGICALLY the original "1", and nothing is smaller than that original "1" quantum as a 'number' representing a FUNDAMENTAL 'unity' state.

It looks vaguely useful, but how will it change the way I count things?

That is why pure abstraction leads to loss of contextual information which is crucial to prevent maths from falling into the unfortunate 'logic/assumption' that a '1' can be decomposed into less-than-one QUANTA in all circumstances/contexts.

You seem to be displaying a poor understanding of numbers. Whole numbers like 1 can be used to count whole objects, like apples and oranges. As soon as you start counting fractional objects, you "decompose" 1 into parts.
If I count some set of objects and "calculate" the total number, I haven't lost any context. This "loss of information" you're banging on about is something you've so far failed to define, at all.
Firstly, you have to define what the information is, which as I explained, is a completely arbitrary thing--one man's information is another man's gibberish.
Then you need to show how this information is lost.

For instance, if I define information as being composed of English or Chinese words, then someone who doesn't understand Chinese will have effectively "lost" information in a message written in Chinese, whereas the message contains the information as defined--Chinese or English words.
If I define information as being composed of strings of 1s and 0s, then a string which is interpreted by some computer as an address will "lose" this information if the same machine interprets the string as an instruction. In either case the string contains the information as defined, however it's interpreted or "given meaning".

You have also implied that since the natural numbers are abstract, hence can be used to count any kind of set or collection of objects, that information about what is being counted is lost, somehow. And yet if I count n camels, I know I haven't counted n apples, or n oranges. What I'm counting is also information, of course. So there is no "problem".
You apparently can't see this.

 

[Undefined]

Hi arfa. Just correcting typos in my last, and saw this. So briefly as I can and then get outta here...

It looks vaguely useful, but how will it change the way I count things?

All new maths when first thought up and introduced/developed seems only 'vaguely useful'. Then the revolution happens and it advances the whole mathematics system to be more effective in domains of applicability not possible/reachable by the maths as it was before that 'vaguely useful' insight/introduction. That's the trajectory of revolutionary advances from individuals which the established 'school of thought' reacts to badly as a first-response. They even called Cantor insane! But you accept his specific treatments for specific contexts as sanguinely as could be. I wonder what you would have made of Cantor and his work/new insights/maths if you had been a contemporary of his in the prevailing school of thought. Would you have called him insane, I wonder? No. I suspect not. I think you are not as 'reactionary' as some I could name.

And like I said earlier, there are many 'maths systems' for different applications/fields/purposes/conveniences/shortcuts etc etc. My making the axiomatic mathematical construct more reality-contextual overall may make all the existing maths systems/applications more 'information dense' than at present. Even your way of 'counting things' may take on whole new 'dimensions' of contextual usefulness and completeness. It will certainly help the real physics modeling no end, hey!

You seem to be displaying a poor understanding of numbers. Whole numbers like 1 can be used to count whole objects, like apples and oranges. As soon as you start counting fractional objects, you "decompose" 1 into parts.
If I count some set of objects and "calculate" the total number, I haven't lost any context. This "loss of information" you're banging on about is something you've so far failed to define, at all.
Firstly, you have to define what the information is, which as I explained, is a completely arbitrary thing--one man's information is another man's gibberish.
Then you need to show how this information is lost.

Numbers as YOU currently know and love them are NOT the same numbers I am talking about. That was the point. Obviously I don't want to limit new insights by just defaulting back to current number understandings. Or what is the point of new insights, hey?

Did you read and understand where I explained at length to Trippy that 1/3 and .333... notations are not yet operations completed, but only 'aspirational statements' of what you want to do. At no stage has the fractional notation actually been completed in reality to identify A fractional entity. At best, fractional notations represent merely another NOTIONAL POINT on a NOTIONAL abstract line/number construct.

That notation OMITS any information which could have given you the understanding that the 'point' is a UNIT of the line. The unfortunate axiom that has 'point' as DIMENSIONLESS effectively destroys all contextual understanding/insight that EACH POINT IF NOT DIMENSIONLESS (ie, as in infinitesimal quantum of effectiveness) would represent the ORIGINAL INDIVISIBLE QUANTUM '1' from which all other numbers are COMPOSED/CONSTRUCTED...analogous to how you think the fractional numbers compose the '1' unitary entity.....without realizing that you are there effectively constructing a COMPOSITE '1" ENTITY from lesser quantum/point entities!

