Deeper than primes - Continuation

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jsfisher said:
Again you show you do not know what a fractal is, what inverse proportionality is, nor did you address that actual point of the post you quoted.

However, you do continue to show that your notational nonsense behaves exactly like, indistinguishably from zero. And you have never, ever shown any different.

Care to try now, just this one?
Click to expand...
Well, I made a mistake about the inverse proportionality stuff in http://www.internationalskeptics.com/forums/showpost.php?p=9327103&postcount=2549, exactly because I used only verbal_symbolic reasoning.

By using also visual_spatial reasoning I have found that the discussed list of numbers is the result of overlap of two different bases, and I showed the relations among them in http://www.internationalskeptics.com/forums/showpost.php?p=9328458&postcount=2559.

Since you are using only verbal_symbolic reasoning, all I show is no more than nonsense to you.

As for fractals, you are invited to show why the place-value method (where base value > 1) does not have a structure of a fractal, (for example, the following base 2 and base 2 structures):

4318895416_366312cf0e_o.jpg


You are also invited to show why there are no infinitely many levels of self similarity in some given fractals (whether this self similarity is trivial, or not).

The general consensus is that theoretical fractals are infinitely self-similar,...
Click to expand...
(http://en.wikipedia.org/wiki/Fractal )
 
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doronshadmi said:
Well, I made a mistake about the inverse proportionality stuff in http://www.internationalskeptics.com/forums/showpost.php?p=9327103&postcount=2549, exactly because I used only verbal_symbolic reasoning.
Click to expand...

There are many reasons you made a mistake. That, however, was not one of them.

By using also visual_spatial reasoning I have found that the discussed list of numbers is the result of overlap of two different bases, and I showed the relations among them in http://www.internationalskeptics.com/forums/showpost.php?p=9328458&postcount=2559.
Click to expand...

They are not numbers, and concluding something from their visual appear to you is irrelevant. This is a difficult concept for you, I realize, but please try to understand that something like 0.356823 is a notational convenience to represent a number. 0.999... is not a number either, just a representation for one. An infinite summation, in this case, which just happens to be equivalent to one.

If you care to trying disproving that, work with the actual summation, not your clumsy attempts to show your ignorance of summations, limits, factals, infinity, and numbers.

Since you are using only verbal_symbolic reasoning, all I show is no more than nonsense to you.
Click to expand...

You assume, presume, and conclude incorrectly.

As for fractals, you are invited to show why the place-value method (where base value > 1) does not have a structure of a fractal
Click to expand...

You made the claim. You provide the proof. Start with nowhere differentiable.


And you are invited to demonstrate that your notational gibberish, 0.000...1, behaves in any manner distinguishable from zero.
 
This pause in the action is because Doronshadmi is busily combing through Wikipedia and exercising his google foo hoping to find something he can contort for his next post.
 
jsfisher said:
So, you don't understand the term, differentiable, either. That's going to be something of an impediment to you showing your stair squiggles to be fractals.
Click to expand...

Let's see:

Plot of Weierstrass Function over the interval [−2, 2]. Like fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot.
Click to expand...

http://en.wikipedia.org/wiki/File:WeierstrassFunction.svg

( http://en.wikipedia.org/wiki/Weierstrass_function )

This function is nowhere differentiable and every zoom is similar to the global from.

The general consensus is that theoretical fractals are infinitely self-similar,...
Click to expand...
(http://en.wikipedia.org/wiki/Fractal )

Now, I'll ask you again, why the following forms are not fractals?

4318895416_366312cf0e_o.jpg
 
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doronshadmi said:
Let's see:

http://en.wikipedia.org/wiki/File:WeierstrassFunction.svg

This function is nowhere differentiable and every zoom is similar to the global from.
Click to expand...

No argument there. The Weierstrass function has the required sense of self-similarity and is nowhere differentiable.

Now, I'll ask you again, why the following forms are not fractals?
Click to expand...

And I say again, it is your claim, so the proof, is yours to provide. Heck, even a simple hand-waving exhibition of the right sort of properties would be a good start. Start with nowhere differentiable.
 
jsfisher said:
This pause in the action is because Doronshadmi is busily combing through Wikipedia and exercising his google foo hoping to find something he can contort for his next post.
Click to expand...

By the way, the very first hit for "nowhere differentiable" is the Wikipedia page for the Weierstrass function.

