Deeper than primes - Continuation

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epix said:
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.
Click to expand...

Nope. You cannot substitute something that's approximately the same and then still expect to get the same result. 1 - 1/10n may approximate 0.999..., but they are not identical.

So the following is all illegitimate:
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.
Click to expand...
 
jsfisher said:
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.
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I claim that your axiomatic system is a flaw version of the considered reality, and so are the proofs that are derived from them, exactly because your axiomatic system deals only with collections, and therefore can't deal with the power of non-composed dimensional spaces > 0, that are inaccessible to any infinite amount of lower dimensional spaces and/or sub-spaces (where a sub-space is a mixture of lower lower spaces with the considered space).

jsfisher said:
This is your latest nonsensical back-fill? How long before you have to stop digging this whole?
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You are invited to do your 9*x test on it.

jsfisher said:
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?
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First demonstrate a number system with base-1 that has a fractal structure.

If you can't do that, then please do not ask irrelevant questions.

By the way, do you know that this is a nice example of base-5 fractal

fractal-fingers.jpg

( http://omnisapien.files.wordpress.com/2008/01/fractal-fingers.jpg )

which its line segments (illustrated as fingers) are irreducible into points, upon infinitely many levels of invariant self-similarity (where points (your limits) are actually a collapse of that self-similarity)?

jsfisher said:
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.
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You do not understand H-trees and their staircases, and how they equivalent to a single path along some base-n (where n > 1) fractal, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=9331947&postcount=2589.
 
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doronshadmi said:
I claim that your axiomatic system is a flaw version of the considered reality, and so are the proofs that are derived from them, exactly because your axiomatic system deals only with collections, and therefore can't deal with the power of non-composed dimensional spaces > 0, that are inaccessible to any infinite amount of lower dimensional spaces and/or sub-spaces (where a sub-space is a mixture of lower lower spaces with the considered space).
Click to expand...

Great!! So the flawed axiomatic system of which you write must lead to an inconsistency. Please, demonstrate that inconsistency. Without that, you don't get the option of adding your own special axioms.

...
First demonstrate a number system with base-1 that has a fractal structure.
Click to expand...

That really was the point, now wasn't it? Next time you hear a rushing sound like that, look up. Be that as it may, I'll take this as an admission your current nonsensical invention cannot be applied it to binary number systems.

...
You do not understand H-trees and their staircases, and how they equivalent to a single path along some base-n (where n > 1) fractal, as explained in http://www.internationalskeptics.com/forums/showpost.php?p=9331947&postcount=2589.
Click to expand...

The thing that makes your misunderstanding of and fascination with fractals so interesting is that they contradict much of the nonsense you have spewed over the past, what is it now? 25+ years?

Well done!! You'd be nothing if not inconsistent.
 
jsfisher said:
Great!! So the flawed axiomatic system of which you write must lead to an inconsistency.
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Great!! So your reading abilities are now equivalent to your spatial abilities.

Maybe if you read my previous post again some of your reading abilities will be improved.

Also I see that you avoiding your 9*x argument on

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.

So please try to use it, or any other way in order to show the inconsistency in the above expressions.

Maybe I have another chance to learn something from you.
 
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doronshadmi said:
Great!! So your reading abilities are now equivalent to your spatial abilities.

Maybe if you read my previous post again some of your reading abilities will be improved.
Click to expand...

Did read it a couple of times. So, what is it you think I missed? You don't mention any self-inconsistencies the axiomatic framework has; you simply reasserted your own non-composed gibberish.

If the axioms are inconsistent, then there must be a statement and its negation that can both be proven within the system. Got anything like that, or will you just be continuing to fire insults without basis?
 
doronshadmi said:
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.
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Why do you include ABS(x) in your sequence? Is it due to yet another difference between OM and the standard math?

For a(real) > 0 and b(base) ≥ 2: ab > ab+1

is how the relation works in standard math. You can use the next to last member of your sequence ABS(0.89 - 0.810) to verify:

0.89 = (8/9)10
0.810 = 8/10

8/9 - 8/10 = (80 - 72)/90 = 8/90 = 4/45

and SIGN(4/35) = 1.

The result is positive and holds through your sequence - no need for absolute values - because the rules of the standard math that relate to ratio comparison say that when the numerators are identical, the denominators decide the result of the comparison. If a < b, then 1/a > 1/b.

Now reveal how it works in OM. And don't forget about your inequality

"2/3 > 0.666... by 0.000...4"

and the way OM converts a ratio into a decimal with a repeating period without including weird sequences that include unexplained number conversions. It must be really different from the standard math, which uses the long division with the result 2/3 = 0.666...
http://www.online-calculator.com/
 
jsfisher said:
Sure.