See? The only difference between my composite '1' and your composite '1' is that mine is constructed from REAL PHYSICALLY REFLECTIVE 'infinitesimal quantum DIMENSIONAL INFORMATION POSSIBLE PHYSICAL ENTITIES, while yours is constructed from UNREAL/PHILOSPHICAL 'point' quantum NON-DIMENSIONAL UN-INFORMATIONAL IMPOSSIBLE NOTIONAL ENTITIES. Short step from one to the other axiomatic understanding; but a BIG step for those already steeped in current 'school of thought' and reacting as humans do to the new, but NOW obvious, insights I point out.

You have also implied that since the natural numbers are abstract, hence can be used to count any kind of set or collection of objects, that information about what is being counted is lost, somehow. And yet if I count n camels, I know I haven't counted n apples, or n oranges. What I'm counting is also information, of course. So there is no "problem".
You apparently can't see this.

Sure, the SUPERFICIAL limited context is as I already explained. But did you forget that I also explained the further subtle and informative complex context regarding what camels may fall pregnant and what the possible changes to that 'count' may be PREDICTED if that further information of PREGNANT was attached to the '1' of a female camel who may be pregnant when the social interactions of REAL camels in REAL time in REAL space. Think of what PHYSICISTS need by way of ALL the necessary REAL information (and not just abstract numbers/counts/analysis math constructs) in order to COMPLETE the REAL (not merely abstracted math modeling) and consistent in all cases contextual maths-physics ToE! Then you'll get the idea where all this is headed and what motivated this effort to review the maths axoms for that ultimate REAL PHYSICAL PURPOSE.

Anyhow, I've tried my best in the short time and limited liberty I have to divulge some of my work on the ToE. If you haven't got at least a sense of the huge potential revolution in maths/physics happening even as we speak of these novel insights, then I must nevertheless leave it at that for obvious reasons pending publication of the whole works! Cheers and than you muchly for your polite discourse, arfa, everyone! Good luck till we speak again. Will go back to read only for a while. Bye for now.

 

[arfa brane]

Did you read and understand where I explained at length to Trippy that 1/3 and .333... notations are not yet operations completed, but only 'aspirational statements' of what you want to do. At no stage has the fractional notation actually been completed in reality to identify A fractional entity. At best, fractional notations represent merely another NOTIONAL POINT on a NOTIONAL abstract line/number construct.

Here you seem to be displaying more poor understanding. The notation 1/3 doesn't imply an incomplete operation, or "something you want to do".
And of course fractions represent notional points on an abstract number line, what's wrong with that?

That notation OMITS any information which could have given you the understanding that the 'point' is a UNIT of the line.

A point can't be a unit. You appear to have a poor understanding of the relation between geometry and the number line. The distance between 0 and 1 is a unit, but the point at 0 and the point at 1 can't be units, that would destroy the logic, or the symmetry.

But did you forget that I also explained the further subtle and informative complex context regarding what camels may fall pregnant and what the possible changes to that 'count' may be PREDICTED if that further information of PREGNANT was attached to the '1' of a female camel who may be pregnant when the social interactions of REAL camels in REAL time in REAL space.

I'm pretty sure animal herders and farmers are aware of all that, and the "complexities" involved. So what does that leave you for a point to make? It doesn't look like there is one.

I'm going to step away from the computer now, and keep my hands where I can see them. Bye.

 

[Undefined]

Here you seem to be displaying more poor understanding. The notation 1/3 doesn't imply an incomplete operation, or "something you want to do".
And of course fractions represent notional points on an abstract number line, what's wrong with that?A point can't be a unit. You appear to have a poor understanding of the relation between geometry and the number line. The distance between 0 and 1 is a unit, but the point at 0 and the point at 1 can't be units, that would destroy the logic, or the symmetry.I'm pretty sure animal herders and farmers are aware of all that, and the "complexities" involved. So what does that leave you for a point to make? It doesn't look like there is one.

I'm going to step away from the computer now, and keep my hands where I can see them. Bye.