May we count this as my MDC preliminary test?
 
doronshadmi said:
Well, I made a mistake about the inverse proportionality stuff in http://www.internationalskeptics.com/forums/showpost.php?p=9327103&postcount=2549, exactly because I used only verbal_symbolic reasoning.
Click to expand...
See? You must use visual_spatial reasoning all the time.

Doron, do you agree or disagree with

0.999... 10 > 0.111... 2

and why?


And now I will teach you something about a proof by contradiction, which is supported by the pillars of consistency - a MUST thing in math.
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent
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As you know, general formulas, which replaced descriptive language very unkind to proof, came to use quite late. But it did, courtesy of
http://www.math.rutgers.edu/courses/436/Honors02/vieta.html

So instead of using foxes or wolves as variables in the same problem, Viete unified the choices into a general, single variable represented by one letter, such as x.

Now let me demonstrate the consistency of algebraic assignment via the proof by contradiction.

a = 0.666...
x = a
10x = 10a
10x - x = 10a - a
9x = 9a
x = 9a/9 = a = 0.666...

The result is consistent with the premise.

Since 0.666... is a repeating 6, and 6 upside down is 9, number 9 is therefore a spatial_visual opposite to 6. But since "opposite" and "contradiction" are siblings, it follows that the consistency must break down in case of a = 0.999... . In other words, the result must become inconsistent in this case.

a = 0.999...
x = a
10x = 10a
10x - x = 10a - a
9x = 9a
x = 9a/9 = a = 0.999...

As you see, there is no sign of inconsistency, which fully justifies Viete's decision. But that would seriously hurt the effectiveness of the proof by contradiction! 6 upside down = 9 is a visual_spatial fact!

Well, we standardized mathematicians, surely have ways to deal with such a problem. Consider for example letter C. That letter is the third in the (English) alphabet and so you make non-arbitrary assignment C = 3. Then you solve equation with 3 parameters, out of which 1 is known - it's the C.

xyC = 3

The conditional solution is as follows: If C stands for Cheating, then

xyC = ZFC

There are surely plenty of options to Chose from. One of them is 0.999... = 1

Don't you consider converting to Standardized Mathematics? Things get less rigorous.

No?

Well, that's your Choice.
 
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jsfisher said:
No argument there. The Weierstrass function has the required sense of self-similarity and is nowhere differentiable.



And I say again, it is your claim, so the proof, is yours to provide. Heck, even a simple hand-waving exhibition of the right sort of properties would be a good start. Start with nowhere differentiable.
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It is actually your claim that the forms that I provided are not fractals ( http://www.internationalskeptics.com/forums/showpost.php?p=9328765&postcount=2565 ), so please support your claim that they are not fractals according to your knowledge about them (no matter what I claim about them).
 
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doronshadmi said:
It is actually your claim that the forms that I provided are not fractals ( http://www.internationalskeptics.com/forums/showpost.php?p=9328765&postcount=2565 ), so please support your claim that they are not fractals according to your knowledge about them (no matter what I claim about them).
Click to expand...

Very telling bit of logic, there Doron. You called them fractals, and I said prove it, but somehow it becomes my claim.

The point stands: You declared them to be fractals. You need to provide some evidence of that. I have suggested nowhere differentiability as a useful start, but apparently your google skills have let you down in understanding what that actually means let alone how to show it for you so-called fractals.

Perhaps you could have a go at self-similarity. Zoom in somewhere other than the very end and show us how that's similar to the whole.
 
BenjaminTR said:
Do I have this right? If we accept your version of mathematics we have to give up the axiom that multiplication distributes over addition? Denying basic arithmetic is not going to gain you many supporters. I recommend you rethink this part.
Click to expand...

That's nothing compared to when he denied basic logic...
 
jsfisher said:
Perhaps you could have a go at self-similarity. Zoom in somewhere other than the very end and show us how that's similar to the whole.
Click to expand...
Do you really missing the fact that in any given scale the place value method is self-similar, such that the whole and the part are actually the same form upon infinitely many scale levels, as found in the following forms?

4318895416_366312cf0e_o.jpg


Zoom in and out in any wished scale and you easily find that the whole and the part remain invariant with respect to each other, no matter if you zoom in or out.

Base 2 is a binary fractal, base 3 is ternary fractal, ... etc.

So jsfisher, whether you like it or not, the place-value method has a fractal structure.
 
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doronshadmi said:
Do you really missing the fact that in any given scale the place value method is self-similar, such that the whole and the part are actually the same form upon infinitely many scale levels, as found in the following forms?
Click to expand...


