On 3/4/2020 12:08 PM, Ganzhinterseher wrote:
> Am Mittwoch, 4. März 2020 17:27:49 UTC+1
> schrieb Peter Percival:
>> 0.999... = 1 isn't a definition at all. It states what the
>> sum of a certain geometric series is. The series being
>> 9/10 + 9/100 + 9/1000 +...
>
> There is no sum but a limit.
An infinite sum is not a sum in the binary-operator sense of
"sum". Any finite number of binary additions ends with a final
sum. Infinitely many additions do not end, because infinite.
The argument for the least-upper-bound sense of sum (that is,
of _infinite_ sum) is that _it CAN'T BE any number OTHER THAN_
_the LUB_
For example, the distance between the set of numbers
for which one of these is true
<3, <3.1, <3.14, <3.141, <3.1415, <3.14159, <3.141592, ...
and the set of numbers for which one of these is true
>4, >3.2, >3.15, >3.142, >3.1416, >3.14160, >3.141593, ...
is smaller than any positive number.
It's essential to this argument that one understands the
difference between
"forall d > 0, exists k e N, d > 1/10^k"
and
"exists k e N, forall d > 0, d > 1/10^k"
The first is true, the second is false.
The first is how we justify saying that, because for any two
distinct numbers x and y, |x - y| > 0, x and y cannot both
be between all of the increasing series 3, 3.1, 4.14, ... and
all of the decreasing series 4, 3.2, 3.15, ...
There cannot be two numbers between those series.
So, there could be one, or there could be none.
If there is one number the value of the infinite sum
could be, we define it to be that one number.
(Yes, we need to define it. Our other definition of "sum"
does not apply here.)
----
That leaves the question of whether there is one or no number
in the gap between the two series. One way to address this is
to treat the _one-point gap itself_ as a point. Conceptually,
we're taking a photographic negative of the number line.
Instead of referring to the point _pi_ , we refer to _all the_
_points which are not pi_ . That would be all the points
<3, <3.1, <3.14, <3.141, <3.1415, <3.14159, <3.141592, or ...
and
>4, >3.2, >3.15, >3.142, >3.1416, >3.14160, >3.141593, or ...
We can define operations on all the points not in the gap
so that, when the point in the gap is rational, the operations
agree.
We can define a least-upper-bound for an arbitrary non-empty
bounded collection of sets-of-points-not-in-the-gap (Dedekind cuts).
This is one of the easier parts of all this: the lower cut
of the LUB is the set union of the lower cuts in the collection.
There is no number less than the LUB which can be the
infinite sum.
There is no number greater than the LUB which can be the
infinite sum.
Whether we agree that the LUB exists or we don't, we agree
(I hope) that _all the other numbers_ on the number line exist.
The behavior of _all the other numbers_ is what we ask for
from the LUB, which is what we ask for from the infinite sum.