You sure about that? Looks like it converges on 1, not e.
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CRAP! Lol. I never noticed that the outer exponent part is missing 10. Sorry.

So once again, hopefully with no further embarrassing omissions.

For n→∞: (1 - 1/10n) = 0.999...

If 0.999... = 1, then

lim. n→∞ (1 - 1/10n)10n = 1/e

cannot be true. But it is.
 
epix said:
CRAP! Lol. I never noticed that the outer exponent part is missing 10. Sorry.

So once again, hopefully with no further embarrassing omissions.

For n→∞: (1 - 1/10n) = 0.999...

If 0.999... = 1, then

lim. n→∞ (1 - 1/10n)10n = 1/e

cannot be true. But it is.
Click to expand...

I am not seeing what you are seeing here. Were you taking the limit inside the parentheses to replace 1 - 1/10n with .999.... That outer exponent prevents it.

That is to say, lim. n→∞ (1 - 1/10n)10n and lim. n→∞ (lim. n→∞ (1 - 1/10n))10n are not equivalent.
 
jsfisher said:
I am not seeing what you are seeing here. Were you taking the limit inside the parentheses to replace 1 - 1/10n with .999.... That outer exponent prevents it.

That is to say, lim. n→∞ (1 - 1/10n)10n and lim. n→∞ (lim. n→∞ (1 - 1/10n))10n are not equivalent.
Click to expand...
You are probably referring to the priorities in evaluation with respect to the limits, but I don't follow the logic of it. I never saw an argument with nested limits.

This expression

for n →∞: 1 - 1/10n = 0.999...

establishes an equivalence in such a way that the right side is a string of nines whose number goes to infinity and 0.999... doesn't represent any limit - it is not equal to 1. If such a number exist, then you can plug it into a formula or an algorithm to see what 0.999... resolves into. For example, if you want to show that 0.999... is different from 1 and employ the long division for that purpose, then the test fails, because

1/9 = 0.111...
0.999... /9 = 0.111...

and that makes 0.999... and 1 equivalent. However, if you take 0.999... and raise it to powers you obtain a result inconsistent with 0.999... = 1

It wasn't Euler who noticed first that

0.910 = 0.348678...
0.99100 = 0.366032...
0.9991000 = 0.366695...

converges toward 1/2.718.... This convergence wouldn't be possible if 0.999... = 1.
 
epix said:
...
However, if you take 0.999... and raise it to powers you obtain a result inconsistent with 0.999... = 1
Click to expand...

No, you don't. 0.999... behaves exactly like 1

It wasn't Euler who noticed first that

0.910 = 0.348678...
0.99100 = 0.366032...
0.9991000 = 0.366695...

converges toward 1/2.718.... This convergence wouldn't be possible if 0.999... = 1.
Click to expand...

The sequence you are noting is not 0.999... raised to any particular power, but something else. You are raising 0.999...9 (n repetitions) to the 1000...0 (n repetitions) power. The exponentiation amplifies the difference between 0.999...9 and 1 faster than 0.999...9 advances towards 1, so, yeah, if converges at 1/e instead of 1.
 
epix said:
It wasn't Euler who noticed first that

0.910 = 0.348678...
0.99100 = 0.366032...
0.9991000 = 0.366695...

converges toward 1/2.718.... This convergence wouldn't be possible if 0.999... = 1.
Click to expand...

You're not working with infinity properly.

0.9...[googolplex of 9s]...9^10...[equally many zeros]...0 = 1/e

.9...[infinitely many nines, no ending 9]...^10...[infinitely many zeros, no ending zero] = 1^∞ = 1

To claim that there is an ending 9 in .9... or that there could be a 0.0...1 is to claim that infinity isn't actually infinite.
 
ehcks said:
You're not working with infinity properly.

0.9...[googolplex of 9s]...9^10...[equally many zeros]...0 = 1/e

.9...[infinitely many nines, no ending 9]...^10...[infinitely many zeros, no ending zero] = 1^∞ = 1

To claim that there is an ending 9 in .9... or that there could be a 0.0...1 is to claim that infinity isn't actually infinite.
Click to expand...
I never made such a claim by even a slight implication. That what you've replied to are the first three members of an infinite sequence, otherwise I wouldn't mention the word convergence.
 
jsfisher said:
If the axioms are inconsistent, then there must be a statement and its negation that can both be proven within the system. Got anything like that, or will you just be continuing to fire insults without basis?
Click to expand...
Let us put it this way:

Objects with mass > 0 can't reach the speed of light, they only approaches to it.

If we use a mathematical abstraction of what is written above, then the length of a point (or a limit point) is equivalent to an object with mass = 0, and the length of a line segment (that can't have 0 length) is equivalent to an object with mass > 0.