See the red circle. Zoom in there. It looks nothing like the whole.
 

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And whereas it may be interesting watching Doron bash into more Mathematics he doesn't understand, fractals now being the latest example, we are still left with the same underlying nonsense.

Doron claims 1 and 0.999... have a non-zero difference (0.999... being the lesser of the two numbers.) Until just recently, Doron would use the notational gibberish of 0.000...1 as representing the value of the difference.

Typical of everything doronetics, all you need is a really good notation to make any need for proof superfluous.

Doron backed himself into a corner with his 0.000...1 when he made an on the fly expansion of his nonsense. Rather than having just the one value, 0.000...1, that he could never show different from zero nor resolve its self-inconsistency, Doron created for himself a gigantic pack of contradiction even he couldn't ignore.

So what did he do?

He did what Doron has done so many times in the past. He has simply abandoned the old notational gibberish, left in shambles, and shifted sideways to a whole new set of invented gibberish and hand waving. Never advance (that would require actually proving something) and never retreat; just rearrange the deck chairs.

Call it what you will, Doron, you still need to show 0.999... and 1 behave differently in some way. You haven't; you won't; you can't.
 
jsfisher said:
See the red circle. Zoom in there. It looks nothing like the whole.
Click to expand...
This is a single path along a base 2 fractal, but also in this case self-similarity upon any scale level is found, for example, please define the difference between your marked form, by mark it in any scale level.
 
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jsfisher said:
And whereas it may be interesting watching Doron bash into more Mathematics he doesn't understand, fractals now being the latest example, we are still left with the same underlying nonsense.

Doron claims 1 and 0.999... have a non-zero difference (0.999... being the lesser of the two numbers.) Until just recently, Doron would use the notational gibberish of 0.000...1 as representing the value of the difference.

Typical of everything doronetics, all you need is a really good notation to make any need for proof superfluous.

Doron backed himself into a corner with his 0.000...1 when he made an on the fly expansion of his nonsense. Rather than having just the one value, 0.000...1, that he could never show different from zero nor resolve its self-inconsistency, Doron created for himself a gigantic pack of contradiction even he couldn't ignore.

So what did he do?

He did what Doron has done so many times in the past. He has simply abandoned the old notational gibberish, left in shambles, and shifted sideways to a whole new set of invented gibberish and hand waving. Never advance (that would require actually proving something) and never retreat; just rearrange the deck chairs.

Call it what you will, Doron, you still need to show 0.999... and 1 behave differently in some way. You haven't; you won't; you can't.
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Jsfisher do not change the subject, you have failed to show that the place-value method is not a fractal, even by your attempt to use a single path of it.

So please try again.

Maybe it will help you to develop your visual-spatial seasoning.
 
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doronshadmi said:
This is a single path along a base 2 fractal, but also in this case self-similarity upon any scale level is found, for example, please define the difference between your marked form, by mark it in any scale level.
Click to expand...

Hahahahaha. You really are lost here, aren't you. Your overall "fractal" has a general staircase appearance with the tread getting narrower and narrower as you move right. You realize this, right? I don't want to hit you with any surprises.

The part within the red circle is basically flat (with part of one step, but even that disappears as you zoom in further). A flat line vs. a staircase. Only in doronetics could those two be considered similar.
 
doronshadmi said:
Jsfisher do not change the subject
Click to expand...

If you read what I wrote carefully, you may eventually note I was commenting on you changing the subject.

You've got that projection thing working pretty well.

you have failed to show that the place-value method is not a fractal, even by your attempt to use a single path of it.
Click to expand...

To anyone who understands the term, nowhere differentiable, it is already obvious your gibberish does not a fractal make. To anyone who understands the term, self-similar, it is already obvious your gibberish does not a fractal make.

So please try again.
Click to expand...

Not my claim, so it was never my responsibility, even though I've done it twice already.

Maybe it will help you to develop your visual-spatial seasoning.
Click to expand...

Can I get it in cajun? I always like cajun.
 
jsfisher said:
You really are lost here, aren't you. Your overall "fractal" has a general staircase appearance with the tread getting narrower and narrower as you move right. You realize this, right?
Click to expand...
Take, for example, this Fractal:

fractal.png


( http://fishofthebay.com/wp-content/themes/fishofthebay/images/fractal.png )

The whole idea of being self-similar upon infinitely many scales is possible exactly because a given form is at most narrower (by using your expression) w.r.t to a given limit.