You have to read/understand all of the the background posts/context/thrust here and elsewhere regarding many of the points made, and especially that 'not yet accomplished' aspect when using 1/3, .333... as if you already have identified the number in reality instead of in your abstract construct which assumes that such is identified as real point on real line. Since your construct is predicated on the philosophical notion of dimensionless point, then all the rest of your 1/3 and .333... is just abstract assumption that you have a dimensionless point there. The 'operation' which you think are implied in that notation is actually not effected at any stage unless you actually calculated out until infinite term, which as you know you can't unless you assume the Limits argument which has been the whole subject of the issue of lack of independent proofs which are not self-referential within the same construct which assumes the operation has already been done.

It's once you get out into an independent perspective that you see (as I explained to Trippy when he too brought 'reality' into the assessment of 1/3, .333... etc notations and what it actually did or did not imply). Like I said, those in independent view are 'aspirational' expressions/nottion, not actually fait accompli statements of facts/proofs of anything that is not already circuitous argument within the current construct. Sorry, must dash. Have said about all I can say at this stage. I (and others too) have given you many starting insights into new perspectives in maths and physics realities for you et al to be going on with. The rest is up to you. Good luck!

 

[hansda]

For what purpose to you wish to prove that?

Just for the simple fact that, $$S_n = 1 - \frac{1}{9} \times T_n$$ for any value of $$n$$.

If $$T_n \ne 0$$ for any value of $$n$$, then $$S_n \ne 1$$ for any value of $$n$$.

1. That does not follow, because if just one member of the series were zero, there would be an unending sequence members past that one member and thus the value of the limit of the sequence of ratios of terms would be unaffected.

1. 
   
   This is just a philosophical imagination but there is no mathematical truth in this, because the moment $$T_n = 0$$, the constant ratio of this infinite geometric series will be either $$0$$ or $$undefined$$. So, it will no longer remain a geometric series and the series will collapse.
   
   Though $$\lim_{n \to \infty} T_n = 0$$, is true but this means as $$n \to \infty$$; $$T_n$$ takes a non-zero value in the neighbour-hood of $$0$$, maintaining the constant ratio of the geometric series and it can continue like this without any end.

 

[rpenner]

The point which i want to prove is that, $$T_n \ne 0$$for any value of $$n$$; where $$n $$ ... [is in the] natural numbers...

For what purpose to you wish to prove that?

Just for the simple fact that, $$S_n = 1 - \frac{1}{9} \times T_n$$ for any value of $$n$$.
If $$T_n \ne 0$$ for any value of $$n$$, then $$S_n \ne 1$$ for any value of $$n$$.

well ...

it was already covered in ...

From [post=3164396]post #915[/post] ... every member of the series $$(T_n)$$ is greater than zero.... [See also] ... [post=3165524]post #990[/post]​

All members of the series $$(T_n)$$ are positive which means none equals zero. But $$T$$ is not defined as a member of the family $$(T_n)$$;

Likewise $$S$$ is not defined as a member of the family $$(S_n)$$. So proving $$S_n \ne 1$$ for every n in the natural numbers is not useful to your side of the debate because I have never said otherwise and often said the stronger statement: $$\forall n \in \mathbb{N} \quad S_n \lt 1$$ (For every n in the natural numbers, $$S_n$$ is a number less than 1).

From the theorem of ratio test, we can see that the constant ratio of the infinite geometric series is maintained throughout the series.

The ratio test doesn't say that. The ratio test says that if the limit $$\lim_{n\to\infty} \left| \frac{T_{n+1}}{T_n} \right|$$ exists and that limit is less than 1 then the related infinite series $$\sum_{n\geq 1} T_n = \lim_{n\to\infty} S_n = S$$ exists in the real numbers.
The ratio test is an implication, not a logical equivalence, because $$\sum_{n\geq 1} \frac{1 + (-1)^n}{n^2} = \frac{\zeta(2)}{2} = \frac{\pi^2}{12}$$

If any member of this series is zero(i.e. if $$T_n = 0$$ for any value of $$n$$), this constant ratio($$\lim_{n \to \infty} \frac{T_{n+1}}{T_n}$$) of the infinite geometric series can not be maintained.

1. That does not follow, because if just one member of the series were zero, there would be an unending sequence members past that one member and thus the value of the limit of the sequence of ratios of terms would be unaffected.
2. ...