EDIT:In other words, line segments can't reach the state of a given limit point, they only approach it, and as a result there is always an irreducible line segment, no matter what magnitude of zoom is used on a given mathematical system.

One of the results of the irreducibility of a line segment into a point upon infinitely many zoom-in levels, is the irreducibility of 1 into 0 by the following expression: 1 - 0.910 - 0.0910 - 0.0910 - ... = 1 - 0.999...10

Since the standard axiomatic system ignores the irreducibility of a line segment into a point, it can be used in order to conclude that 1 and 0.999...10 are actually the same mathematical object. You will not find any inconsistency about the claim that 1 and 0.999...10 are actually the same mathematical object, within an axiomatic system that ignores the irreducibility of a line segment into a point.

But it does not not mean that this axiomatic system is right about the considered subject, it simply means that such an axiomatic system is a degenerate framework with respect to axiomatic systems that actually deal with the irreducibility of a line segment into a point (where one possible non-degenerate axiomatic system is Organic Mathematics, which according to it, the irreducibility of a line segment into a point (or more generally, the irreducibility of a non-composed dimensional space > 0 into 0 dimensional space) is an axiom.

According to this axiomatic reasoning, actual infinity is not less than non-composed dimensional space > 0, where potential infinity is at most a collection of lower dimensional spaces or sub-spaces (where a sub-space is a mixture of the given space with lower spaces).

The degenerate axiomatic reasoning (the standard one) gets actual infinity only at the level of collections, and as a result an expression like 1 > 0.999...10 can't be deduced by it.

According to a non-degenerate axiomatic framework an expression like 1 > 0.999...10 by 0.000...(1 - .910) simply asserts that the potential infinity of xxx... expressions (which are at the level of collections) can't reach the actual infinity of the (1-.910) non-composed space (where this non-composed space is actually the self-similarity upon infinitely many zoom-in levels (of base 10 fractal, in this particular case)).

ehcks said:
To claim that there is an ending 9 in .9... or that there could be a 0.0...1 is to claim that infinity isn't actually infinite.
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By the standard framework, which is based only on collections, 0.0...1 is impossible.

By the non-degenerate axiomatic reasoning 0.0...1 is simply the inaccessibility of the potential infinity at the level of collections, to the non-composed higher dimensional space of actual infinity.
 
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doronshadmi said:
...irrelevance snipped...

In other words, line segments can't reach the limit point, they only approach it, and as a result there is always an irreducible line segment, no matter what magnitude of zoom is used on a given mathematical system.
Click to expand...

Someday you may finally have a grasp of limits, and quite possibly infinity, too. Today isn't that day. Were you to understand limits at all, you'd know they do not include "reach the limit point" as any sort of requirement. The exact opposite is true, in fact. But please continue. Post all the ignorance you will; 0.999... will still be exactly the same number as 1.
 
jsfisher said:
Someday you may finally have a grasp of limits, and quite possibly infinity, too.
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Some day you may grasp the non-finally of xxx... potential infinity, and quite possibly the non-composed mathematical space > 0 if actual infinity, too (and maybe also fractals).
 
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doronshadmi said:
By the degenerate axiomatic reasoning, this is indeed what we get.
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Your dislike for Mathematics has long been established. That doesn't make it degenerate in any way, though, just not liked by Doron.

doronshadmi said:
Some day you may grasp the non-finally of xxx... potential infinity, and quite possibly the non-composed mathematical space > 0 if actual infinity, too (and maybe also fractals).
Click to expand...

...nor does it grant you leave to redefine things at will as a substitute for understanding.

You can construct something entirely new of you like, but that doesn't change that which you dislike but do not understand, nor has it worked out very well for you in your past attempts to define, well, anything.
 
jsfisher said:
Your dislike for Mathematics has long been established. That doesn't make it degenerate in any way, though, just not liked by Doron.
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Your dislike of infinitesimals do make them flawed, and so is the case about fractals with invariant self-similarly upon infinitely many scales, which are at most potential-infinity w.r.t the actual infinity of non-composed dimensional spaces > 0 (Potential infinity and actual infinity can't be deduced by the degenerate axiomatic reasoning, because it is based only on collections (that are at most potential infinity by the non-degenerate axiomatic reasoning)).

jsfisher said:
You can construct something entirely new of you like, but that doesn't change that which you dislike but do not understand, nor has it worked out very well for you in your past attempts to define, well, anything.
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You can stay in the standard scope of degenerate axiomatic reasoning, but that doesn't change that which you dislike (infinitesimals) but do not understand (the potential infinity of the amount of numbers, which are found along base n(n>1) fractals), it did not work out very well for you in your past attempts to get anything beyond the standard degenerate axiomatic reasoning.

