Am Freitag, 6. Oktober 2017 02:54:40 UTC+2 schrieb Jim Burns: > On 10/5/2017 3:12 PM, netzweltler wrote: > > Am Donnerstag, 5. Oktober 2017 17:59:25 UTC+2 > > schrieb Jim Burns: > >> On 10/5/2017 10:00 AM, netzweltler wrote: > >>> Am Donnerstag, 5. Oktober 2017 15:22:35 UTC+2 > >>> schrieb Jim Burns: > > >> [...] > >>>> _We don't do what you're describing_ > >>> > >>> Nevertheless, > >> > >> "Nevertheless"? > >> Do you agree that what you're describing > >> is not what we're doing? > > *NETZWELTLER* > DO YOU AGREE THAT WHAT YOU'RE DOING > IS NOT WHAT WE'RE DOING?
Let's say I agree. Doesn't mean that it is obvious to me what *you* are doing. All I've seen so far is, that you define 0.999... to be the LUB or limit (I guess you use these expressions interchangeably in this context) of the sequence (1-1/10^n)n?N. That's it.
Do we agree that a LUB or a limit is a point on the number line - or to simplify it in case of a positive number - a point on a geometric line from 0 to +infinity? I haven't seen any proof for that yet other than the claim that it is. For some reason coincident with point 1.
> I think you do agree. > This a pretty fundamental requirement: > When you criticize what someone is doing, > criticize _what they are doing_ and not something else. > > >>> the process > >>> 0 |-> write 0.9 > >>> 1 |-> append another 9 (to the 0.9 already written) > >>> 2 |-> append another 9 (to the 0.99 already written) > >>> ... > >>> results in 0.999... > >>> > >>> Whereas the process you specified earlier > >>> 0 |-> 0.9 > >>> 1 |-> 0.99 > >>> 2 |-> 0.999 > >>> ... > >>> is nothing else but an infinite list of terminating decimals. > >> > >> Right. Nothing else but an infinite list of terminating decimals, > >> which presents no problem, right? > >> > >> And we (meaning _we_ whether or not you include yourself) > >> assign the value of the least upper bound of that list > >> to the non-terminating decimal 0.999... > >> > >> I'm guessing you don't have a problem with the LUB either, > >> because you talk about other things instead. > >> _But this is what we do_ > > > > We obviously agree that the process you specified earlier > > 0 |-> 0.9 > > 1 |-> 0.99 > > 2 |-> 0.999 > > ... > > is nothing else but an infinite list of terminating decimals. > > It think it is also obvious that you have no problem with > an infinite list of terminating decimals. > > > What you don't want to see is, that the process > > 0 |-> write 0.9 > > 1 |-> append another 9 (to the 0.9 already written) > > 2 |-> append another 9 (to the 0.99 already written) > > ... > > results in 0.999... > > > > Maybe you cannot see that I am not writing a new number > > in a new line at each step - as in your process. I am > > appending the 9s in the same line. So I am not creating > > an infinite list of terminating decimals. I am creating > > a single non-terminating decimal. The append operations > > are representing addition operations - infinitely many > > addition operations. > > Suppose, for the sake of argument, that _everything_ that > you have said about what *you* mean by 0.999... is true. > Why does it matter, if it doesn't apply to what *we* mean > by 0.999... ? > > *You* give a meaning to 0.999... that involves infinitely > many addition operations, and then *you* find a problem with > the meaning that *you* gave 0.999... -- a meaning which is > *NOT* the meaning *we* give to 0.999... So what? > > I mean, fine. Whatever. Let me grant, for the sake of argument, > _every error_ that you point out about what *you* mean is > in fact an error. Whoopsie! We'll just have to fix that right > now: We "now" evaluate infinite decimals in a way that avoids > infinite multiplications, whatever they may be. End of problem. > Of course, we already didn't do what your argument suggests > we shouldn't do before you made your argument, but never mind. > _There is no problem_ > > Let me remind you what we're talking about: > > <Burns<netzweltler>> > > > 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. > > </Burns<netzweltler>>