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 , contradiction .