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

Ok, let's continue to develop the number system of base n(n>1) fractals (which are known as numerals by the degenerate axiomatic reasoning that can't distinguish between potential infinity (at the level of collections) and actual infinity (at the level of non-composed dimensional spaces > 0)).

For example, the case of number 0.999...10 w.r.t number 1, is understood as follows:

1 - 0.910 - 0.0910 - 0.00910 - ... ,such that this long subtraction is non-terminate (there is no number = 0 in that long subtraction, exactly because the actual infinity of the non-composed 1-dimesional space is inaccessible to the potential infinity of the amount of the numbers in the collection of long subtraction (it has to be stressed that if number 0 is included in this long subtraction, it must be the last number of it and as a result this long subtraction is finitely long).
 
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Some correction of the previous post, the first part has to start by:

"Your dislike of infinitesimals do not make them flawed,..."

instead of the wrong one:

"Your dislike of infinitesimals do make them flawed,..."
 
Let's demonstrate the rigorous abstract ability of verbal_symbolic AND visual_spatial reasoning.

In order to do that we are using a binary fractal tree that is unbounded below:

Code: 
  |\
  | \
  |  \
  |   \
  |    \
  |     \
  |      \
  |       \
  |        \
  |         \
  |          \
  |           \
  |            \
  |             \
  |              \
  |               \
  |                \
  |                 \
  |                  \
  |                   \
  |                    \
  |                     \
  |                      \
  |                       \ 
  |                        \ 
  |                         \
  |                          \                                      
  |                           \                                        
  |                            \                                     
  |                             \                                    
  |                              \                                   
  0                               1                                  
  |\                              |\                                 
  | \                             | \                                
  |  \                            |  \                               
  |   \                           |   \                              
  |    \                          |    \                             
  |     \                         |     \                            
  |      \                        |      \                           
  |       \                       |       \                          
  |        \                      |        \                         
  |         \                     |         \                        
  |          \                    |          \                       
  |           \                   |           \                      
  |            \                  |            \                     
  |             \                 |             \                    
  |              \                |              \                   
  0               1               0               1                  
  |\              |\              |\              |\                 
  | \             | \             | \             | \                
  |  \            |  \            |  \            |  \               
  |   \           |   \           |   \           |   \              
  |    \          |    \          |    \          |    \             
  |     \         |     \         |     \         |     \            
  |      \        |      \        |      \        |      \           
  0       1       0       1       0       1       0       1          
  |\      |\      |\      |\      |\      |\      |\      |\         
  | \     | \     | \     | \     | \     | \     | \     | \        
  |  \    |  \    |  \    |  \    |  \    |  \    |  \    |  \       
  0   1   0   1   0   1   0   1   0   1   0   1   0   1   0   1      
  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\     
  0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1    

                                ...

Let 000... be the non-composed 0 space level of that tree.

Let any 100... be a non-composed space (which is not 000...) of that tree.

Let any 111... be some composed space of that tree.

By translating this tree into a number system, given any 100... it is > its immediate left 111... simply because this number system is based on a binary fractal tree that is unbounded below (which means that no branch (composed or non-composed) of that tree is accessible to any other branch of that tree, downward.

Without loss of generality, this conclusion is true for all number systems that are based on unbounded below fractal trees > 2.

These conclusions are not in the scope of the standard degenerate axiomatic reasoning.

B.t.w, given some level of a non-composed space > 0 on a given branch of some unbounded below fractal tree, this level is inaccessible to the composed spaces of this branch.
 
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jsfisher said:
I'm not sure why you write "approximate" though.
epix Quote:
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...
??
Click to expand...
The exact and approximate format is used with CAS and other math software that displays results other than it used to be when you could see only decimal representation. By definition, any result that includes a radix point is a result in the approximate format and any expression that is free of radix point is an expression in the exact format. So when you enter for example

1/2 + 1/3

the result is 5/6. However, it takes only a single inclusion of the radix point after one of the above integers, such as

1./2 + 1/3

and the expression is considered being in the approximate format, even though "1." is an integer that doesn't invoke the adjective approximate at the slightest. The result is then displayed in the approximate format, that means all terms in the result are numerical arrays that include the radix point.

1./2 + 1/3 =0.8333333333333333

Wolfram Alpha is directly tied with Matematica, so you can verify my explanation.

ow3n.png


Note that the result 0.8333333333333333 in the approximate format doesn't include ellipses. The result is obtained by the long division, which is a loop that doesn't terminate unless you set a condition under which the looping stops. You can set up the degree of precision and make your judgement regarding a possible periodicity. It would be nice if the script would include a period recognition lines, but that's a very cumbersome process.