In order to understand it, you can choose some arbitrary single path upon its infinitely many scale levels (as you did in the case of my form).

This single path does not look like the self-similar pattern of this fractal, yet it is clear that exactly as the self-similar pattern is not vanished into a point upon infinitely many scale levels (otherwise the reasoning of being self-similar upon infinitely many scale levels does not hold) so is the case with the line segments of the arbitrary chosen single path. It always have line segments upon infinitely many levels, which are irreducible into a limit point.

Things do not change even if you take the arbitrary chosen single path and give it a shape of a straight line.



jsfisher said:
To anyone who understands the term, nowhere differentiable, it is already obvious your gibberish does not a fractal make. To anyone who understands the term, self-similar, it is already obvious your gibberish does not a fractal make.
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Jsfisher, you are unable to understand the essence of being self-similar upon infinitely many scale levels, and this essence is simply the irreducibility of, at least, 1-dimensional space into 0-dimensional space (there is no homomorphism among them).

You can use the power of infinite zoom, but it does not change the fact that no 1-dimensional space is reducible into 0-dimensional space.

Keep trying to develop your visual-spatial reasoning. Maybe some day you will correctly use it. But this day is not today.
 
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doronshadmi said:
Take, for example, this Fractal:
Click to expand...

Interesting how your argument shifts. First you didn't understand what self-similarity even meant, and now you are about to redefine it in order to bend reality to cover your blunders.

The generally accepted characteristics required of a fractal are (1) self-similarity, (2) nowhere differentiability, and (3) having a fractal dimension (which is a measure of change in complexity with scale).

Your diminishing staircase meets exactly none of those characteristics.

The term, fractal, has become widely abused, probably due to fractals' inroads into popular culture via their influence in art and especially computer animation. Many things get called fractals even though there are not.

For example, this is not: http://fishofthebay.com/wp-content/themes/fishofthebay/images/fractal.png

Be that as it may, it is still wildly oblique from the real topic of discussion. That would be the topic from which you so desperately try to distance yourself:
Whether 0.999... = 1​

You have boldly claimed the two differ by a non-zero amount denoted by the symbolic absurdity, 0.000...1. That is, 1 = 0.999... + 0.000...1. Although you may have abandoned the 0.000...1 notation in favor of other nonsense, the underlying point remains as to whether 0.999... and 1 represent the same number.

Now, Doron, if you think your new-found source of mathematical wisdom, FishoftheBay.com, has something significant and relevant to add, have at it. Meanwhile, we still await you providing any demonstration whatsoever that 1 and 0.999... can be distinguished mathematically.
 
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jsfisher said:
Interesting how your argument shifts. First you didn't understand what self-similarity even meant, and now you are about to redefine it in order to bend reality to cover your blunders.

The generally accepted characteristics required of a fractal are (1) self-similarity, (2) nowhere differentiability, and (3) having a fractal dimension (which is a measure of change in complexity with scale).

Your diminishing staircase meets exactly none of those characteristics.

The term, fractal, has become widely abused, probably due to fractals' inroads into popular culture via their influence in art and especially computer animation. Many things get called fractals even though there are not.

For example, this is not: http://fishofthebay.com/wp-content/themes/fishofthebay/images/fractal.png
Click to expand...

Well, it is known as H-fractal (as can be found in http://en.wikipedia.org/wiki/H-fractal or http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension ) and it is arbitrarily close to Hausdorff dimension 2.

So we can ignore your "understanding" of fractals, which does not able to capture that the place-value method is actually a fractal.

Since you do not understand what fractals are, you are also unable to capture http://www.internationalskeptics.com/forums/showpost.php?p=9330591&postcount=2586.

Probably you will claim that since (according to Standard Mathematics) it has Hausdorff dimension 2, it means that a collection of lower dimensional spaces can completely cover a higher dimensional space.

But then we return to stage one, where I axiomatically claim (by using verbal_symbolic AND visual_spatial reasoning) that collections of lower dimensional spaces are not extensible into a non-composed higher dimensional space, and a non-composed higher dimensional space is irreducible into lower dimensional spaces.

One of the results of this axiomatic approach is that 1 > 0.999... , but know I carefully have to check how to consistently express this inequality also by verbal_symbolic reasoning.

-----------------

Before I continue, I which to clarify some things.

