Date: Oct 2, 2017 2:35 PM
Author: Guest
Subject: Re: It is a very bad idea and nothing less than stupid to define 1/3<br> = 0.333...

On 10/2/2017 1:58 PM, netzweltler wrote:
> Am Montag, 2. Oktober 2017 17:59:21 UTC+2 schrieb Jim Burns:
>> On 10/1/2017 3:22 AM, netzweltler wrote:

>>> Do you agree that 0.999... means infinitely many commands
>>> Add 0.9 + 0.09
>>> Add 0.99 + 0.009
>>> Add 0.999 + 0.0009
>>> ...?

>>
>> 0.999... does not mean infinitely many commands.

>
> But that's exactly what it means.


That's not the standard meaning.

You give it some other meaning, and then you find a problem
with the meaning you gave it. Supposing I wanted to sort out
what that other meaning was, and how to make sense of it, my
attention to your meaning would not affect the standard meaning.

I am not a math historian, but the impression I have
is that great care was taken in choosing the standard meaning
in order to avoid problems like the ones you are finding.

You have the ability to create and then wallow in whatever
problems you choose. No one is able to take that power away
from you. But you can't "choose" by an act of your will to
make your created problem relevant to what everyone else
is doing. You are not the boss of us.

> Infinitely many commands. Infinitely many additions.
> Infinitely many steps trying to reach a point on the number line.
>

>> There is a set of results of certain finite sums, a set of
>> numbers. We can informally write that set as
>> { 0.9, 0.99, 0.999, ... }
>> That is an infinite set, but we can give it a finite description.
>>
>> (Our finite description won't use '...'. The meaning of
>> '...' depends upon it being obvious. If we are discussing
>> what '...' means, it must not be obvious, so we ought to
>> avoid using '...')
>>
>> There is number which is the unique least upper bound of that set.
>> The least upper bound is a finite description of that number.
>>
>> 0.999... means "the least upper bound of the set
>> { 0.9, 0.99, 0.999, ... }".
>> That number can be show to be 1, by reasoning in a finite manner
>> from these finite descriptions of what we mean.
>>
>> If you give 0.999... some meaning other than what we mean,
>> and then it turns out there are problems of some sort with
>> your meaning, than that is your problem, not ours.