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Topic: Matheology � 258
Replies: 104   Last Post: May 5, 2013 2:26 PM

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 dan.ms.chaos@gmail.com Posts: 409 Registered: 3/1/08
Re: Matheology § 258
Posted: May 3, 2013 2:45 AM

On May 2, 11:38 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 2 Mai, 19:51, Dan <dan.ms.ch...@gmail.com> wrote:
>

> Didn't you understand yet the argument? Every power of 10 with natural
> exponent n can be reflected: 10^n reflected gives 10^-n.

The natural number corresponding to 1/9 doesn't exist .
That doesn't show 1/9 doesn't exist , it only dosen't exist on the
list, a natural consequence of it being the anti diagonal of the
list.

> For the infinite sequence 0.111... this is not possible. Therefore it
> does not exist as sequence of only natural powers of 10. (But
> supernatural powers are not element of mathematics.)

111 = 10^0 + 10^1 + 10^2 + 10^3

0.1111 .... < 1
There are no 'supernatural powers of 10 in the expansion of 1/9 .
However ,there are NEGATIVE powers .

While positive powers of 10 (1 , 10 , 100 .... ) can correspond to
discrete magnitudes , and the expression
10^0 + 10^1 + 10^2 + ..... = 1 +10 + 100 + .... is a DIVERGENT
expression .
the negative powers of 10 (1/10 , 1/100 , ) correspond ONLY to
CONTINUOUS magnitudes , and the expression
10^(-1) + 10^(-2) + ...... = 0.1 + 0.01 + ..... = 1/9 is a CONVERGENT
expression .
We know that because it also has OTHER NAMES besides its digit
expansion that allow us to make sense of it (it's the ninth part of a
unit) .

"Natura non saltum facit"

You can divide a [0,1) segment in half: [0,1/2) + [1/2,1 ) pick one
of the halves ,divide it in half , and so on .
To each of your divisions, you can associate a natural number :

to your first division and choice 1 : You've picked a segment of
length 1/(2^1)
to your second division and choice 2 : You've picked a segment of
length 1/(2^2)
to your third division and choice 3 : You've picked a segment of
length 1/(2^3)

After you've divided and choose a finite number of times , let's say
456 times , you still have a segment on your hands (a very small
segment, but a segment nonetheless . An aggregate , and not an atom .
Only after you've made a division and choice for every natural
number : 1 , 2 , 3 .......
would you be able to get to the true 'atom of the continuum' , the
point . The point is indivisible .
This sums up the behavior of continuous magnitudes .

On the other hand, let's say you have 8 apples .
You can divide them into 2 groups of 4 apples , (1'th
division) ,then choose one of the groups and
divide it into 2 groups of 2 apples , (2'nd division) ,then
choose one of the groups and
divide it into 2 groups of 1 apples , (3'nd division) ,then
choose one of the groups and
Now you've reached 'the apple' , the 'atom' of your discrete
quantity . And you've only used the first 3 natural numbers .
This sums up the behavior of discrete magnitudes .

Its the difference between 'arbitrarily large, but finite divisibility
(uses all natural numbers up to a specified n)' , and
'infinite divisibility (uses all natural numbers , nothing more ,
nothing less)' .

You can't give the discrete the property of 'infinite divisibility
' (11111..... 11111 is not a number)
and you can't hope to explain the continuum by 'arbitrarily large, but
finite divisibility' :

(1-10^k) / 9 = 0.1111.... 1 (k'th ones) it's a valid continuous
magnitude , for any natural number k .
1/9 = 0.1111 ..... (infinity ones) it's ALSO a valid continuous
magnitude .

Your argument is based on the fact that your notions of infinity are
confused . You restrict the continuum to "discrete notions of infinity
" (potential infinity , arbitrarily large but finite) , and you try to
apply "actual infinity" to natural numbers , where it doesn't belong
(111.... 1 is not a number ) . You reverse the domains of
applicability , and then pride yourself that you get a contradiction .
Fish don't fly and birds don't live underwater . Therefore ,

Date Subject Author
4/29/13 Virgil
4/29/13 mueckenh@rz.fh-augsburg.de
4/29/13 Virgil
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 dan.ms.chaos@gmail.com
4/30/13 mueckenh@rz.fh-augsburg.de
4/30/13 Virgil
5/1/13 dan.ms.chaos@gmail.com
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5/1/13 dan.ms.chaos@gmail.com
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5/1/13 Virgil
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5/1/13 mueckenh@rz.fh-augsburg.de
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