This is just a philosophical imagination but there is no mathematical truth in this, because the moment $$T_n = 0$$, the constant ratio of this infinite geometric series will be either $$0$$ or $$undefined$$. So, it will no longer remain a geometric series and the series will collapse.

Aha, but have you considered that it was not I that first stepped out into contra-mathematical philosophical speculation, as we both agree that because the positive numbers are closed under multiplication (i.e. positive times positive equals positive), then no member of the particular sequence $$(T_n)$$ could be zero. So you later changed the topic to the ratio test (which was proved in your reference that I quoted the proof of in [post=3170118]post #1079[/post]) but misunderstood what the ratio test was saying. I was just pointing out that if you had a different series, that differed from the $$(T_n) = (9 \times 10^{-n})$$ formerly under discussion at just one point where it was zero (which does seem to be your hypothesis in your discussion about the ratio test), then that would not be a geometric series but the ratio test would still say the new series would converge.

For example

$$ 0.999099999... = \sum_{n\geq 1} \left{ \begin{array}{lcl} 0 & \quad \quad \quad & \textrm{if} \; n = 4 \\ 9 \times 10^{-n} & & \textrm{otherwise} \end{array} \right.$$​

is a series where the limit of the absolute value of the ratio of successive terms is a number less than 1 and therefore this series converges under the limit test. Indeed its value is $$\frac{9991}{10000}$$.

Logically, $$(T_n) = (9 \times 10^{-n})$$ and its cousin which is zero at just one term are distinct from each other, but I followed your practice of calling the second one $$(T_n) $$ in your discussion of the limit test, just as I can keep track of two different people both named "Fred", because to assume $$(T_n) $$ could only stand for one and only one series, I would have to assume you were saying a particular term could both be positive and equal to zero, which would be nonsense. Therefore I assumed you were talking about two sequences both named $$(T_n)$$, and focused on your misunderstanding of the ratio test and limits.

Though $$\lim_{n \to \infty} T_n = 0$$, is true but this means as $$n \to \infty$$; $$T_n$$ takes a non-zero value in the neighbour-hood of $$0$$, maintaining the constant ratio of the geometric series and it can continue like this without any end.

You are close to understanding limits and thus the real numbers.

The proof that $$\sqrt{2}$$ is in the real numbers follows directly from the definition in terms of rational numbers: $$\sqrt{2} = \text{sup} \left{ x \in \mathbb{Q} : x^2 \lt 2 \right}$$ (The square root of two is the supremum of all rational numbers with the property that 2 is larger than their squares). From this it follows that there is a sequence of rational numbers $$\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \frac{99}{70}, ..., \frac{p_n}{q_n}, \frac{p_{n+1}}{q_{n+1}}, \frac{2 p_{n+1} + p_n}{2 q_{n+1} + q_n}, ... $$ that converges to $$\sqrt{2}$$ even though $$\sqrt{2}$$ is not rational. Limits in analysis are related to limit points in geometry which are, as you say, about what points exist that have an unending set of neighbors in the series no matter how small you assume the neighborhood to be.

 

[rpenner]

Aside on $$\sqrt{2}$$.

Let
$$M = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{\pmatrix} = \begin{pmatrix} - \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & 2^{\tiny - \frac{3}{4}} \end{\pmatrix} \begin{pmatrix} 1 - \sqrt{2} & 0 \\ 0 & 1 + \sqrt{2} \end{\pmatrix} \begin{pmatrix} -2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} \end{\pmatrix}$$

Then since
$$\begin{pmatrix} -2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} \end{\pmatrix} \begin{pmatrix} - \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & 2^{\tiny - \frac{3}{4}} \end{\pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{\pmatrix}$$
it follows that
$$M^n = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{\pmatrix}^n = \begin{pmatrix} - \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & 2^{\tiny - \frac{3}{4}} \end{\pmatrix} \begin{pmatrix} \left(1 - \sqrt{2}\right)^n & 0 \\ 0 & \left(1 + \sqrt{2}\right)^n \end{\pmatrix} \begin{pmatrix} -2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{4 + 3 \sqrt{2}} \\ 2^{\tiny - \frac{3}{4}} & \frac{1}{2} \sqrt{-4 + 3 \sqrt{2}} \end{\pmatrix}
= \begin{pmatrix} \frac{ \left(1 + \sqrt{2}\right)^{n+1} - \left(1 - \sqrt{2}\right)^{n+1} }{ 2 \sqrt{2} } & \frac{ \left(1 + \sqrt{2}\right)^n - \left(1 - \sqrt{2}\right)^n }{ 2 \sqrt{2} } \\ \frac{ \left(1 + \sqrt{2}\right)^n - \left(1 - \sqrt{2}\right)^n }{ 2 \sqrt{2} } & \frac{ \left(1 + \sqrt{2}\right)^{n-1} - \left(1 - \sqrt{2}\right)^{n-1} }{ 2 \sqrt{2} } \end{\pmatrix} $$