During my work I make mistakes, where my last mistake was that I did not carefully checked the following expressions, which are wrong

1 - 9/9 = 0/9 = 0
1 - 8/9 = 1/9 > 0.1111 ... by 0.000...910
1 - 7/9 = 2/9 > 0.2222 ... by 0.000...810
1 - 6/9 = 3/9 > 0.3333 ... by 0.000...710
1 - 5/9 = 4/9 > 0.4444 ... by 0.000...610
1 - 4/9 = 5/9 > 0.5555 ... by 0.000...510
1 - 3/9 = 6/9 > 0.6666 ... by 0.000...410
1 - 2/9 = 7/9 > 0.7777 ... by 0.000...310
1 - 1/9 = 8/9 > 0.8888 ... by 0.000...210
1 - 0/9 = 9/9 > 0.9999 ... by 0.000...110

as shown by jsfisher, by using the expression 9*(1/9) = 9*(0.111..10+0.000...910), which leads to inconstancy (x can't be both 0.000...110 AND 0.000...8110).

I stupidly tried to defend my argument by using nonsense like inverse proportionality among two given expressions.

But then I have noticed that I made a fundamental mistake, which is: to use only verbal_symbolic reasoning, in order to deal with the expressions above.

By using also visual_spatial reasoning ( as can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=9328458&postcount=2559 ) I have found that the expressions above are actually relations among different bases, so at this stage I am working to define the right way to express these relations consistently also by verbal_symbolic reasoning.

I hope that I have learned my lesson, so from now on I am going to check my expressions by using not less than verbal_symbolic AND visual_spatial reasoning, which is, after all, my fundamental claim about any mathematical development.
 
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doronshadmi said:
Before I continue, I which to clarify some things.

During my work I make mistakes, where my last mistake was that I did not carefully checked the following expressions, which are wrong

...

I stupidly tried to defend my argument by using nonsense like inverse proportionality among two given expressions.
Click to expand...

You make this sound like just a one-off event.

You are transparent, Doron. You have a fixed conclusion borne out of ignorance. You support that fixed conclusion with more ignorance and invention. Then you defend your ignorance and invention with even more ignorance and invention until something is so badly impossible that even you cannot ignore it.

Then you try to slough it off as a casual misstatement and that you really meant something else. Lather, rinse, repeat.

...
By using also visual_spatial reasoning ( as can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=9328458&postcount=2559 ) I have found that the expressions above are actually relations among different bases, so at this stage I am working to define the right way to express these relations consistently also by verbal_symbolic reasoning.
Click to expand...

Just getting ready for the next lather.

How about just skipping all the nonsense and attack the problem head-on? Show 0.999... isn't the same as 1. Work from the fundamentals; you know, the thing 0.999... actually means.
 
jsfisher said:
How about just skipping all the nonsense and attack the problem head-on? Show 0.999... isn't the same as 1. Work from the fundamentals; you know, the thing 0.999... actually means.
Click to expand...
What about your fundamental misunderstanding about fractals?

Edit:When you develop also your visual_spatial reasoning, you will also understand the 0.999... thing, which is a single path along a base 10 fractal, with infinitely many scale levels, that are irreducible into 0-dimesional space (your illusionary limit point, ho sorry it is not a point but it is a degenerate circle ( http://www.internationalskeptics.com/forums/showpost.php?p=8688626&postcount=2044 ), isn't it jsfisher?).
 
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doronshadmi said:
What about your fundamental misunderstanding about fractals?
Click to expand...

I stand by my statements, but they aren't really relevant to the topic of this thread.

When you develop also your verbal_symbolic reasoning, you will also understand the 0.999... thing.
Click to expand...

More proof by name-calling? It doesn't work that way, and no matter what you'd like to assume of others, that doesn't create impediments to their reasoning abilities.

Come on, just a little demonstration that 0.999... and 1 behave differently in some way.
 
jsfisher said:
I stand by my statements, but they aren't really relevant to the topic of this thread.
Click to expand...
Yes, H-fractal is not a fractal but it is a frog or what ever.

Your "stand by my statements" do not hold water, in this case, which prevents from you to understand the entropic-free realm, which is very very relevant to the topic of this thread.


jsfisher said:
Come on, just a little demonstration that 0.999... and 1 behave differently in some way.
Click to expand...
Come on, just a tiny step into visual_spatial reasoning is needed, in order to realize that 0.999... and 1 behave differently in some way.
 
doronshadmi said:
Yes, H-fractal is not a fractal but it is a frog or what ever.