When developing CAS, the math folks ran into similar issues that bug Doron, but unlike Doron, they resolved the format conversion in a usable manner. Of course, the developers are very much familiar with the language that standard math speaks, which is provably not the case with Doron. I would even dare to claim that Doron ignores the word limit in

limit n→∞ Sn = a

and considers only the

n→∞ Sn = a

part. Then he invokes different number bases, fractals, spacial visuals - local or non-local - to help him show that Sn ≠ a. His statement that 0.666...(approximate) is not a rational number proves that he doesn't know how to convert periodic decimals into p/q (exact). That's a strategic deficiency that sends Organic Mathematics well back in time.
 
epix said:
When developing CAS, the math folks ran into similar issues that bug Doron, but unlike Doron, they resolved the format conversion in a usable manner. Of course, the developers are very much familiar with the language that standard math speaks, which is provably not the case with Doron. I would even dare to claim that Doron ignores the word limit in

limit n→∞ Sn = a

and considers only the

n→∞ Sn = a

part. Then he invokes different number bases, fractals, spacial visuals - local or non-local - to help him show that Sn ≠ a. His statement that 0.666...(approximate) is not a rational number proves that he doesn't know how to convert periodic decimals into p/q (exact). That's a strategic deficiency that sends Organic Mathematics well back in time.
Click to expand...
Nice try epix, but numbers of the form .###... (where # is a placeholder for some digit in some base > 1) are not members of R or Q sets.

By the degenerate axiomatic reasoning .###... is not a number, but it is a numeral of some member of R or Q sets.

So, nothing goes back in time because I do not ignore limits, what I say is that a number of the form .###... is inaccessible to any given Q or R member, which is considered as its limit.

In other words, it is always the case that ABS(Q member > 0) OR ABS(R member > 0) > number of the form .###..., if ABS(Q member > 0) OR ABS(R member > 0) are considered as its limit.

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

As long as you try to deduce Organic Mathematics in terms of the degenerate axiomatic reasoning that excludes infinitesimals, you are not in the scope of Organic Mathematics, and therefore can't conclude anything about it like, for example:
epix said:
That's a strategic deficiency that sends Organic Mathematics well back in time.
Click to expand...
You and jsfisher are using a framework that does not distinguish between potential infinity (which is the level of collections) and actual infinity (which is the level of a given non-composed space > 0).
 
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Some explanation of the following part of post http://www.internationalskeptics.com/forums/showpost.php?p=9341057&postcount=2622 :

doronshadmi said:
B.t.w, given some level of a non-composed space > 0 on a given branch of some unbounded below fractal tree, this level is inaccessible to the composed spaces of this branch.
Click to expand...

By translating a given unbounded below fractal tree with base > 1, a floating point distinguishes between the whole level and the fractional levels. According to this term, no fractional level is accessible into the whole level.
 
doronshadmi said:
Nice try epix, but numbers of the form .###... (where # is a placeholder for some digit in some base > 1) are not members of R or Q sets.
Click to expand...

Really? Define Q at least.

doronshadmi said:
By the degenerate axiomatic reasoning .###... is not a number, but it is a numeral of some member of R or Q sets.
Click to expand...

Gibberish.

doronshadmi said:
So, nothing goes back in time because I do not ignore limits, what I say is that a number of the form .###... is inaccessible to any given Q or R member, which is considered as its limit.
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Inane gibberish.

doronshadmi said:
In other words, it is always the case that ABS(Q member > 0) OR ABS(R member > 0) > number of the form .###..., if ABS(Q member > 0) OR ABS(R member > 0) are considered as its limit.
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More inane gibberish. ABS(Q member > 0)? :eye-poppi

doronshadmi said:
As long as you try to deduce Organic Mathematics in terms of the degenerate axiomatic reasoning that excludes infinitesimals,<snip>
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LOL. You can't even understand a simple concept like 1/3=0.333... , forget about infinitesimals.

doronshadmi said:
<cont'd> you are not in the scope of Organic Mathematics, and therefore can't conclude anything about it like, for example:

You and jsfisher are using a framework that does not distinguish between potential infinity (which is the level of collections) and actual infinity (which is the level of a given non-composed space > 0).
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Well, one more post full of proof that you don't understand the concepts you think you're improving on. Oh, and gibberish. Lots of it. I guess that counts for something.
 
QUOTE=doronshadmi;9341057]Let's demonstrate the rigorous abstract ability of verbal_symbolic AND visual_spatial reasoning.

[...]



Let 000... be the non-composed 0 space level of that tree.



Let any 100... be a non-composed space (which is not 000...) of that tree.



Let any 111... be some composed space of that tree.



By translating this tree into a number system, given any 100... it is > its immediate left 111... simply because this number system is based on a binary fractal tree that is unbounded below (which means that no branch (composed or non-composed) of that tree is accessible to any other branch of that tree, downward.



Without loss of generality, this conclusion is true for all number systems that are based on unbounded below fractal trees > 2.