And thus:
$$M^n \begin{pmatrix} p_{m+1} & q_{m+1} \\ p_{m} & q_{m} \end{\pmatrix} = \begin{pmatrix} p_{n+m+1} & q_{n+m+1} \\ p_{n+m} & q_{n+m} \end{\pmatrix}$$
Giving closed form sequences:
$$ p_n = \frac{ \left(1 + \sqrt{2}\right)^n + \left(1 - \sqrt{2}\right)^n }{ 2 }
q_n = \frac{ \left(1 + \sqrt{2}\right)^n - \left(1 - \sqrt{2}\right)^n }{ 2 \sqrt{2} }
\frac{p_n}{q_n} = \sqrt{2} \frac{ \left(1 + \sqrt{2}\right)^n + \left(1 - \sqrt{2}\right)^n }{ \left(1 + \sqrt{2}\right)^n - \left(1 - \sqrt{2}\right)^n } = \sqrt{2} \left(\frac{2}{1 - \left(2 \sqrt{2}-3\right)^n}-1 \right)$$

Given any positive number $$\epsilon$$ we can compute $$N = \left\lceil \frac{\ln \, \epsilon \quad - \quad \ln \, \left( \epsilon + \sqrt{8} \right) }{\ln \, \left( 3 - \sqrt{8} \right) }\right\rceil$$ and thus formally establish the proposition that $$\lim_{n\to\infty} \frac{p_n}{q_n} = \sqrt{2}$$.

Thus, for example $$\frac{p_{13}}{q_{13}} = \frac{47321}{33461}$$ is accurate to one part in a billion as predicted from the above formula.
Likewise $$\frac{p_{40}}{q_{40}} = \frac{1023286908188737}{723573111879672}$$ is accurate to better than one part in $$10^{30}$$, and successive rational terms just keep on getting closer.

 

[hansda]

Aha, but have you considered that it was not I that first stepped out into contra-mathematical philosophical speculation, as we both agree that because the positive numbers are closed under multiplication (i.e. positive times positive equals positive), then no member of the particular sequence $$(T_n)$$ could be zero.

So, $$T_n \ne 0$$ for any value of $$n$$; though $$\lim_{n \to \infty} {T_n} = 0$$.

 

[arfa brane]

The key here is understanding what "n approaches infinity" means. $$ T_n \ne 0 $$ is true because n is a finite number; $$ n \to \infty $$ does not mean $$ n = \infty $$.

It's about what's logically true, not about equality.

 

[Undefined]

The key here is understanding what "n approaches infinity" means. $$ T_n \ne 0 $$ is true because n is a finite number; $$ n \to \infty $$ does not mean $$ n = \infty $$.

It's about what's logically true, not about equality.

Hi arfa.

That is 'logically true' only within the construct and conventions of that construct which uses that argument within that context assuming it's true. Yes?


Here is something possibly relevant for your further cogitations here and in your P^M thread on fractional calculus.

The relationship of an increasing fractal dimension with space-filling might be taken to mean fractal dimensions measure density, but that is not so; the two are not strictly correlated. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity. These features are evident in the two examples of fractal curves. Both are curves with topological dimension of 1, so one might hope to be able to measure their length or slope, as with ordinary lines. But we cannot do either of these things, because fractal curves have complexity in the form of self-similarity and detail that ordinary lines lack. The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. The length between any two points on these curves is undefined because the curves are theoretical constructs that never stop repeating themselves. Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. These are not rectifiable curves, meaning they cannot be measured by being broken down into many segments approximating their respective lengths. They cannot be characterized by finding their lengths or slopes. However, their fractal dimensions can be determined, which shows that both fill space more than ordinary lines but less than surfaces, and allows them to be compared in this regard.