Your "stand by my statements" do not hold water, in this case, which prevents from you to understand the entropic-free realm, which is very very relevant to the topic of this thread.
Click to expand...

My statements were appropriately qualified. They stand correct, and I stand by them.

Come on, just a tiny step into visual_spatial reasoning is needed, in order to realize that 0.999... and 1 behave differently in some way.
Click to expand...

As laca noted, you need to prove that.

Why is this so hard? You are deeply convinced 1 and 0.999... are different numbers, yet, you cannot defend that view in any meaningful way. Instead, all we get is, "You don't get it."

The trouble is, we do get it, and you are again guilty of projection. And you are utterly wrong.
 
jsfisher said:
My statements were appropriately qualified. They stand correct, and I stand by them.



As laca noted, you need to prove that.

Why is this so hard? You are deeply convinced 1 and 0.999... are different numbers, yet, you cannot defend that view in any meaningful way. Instead, all we get is, "You don't get it."

The trouble is, we do get it, and you are again guilty of projection. And you are utterly wrong.
Click to expand...
You ask for "defence" that is based on symbolic reasoning, by rejecting the axiom that a given non-composed dimensional space > 0 can't be reached by any collection of lower spaces or sub-spaces (where a sub-space is a mixture of the given space and lower spaces).

All you have do do is to use also your visual_spatial reasoning in order to define the complement infinitesimal > 0, which is based on the ratio among one or more bases.

I did not check it carefully yet, so I still can make mistakes by translating it into symbolic reasoning but anyway let us try to illustrate the outlines of it as follows, by using again the base 10 and base 9 expressions:

EDIT:

1 - 9/9 = 0/9 = 0
1 - 8/9 = 1/9 > 0.111...10 by 0.000...ABS(0.19-0.110)
1 - 7/9 = 2/9 > 0.222...10 by 0.000...ABS(0.29-0.210)
1 - 6/9 = 3/9 > 0.333...10 by 0.000...ABS(0.39-0.310)
1 - 5/9 = 4/9 > 0.444...10 by 0.000...ABS(0.49-0.410)
1 - 4/9 = 5/9 > 0.555...10 by 0.000...ABS(0.59-0.510)
1 - 3/9 = 6/9 > 0.666...10 by 0.000...ABS(0.69-0.610)
1 - 2/9 = 7/9 > 0.777...10 by 0.000...ABS(0.79-0.710)
1 - 1/9 = 8/9 > 0.888...10 by 0.000...ABS(0.89-0.810)
1 - 0/9 = 9/9 > 0.999...10 by 0.000...ABS(1-0.910)

where ABS(x) is an absolute function.

You are invited to check it if you wish.

I tried to write it according to what can be seen in the following diagram:

Available in this post.

Anyway it is still under construction.


Edited by Loss Leader: 
Reposted diagram removed as per recent mod box. Link added. Do not spam this thread with the same images.
 
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doronshadmi said:
You ask for "defence" that is based on symbolic reasoning
Click to expand...

I asked for no such thing. I asked for proof. Mathematical proof that 1 and 0.999... are different numbers. Gibberish of the several sorts you produce aren't proof.

...by rejecting the axiom that a given non-composed dimensional space > 0 can't be reached by any collection of lower spaces or sub-spaces (where a sub-space is a mixture of the given space and lower spaces).
Click to expand...

You'd like that to be an axiom? So, you want to prove your assertion by changing reality to match. No, that's not how you prove things, try though you may.

All you have do do is to use your visual_spatial reasoning in order to define the complement infinitesimal > 0 which is based on the ratio among, at least, two bases.
Click to expand...

This is your latest nonsensical back-fill? How long before you have to stop digging this whole?

I did not check it carefully yet, so I still can make mistakes by translating it into symbolic reasoning but anyway let us try to illustrate the outlines of it as follows, by using again the base 10 and base 9 expressions....
Click to expand...

Of course you did not check it. Like everything else, you simple assert without any thought to much of anything. How about showing us a more interesting example. Show how this would work for 0.111... in base-2. You'd use base-2 and base-1 expressions for this, right?