These conclusions are not in the scope of the standard degenerate axiomatic reasoning.[/QUOTE]

What you have shown is that a string of digits ending in a 1 followed by infinitely many zeroes is a different string of digits from a string otherwise the same but ending in a 0 followed by an infinite string of 1's. Believe it or not, axiomatic systems can prove this fact as well.



Now we just need to determine what numbers the strings of symbols represent and show whether they are different.
 
laca said:
Define Q. I bet you don't even have the slightest idea how to go about doing it.
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Q is the set of rational numbers (the ratio p/q among two integers, where q is not equal to 0).
 
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BenjaminTR said:
What you have shown is that a string of digits ending in a 1 followed by infinitely many zeroes is a different string of digits from a string otherwise the same but ending in a 0 followed by an infinite string of 1's. Believe it or not, axiomatic systems can prove this fact as well.
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Exactly the opposite, 100... starting with 1 and followed by infinitely many zeroes and it represents a non-composed dimensional space, which its level is inaccessible to its immediate left composed 111...


BenjaminTR said:
Now we just need to determine what numbers the strings of symbols represent and show whether they are different.
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100... and its immediate left composed 111... are not the same numbers according to their different levels in the same tree ( the unbounded below fractal tree can be seen in http://www.internationalskeptics.com/forums/showpost.php?p=9341057&postcount=2622 ).

For example, let's use base-2.

As can be seen in the following visual_spatial AND verbal_symbolic expression, the composed number 0.111...2 and the non-composed number 1.000... are in different levels of the considered mathematical space:

Code: 
0               0               1               1                   
0               [B]1[/B]               0               1      Integers     
[B].[/B]---------------[B].[/B]---------------.---------------.-----------------  
                                                       Fractions    
|\              [B]|[/B]\              |\              |\                  
| \             [B]|[/B] \             | \             | \                 
|  \            [B]|[/B]  \            |  \            |  \                
|   \           [B]|[/B]   \           |   \           |   \               
|    \          [B]|[/B]    \          |    \          |    \          ... 
|     \         [B]|[/B]     \         |     \         |     \             
|      \        [B]|[/B]      \        |      \        |      \            
0       [B]1[/B]       [B]0[/B]       1       0       1       0       1           
|\      |[B]\[/B]      [B]|[/B]\      |\      |\      |\      |\      |\          
| \     | [B]\[/B]     [B]|[/B] \     | \     | \     | \     | \     | \         
|  \    |  [B]\[/B]    [B]|[/B]  \    |  \    |  \    |  \    |  \    |  \        
0   1   0   [B]1[/B]   [B]0[/B]   1   0   1   0   1   0   1   0   1   0   1       
|\  |\  |\  |[B]\[/B]  [B]|[/B]\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\  |\      
0 1 0 1 0 1 0 [B]1[/B] [B]0[/B] 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1     
                                                                    
                              ...

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=9338187&postcount=2620 and http://www.internationalskeptics.com/forums/showpost.php?p=9338744&postcount=2621.
 
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doronshadmi said:
Nice try epix, but numbers of the form .###... (where # is a placeholder for some digit in some base > 1) are not members of R or Q sets.
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I believe that there is a reason for you to claim so. First, let's consider a finite case and select number 0.372 as a guinea pig for the purpose of showing how things can get treacherous when someone wanders off the path right into the wooded area of Organic Mathematics (OM). Standard Math (SM) says that 0.372 is a rational number. The adjective is derived from the word ratio, which describes this symbol: p/q. SM establishes this equivalency:

A) 0.372 = p/q

where p and q are non-zero integers. So, what are the p and q?

There is no way of finding out, because (A) is one equation with two different variables. You can employ all the rules that govern over equations solving, but to no avail.

See, Doron? One considers just a finite case and runs straight into a legitimate obstacle. What is the problem here apart from that one described above?

The problem is that you are attempting to solve an equation instead of making a progress toward converting 0.372 into p/q . The process of conversion is similar to equation solving, but it includes steps that are not a part of equation solving. Let me demonstrate...

First, you dismiss this part of (A): "= p/q".

That move leaves you with 0.372. Since you know that p and q are supposed to be integers and 0.372 is not, you start to scheme and ask this question: Can I convert 0.372 into an integer?

Of course you can. There are actually many ways to do so. You can add 0.628 to 0.372 and voila! 0.372 + 0.628 = 1 with 1 being a nice integer. But if you go other way than multiplication, you hit the dead end. Trust me. See, if you use multiplication as a tool and multiply 0.372 by 103, where the exponent is equal to the number of digits in the fractional part of 0.372, then you obtain a wonderful integer.