There are some cases where the notion of 'dimensionless points' is made moot because one cannot actually identify mathematically an actual 'point' at all within the maths treatment involved, as in above case.

Then there is the other example that makes the 'dimensionless point' notion a non-sequitur which is demonstrated by a certain topology/geometry example I have alluded to before but which forms part of my ToE examples/explanation points so I can't say any more on that at this time.

Good luck and enjoy your further explorations/discussions, arfa, everyone.

 

[rpenner]

You are not reading that correctly. It is saying the points are well-defined but the length between them isn't a simple number.

Take two endpoints on a line of length 1. By defining a self-similar fractal system where the middle third of every straight line is replaced with two sides of an equilateral triangle, we see that the approximate arc length measured with dividers of length $$3^{-n}$$ has $$4^n$$ segments and therefore this curve (part of a Koch snowflake) is assigned fractional dimension $$\frac{\ln 4}{\ln 3}$$ which is intermediate between that of a line and a plane. Indeed, a continuous mapping between the interval [0,1] and the x and y coordinates of points on a Koch snowflake is describable in the field of real analysis. (This is why the curve is simple to implement as a Iterated Function System (IFS) fractal. http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm

 

[Undefined]

Hi rpenner.

Thanks for your trouble in replying. Much appreciated.

You are not reading that correctly. It is saying the points are well-defined but the length between them isn't a simple number.

Take two endpoints on a line of length 1. By defining a self-similar fractal system where the middle third of every straight line is replaced with two sides of an equilateral triangle, we see that the approximate arc length measured with dividers of length $$3^{-n}$$ has $$4^n$$ segments and therefore this curve (part of a Koch snowflake) is assigned fractional dimension $$\frac{\ln 4}{\ln 3}$$ which is intermediate between that of a line and a plane. Indeed, a continuous mapping between the interval [0,1] and the x and y coordinates of points on a Koch snowflake is describable in the field of real analysis. (This is why the curve is simple to implement as a Iterated Function System (IFS) fractal. http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm

Yes, I know what 'aspect' you assume I have concentrated on, but that's not the same 'aspect' I actually considered further as per my further perspective.

I read more and further into it than you may have. I not only already understood what your reading says about the points 'definition', but I also read that since the length was not well-defined, then so were the number of points also effectively impossible to well define perforce of the length itself not being well defined. One affects the other logically and in reality, since although you may ASSUME an 'inifinity of dimensionless points' within the length terminated by the end-points involved, we cannot say what those points actually are until we define the length properly of the line. Just assuming an infinity of points does not actually 'well-define' them except in a 'collective sense' as 'SOME collection of infinite number of points on SOME length of undefined line'.

That's the further thrust of my reading, and not just as per current conventions/assumptions in current maths construct.

Thanks again for your trouble and info, rpenner. Cheers!

 

[hansda]

So, $$T_n \ne 0$$ for any value of $$n$$; though $$\lim_{n \to \infty} {T_n} = 0$$.

The key here is understanding what "n approaches infinity" means.

Infinity means not finite or no end. So, "n approaches infinity" means $$n$$ is not finite or $$n$$ has no end.

You can also see these links: 1, 2.

$$ T_n \ne 0 $$ is true because n is a finite number;

True.

Also, $$T_n \ne 0$$ as $$n \to \infty$$(meaning "n approaches infinity").

$$ n \to \infty $$ does not mean $$ n = \infty $$.

I think this link can clarify your doubts.

It's about what's logically true, not about equality.

It has to be mathematically true.

 

[rpenner]

I also read that since the length was not well-defined, then so were the number of points also effectively impossible to well define

Not at all -- the cardinality of the points of the Koch curve is identical to the cardinality of the real line segment [0,1] which is equal to the cardinality of $$\mathbb{R}$$.

I challenge you to cite and quote this supposed source that claims the number of points is impossible to well define -- for I believe that is another unreliable implication you have assumed because you don't know math.