By the way, if it is of any consolation, I was relying on some of Mandelbrot's original requirements for what constituted a fractal. In his terminology, fractals could be divided into parts such that each part was similar to the whole. That's not the same as self-similar, which is what I kept calling it, since self-similar only requires one part be similar to the whole. So, my claim about fractals needs correction in line with all this. That doesn't help your staircase at all, but it makes things more lenient for things like H-trees.
 
jsfisher said:
Doron claims 1 and 0.999... have a non-zero difference (0.999... being the lesser of the two numbers.) Until just recently, Doron would use the notational gibberish of 0.000...1 as representing the value of the difference.
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Do you have any other suggestion how to convert 1/10n, where n→∞ and n in N1, into the approximate format?

Similar situation exists when exact 1 - 1/10n, with n→∞ and n in N1, is being converted to 0.999... approximate - without a reference to gibberish by Wikipedia writers.

The difference between the approximate and the exact result is that the former includes the radix point, whereas the latter doesn't. So the difference between both formats is free of any precision consideration, as the meaning of both adjectives misleadingly implies in this case.

Doron makes his claim without stipulating

For (...)→∞: 1 - 0.999... ≠ 0

The condition inclusion wouldn't create a problem, because the case of no condition given defaults to 1 - 0.999... = 0. Written in full and always in the exact format:

lim. n→∞ 1 - 10-n = 0

How would the limiting condition looked like in the approximate format 1 - 0.999...?

I'm not sure. Maybe

For lim. (...)→∞: 1 - 0.999... = 0
 
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epix said:
Do you have any other suggestion how to convert 1/10n, where n→∞ and n in N1, into the approximate format?
Click to expand...

I'm not sure where you were going with this. Are you trying to lend meaning to Doron's 0.000...1 nonsense? With the minor addition of a notation to make it clear there are n-1 zeroes between the radix point and the 1, then sure, that would represent 1/10n.

Of course, Doron never meant a finite number of 0's. For him, there were infinitely many, then the 1.

Similar situation exists when exact 1 - 1/10n, with n→∞ and n in N1, is being converted to 0.999... approximate - without a reference to gibberish by Wikipedia writers.
Click to expand...

The limit of 1 - 1/10n as n→∞ readily translates to the limit sum 9/10n as n→∞, no? And that readily becomes 0.999....

I'm not sure why you write "approximate" though.

The difference between the approximate and the exact result is that the former includes the radix point, whereas the latter doesn't. So the difference between both formats is free of any precision consideration, as the meaning of both adjectives misleadingly implies in this case.
Click to expand...

??

Doron makes his claim without stipulating

For (...)→∞: 1 - 0.999... ≠ 0

The condition inclusion wouldn't create a problem, because the case of no condition given defaults to 1 - 0.999... = 0. Written in full and always in the exact format:

lim. n→∞ 1 - 10-n = 0

How would the limiting condition looked like in the approximate format 1 - 0.999...?

I'm not sure. Maybe

For lim. (...)→∞: 1 - 0.999... = 0
Click to expand...

The 0.999... already has the limit built into it. It sounds like you may be using the notation 0.999..., here, for the case where there are a finite number of 9's after the radix point. Is that correct? If so, you'd need more notation to indicate that.
 
jsfisher said:
The 0.999... already has the limit built into it. It sounds like you may be using the notation 0.999..., here, for the case where there are a finite number of 9's after the radix point. Is that correct?
Click to expand...
No. That's not what I meant. I'm just making a case for 0.999... ≠ 1 and for the term 0.000...1 being the result of the subtraction.

You are basically saying that 0.999... = 1 applies under any circumstance and that means you can plug either side of the identity into an equation with the same result. But this is not so. Consider

S(1) = lim. n→∞ (1 + 1/n)n = e

When you let 1/n = 1/10n, the sequence becomes

S(2) = lim. n→∞ (1 + 1/10n)n = e

with the difference of S(2) converging much faster toward e.

Now change the sign in S(2). That affects the limit this way.

S(3) = lim. n→∞ (1 - 1/10n)n = 1/e

The expression 1 - 1/10n is nothing but 0.999... in the approximate format. So if 0.999... = 1, you can make the substitution.

S(4) = lim. n→∞ 1n = 1/e

Nope. That's not correct, because

S(4) = lim. n→∞ 1n = 1

That only documents that 0.999... = 1 cannot be globally true and that 0.999... ≠ 1 does exist. Actually the inequality became a very important instrument in the discovery and introduction of e into math as a widely used base for logarithms and as an important constant in the solution of differential equations. Here is the difference once again.

lim. n→∞ 0.999... n = 1/2.718...

whereas

lim. n→∞ 1n = 1

whatever that latter may mean.
 
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