0.372 * 103 = 372

Since 372 is an integer and p is supposed to be integer as well, you can set 372 = p. But you need q as well on the right side of the identity. You get it by catching two flies in one swipe. It would be more than advisable to get rid of that "*103" helper after it did its job. But how? Well, you divide both sides of the identity by 103.

Left: 0.372 * 103/103 = 0.372
Right: 372/103 = 93/250

Written in one line, the result looks like this.

372 = 93/250

where 93 and 250 are integers organized in ratio p/q.

So remember that conversion of numbers displayed in different formats is not identical to equation solving, even though SM says 372 = p/q, which is an equation.

Now when we learn the finite case, we apply our knowledge in the infinite case. Get this guy: 0.666...

So we repeat the steps taken in the finite case. We convert 0.666... into an integer by multiplying the number by 10n where n is the number of repeating sixes in the fractional part of 0.666...

And here is the problem that OM cannot solve: n→∞, and since infinity is not a point on R - it is not a number - there is no way to multiply and convert 0.666... into an integer. It follows that 0.666... cannot be converted to its equivalent p/q and that means that 0.666... and all 0.###... are not rational numbers, exactly as OM claims. That claim rests on a firm logical ground:

Finite is to Can as Infinite is to Cannot.

But there is something that may undermine all the rigor:boggled: that OM paddles on. You may come accidentally across a particular long division case involving 2 in the numerator p, and 3 in the denominator q:

[p/q = 2/3] => [LD(2, 3) = 0.666...]

Is it possible that if a conversion from 2/3 into 0.666... exists via an algorithm, then there is an algorithm capable of converting 0.666... into 2/3 ?

Since OM claims that all 0.###... are not in Q, you have not found such an algorithm. Maybe you didn't look with powerful enough spatial_visual binoculars.

I know the argument by heart:
If a/b = c, then a = b*c
and since there is no last digit in 0.666... the check by multiplication cannot be done. But that doesn't prove that 2/3 = 0.666... is wrong - infinity merely prevents to perform the check.

Your claim is a proof-free statement. It lacks the necessary intro: Suppose that 0.###... is in Q. If it is so ... and a contradiction would follow.
 
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epix said:
Your claim is a proof-free statement. It lacks the necessary intro: Suppose that 0.###... is in Q. If it is so ... and a contradiction would follow.
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My claim is rigorously rooted and farmed in the ground of visual_spatial AND verbal_symbolic reasoning, as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=9342098&postcount=2632.

So any "Suppose that ..." holds only it is done by using visual_spatial AND verbal_symbolic reasoning.

epix said:
But that doesn't prove that 2/3 = 0.666... is wrong - infinity merely prevents to perform the check.
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According to visual_spatial AND verbal_symbolic reasoning the inequality among 2/3 and 0.666...10 is an axiomatic state, no proof is needed.

The "Suppose that ..." and "But that doesn't prove that 2/3 = 0.666... is wrong" are typical signatures of the inability of verbal_symbolic-only reasoning to deal with the considered subject.
 
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Some correction of the previous post:

Instead of:

So any "Suppose that ..." holds only it is done by using visual_spatial AND verbal_symbolic reasoning.

it has to be:

So any "Suppose that ..." is derived from verbal_symbolic-only reasoning.
 
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doronshadmi said:
My claim is rigorously rooted and farmed in the ground of visual_spatial AND verbal_symbolic reasoning, as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=9342098&postcount=2632.
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If the inequality is in "axiomatic state," then it means that Organic Mathematics doesn't use the long division to convert p/q into its decimal equivalent in special cases involving p<q. If one of your previous statement said that

2/3 - 0.666... = 0.000....4

does that mean that the identity lies in axiomatic state and Organic Mathematics has infinitely many axioms to cover stuff going inside Q? (I know, 0.666... is not in Q nor R according to OM.)

Can you define your new term composed space? See "organic" relates to "living" - we have organic chemistry that studies compounds that living matter is made of. Since the definition of the adjective says

Composed: Having one's feelings and expression under control; calm.

there is a comprehensive link between Organic Mathematics and composed space. But there is also the noun "composition."

1) not composed
2) composed

But Mozart is no longer with us - he is no longer living. And that's throws a few pounds of confusion into the barrel of understanding.
 
doronshadmi said:
Some correction of the previous post:

Instead of:

So any "Suppose that ..." holds only it is done by using visual_spatial AND verbal_symbolic reasoning.

it has to be:

So any "Suppose that ..." is derived from verbal_symbolic-only reasoning.
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An incorrect way to use the proof by contradiction:

Let A be infinite series

A = 6/101 + 6/102 + 6/103 + 6/104 + . . .

It is said that sum S of A converges toward a point on the real line called the limit of A. The property of the limit L is such that S<L. In particular, the limit of A is said to be L = 2/3

Now SUPPOSE that S=L. Since S = 0.666... in the decimal (radix) format and 2/3 = 0.666..., the stated property of L is false.