Equinumerosity is an equivalence relation between sets where it follows that there exists a one-to-one onto mapping between one set and another, like pairing off dance partners in the 1950's as a proof that there are the same number of girls and boys in the class. Thus it follows that if you have a one-to-one mapping of set A into a set B (where the image of A under the mapping is a subset of B ) then the cardinality of B is greater than or equal to A.
Because I can establish a procedure which maps a set with the cardinality of the continuum [0,1] one-to-one into the Koch curve K it follows that $$ [0,1] \preceq K $$.
And because I know that the Koch curve is a subset of plane, it follows that $$ [0,1] \preceq K \preceq \mathbb{R}^2 $$.
But because the plane also has the cardinality of the continuum it follows that all of these sets are equinumerous $$ [0,1] \approx K \approx \mathbb{R}^2 $$.

 

[rpenner]

Aside on continuous mapping between [0,1] and the Koch curve.

Let

$$M_a = \begin{pmatrix} \frac{1 + \delta_{a,0} + \delta_{a,3}}{6} & \frac{\delta_{a,2} - \delta_{a,1}}{\sqrt{12}} & \frac{2 \delta_{a,1} + 3 \delta_{a,2} + 4 \delta_{a,3}}{6} \\ \frac{\delta_{a,1} - \delta_{a,2}}{\sqrt{12}} & \frac{1 + \delta_{a,0} + \delta_{a,3}}{6} & \frac{\delta_{a,2} }{\sqrt{12}} \\ 0 & 0 & 1 \end{pmatrix}$$​

where $$a \in \left{0, 1, 2, 3 \right}$$. Then $$\det \; M_a \; = \; \frac{1}{9}$$.

$$M_3^n = \begin{pmatrix} 3^{-n} & 0 & 1 - 3^{-n} \\ 0 & 3^{-n} & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ therefore $$\lim_{n\to\infty} M_3^n = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$. Similarly, $$\lim_{n\to\infty} M_0^n = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$. And since it follows that $$M_0 \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} = M_1 \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$, $$M_1 \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} = M_2 \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ and $$M_3 \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} = M_2 \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$, it follows that we have a continuous mapping from any number t in [0,1] and $$M_t = \prod_{k\geq 1} M__{a_k}$$ where $$a_k$$ is the kth digit in the base-4 expansion of the number. And from $$M_t$$ it is trivial to read off the x and y coordinates of the point on the Koch curve.

$$\begin{array}{c|cc} t & x & y \\ \hline
0 & 0 & 0
\frac{1}{15} & \frac{17}{146} & \frac{\sqrt{3}}{146}
\frac{2}{15} & \frac{27}{146} & \frac{7\sqrt{3}}{146}
\frac{1}{5} & \frac{1}{4} & 0
\frac{1}{4} & \frac{1}{3} & 0
\frac{4}{15} & \frac{51}{146} & \frac{3 \sqrt{3}}{146}
\frac{1}{3} & \frac{5}{14} & \frac{\sqrt{3}}{14}
\frac{3}{8} & \frac{3}{9} & \frac{\sqrt{3}}{9}
\frac{387}{1024} & \frac{83}{243} & \frac{27 \sqrt{3}}{243}
\frac{48}{127} & \frac{747}{2186} & \frac{243 \sqrt{3}}{2186}
\frac{1}{\sqrt{7}} & 0.341825594566041 & 0.192542562279971 & \textrm{(approximate)}
\frac{2}{5} & \frac{3}{8} & \frac{\sqrt{3}}{8}
\frac{7}{15} & \frac{65}{146} & \frac{21 \sqrt{3}}{146}
\frac{1}{2} & \frac{1}{2} & \frac{\sqrt{3}}{6}
\frac{8}{15} & \frac{81}{146} & \frac{21\sqrt{3}}{146}
\frac{3}{5} & \frac{5}{8} & \frac{\sqrt{3}}{8}
\frac{2}{3} & \frac{9}{14} & \frac{\sqrt{3}}{14}
\frac{11}{15} & \frac{95}{146} & \frac{3 \sqrt{3}}{146}
\frac{3}{4} & \frac{2}{3} & 0
\frac{4}{5} & \frac{3}{4} & 0
\frac{13}{15} & \frac{119}{146} & \frac{7 \sqrt{3}}{146}
\frac{14}{15} & \frac{129}{146} & \frac{\sqrt{3}}{146}
1 &1 & 0
\end{array}$$
Here analysis agrees with geometry as $$t \to 1 - t$$ results in mirroring the Koch curve about the line $$x = \frac{1}{2}$$