Date: Oct 2, 2017 2:47 PM Author: netzweltler Subject: Re: It is a very bad idea and nothing less than stupid to define 1/3<br> = 0.333... Am Montag, 2. Oktober 2017 20:35:56 UTC+2 schrieb Jim Burns:

> 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.

So, you disagree that

0.999... = 0.9 + 0.09 + 0.009 + ... ?

> 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.