But if for some reason the property of L still holds true, it follows that

2/3 > 0.666...

(Btw, which case is which, Doron?)
 
epix said:
2/3 - 0.666... = 0.000....4
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Wrong.

1 - 3/9 = 6/9 > 0.666...10 by 0.000...ABS(0.69-0.610)

epix said:
Can you define your new term composed space?
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Organic Mathematics is based on the different levels of non-composed spaces, and their associations among them, which are composed.

By using a common mathematical universe (like an unbounded below fractal) for composed and non-composed forms, it is discovered that no composed from of the structure 0.xxx..., is accessible to the level of its immediate right non-composed form of the structure x.000...

Furthermore, given a non-composed structure x000... on some unbounded below fractal's branch of base > 2, any composed xxx.. on the same fractal's branch of base > 2, is inaccessible to the level of the non-composed x000... on that branch.
 
epix said:
An incorrect way to use the proof by contradiction:

Let A be infinite series

A = 6/101 + 6/102 + 6/103 + 6/104 + . . .

It is said that sum S of A converges toward a point on the real line called the limit of A. The property of the limit L is such that S<L. In particular, the limit of A is said to be L = 2/3

Now SUPPOSE that S=L. Since S = 0.666... in the decimal (radix) format and 2/3 = 0.666..., the stated property of L is false.

But if for some reason the property of L still holds true, it follows that

2/3 > 0.666...

(Btw, which case is which, Doron?)
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Our discussion on this subject is done on different levels.

According to verbal_symbolic AND visual_spatial reasoning 2/3 > 0.666...10 is derived form the axiom that a non-composed number's level is inaccessible to its immediate composed number, according to a given base > 2.

So, no proof (including a proof by contradiction) is used in this case.

As long as you are using verbal_symbolic-only reasoning you force 2/3 and 0.666...10 to be expressions on the same levels, which leads to a framework that has to exclude 0.666...10 expressions as legitimate numbers, in order to avoid contradiction.

The solution of the verbal_symbolic-only degenerate axiomatic reasoning is to define an expression like 0.666...10 as a numeral and not as a number.

In this case 2/3 = 0.666...10 is the only option (in base 10 case), and so is the case in 1 = 0.999...10
 
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doronshadmi said:
Wrong.

1 - 3/9 = 6/9 > 0.666...10 by 0.000...ABS(0.69-0.610)
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You included the ABS(x) function once again even though it is not needed. But let's compare notes. According to mine

0.69 - 0.610 = 0.666...10 - 0.610 = 0.0666...10: SIGN(0.066...10) = +1.

So you are saying that

1 - 3/9 = 6/9 = 2/3 > 0.666...10 by 0.000...666...10.
Is this your handwriting?
doronshadmi said:
As for multiplication of non-local numbers, for example 0.333...10*2 = 0.666...10, where 0.666...10 < 2/3 by 0.000...410, as clearly given in http://www.internationalskeptics.com/forums/showpost.php?p=9313916&postcount=2454.
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When you make up your mind, let me know.


Organic Mathematics is based on the different levels of non-composed spaces, and their associations among them, which are composed.

By using a common mathematical universe (like an unbounded below fractal) for composed and non-composed forms, it is discovered that no composed from of the structure 0.xxx..., is accessible to the level of its immediate right non-composed form of the structure x.000...

Furthermore, given a non-composed structure x000... on some unbounded below fractal's branch of base > 2, any composed xxx.. on the same fractal's branch of base > 2, is inaccessible to the level of the non-composed x000... on that branch.
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You either misunderstood my question or can't do better than that. You mentioned the term "composed form." I googled up the term to get some clue of what that might resemble, but the only answer that ties "form" with "composed" is once again of musical nature.
Form:
In music theory about musical form, the term through-composed means that the music is relatively continuous, non-sectional, and/or non-repetitive. A song is said to be through-composed if it has different music for each stanza of the lyrics. This is in contrast to strophic form, in which each stanza is set to the same music. Sometimes the German durchkomponiert is used to indicate the same concept.
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http://en.wikipedia.org/wiki/Through-composed#Form

I do believe that there is some comprehensive link between the music and your terminology, because the adjective in "Organic Mathematics" relates to the word organ, which in turn is strongly linked with music - the nice music in particular:


That particular organ playing is comprehensively linked with the concept of fractals, which play (no pun intended) a leading role in your composed or non-composed spaces, because the word "fractal" is etymologically married to "fracture" or to "break apart."

I think that Organic Mathematic terminology is too demanding as far as an association formation is concerned - it calls for too much visual_spacial work. Can you come up with something less complicated?
 
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