Am Donnerstag, 5. Oktober 2017 15:02:00 UTC+2 schrieb Jim Burns: > On 10/4/2017 3:58 PM, netzweltler wrote: > > Am Mittwoch, 4. Oktober 2017 18:27:18 UTC+2 > > schrieb Jim Burns: > >> On 10/4/2017 4:19 AM, netzweltler wrote: > > >>> To me it looks like that we don't even agree, that there > >>> are infinitely many 9s following. > >> > >> Maybe we agree, maybe we don't. We might be using the same > >> words and meaning different things by them. > >> > >> I say there are infinitely many nines following the '.' > >> What I mean by "infinitely many" here is that there is a > >> map, one-to-one but not onto, from those after-dot decimal > >> places to those after-dot decimal places. And '9' is in every > >> place. > >> > >> I could say more, and I should, in order to say what 0.999... > >> means, but that is what "infinitely many 9s following" means. > >> > >> What do you mean? That there is a '...' at the end? > > > > There is no end. > > I mean a '...' at the end of the description. > When one writes > 0.9, 0.99, 0.999, ... > one puts '...' at the end of _that_ but what does it mean?
> What I mean by "infinitely many 9s following" is broken down > into concepts that we already share in order to explain what > I mean -- which might not be the same as what you mean, even > though we use the same words, "infinitely many 9s following". > > You raised this question. Do we agree? This is a question > which we can answer. > > > Nothing follows after infinitely many 9s. > > "infinitely many 9s following" replaces '...'. > > How do you say "infinitely many 9s following" without merely > trading one thing that needs explaining for another thing that > needs explaining? > > I'll give another example of what I'm talking about, taking > some concept and expressing it using only more basic concepts. > Suppose we want to say that a real function f: R -> R is > continuous. > > Here's one way: > A function f: R -> R is _continuous at b_ if > for every eps > 0, there exists del > 0 such that > for all x, abs(x - b) < del -> abs(f(x) - f(b)) < eps > > This might not be an obvious thing to mean by "continuous", > but there are good reasons for it which can be explored. > An important part of what mathematicians do is hash out > definitions which refer to what we want our higher-level concepts > to refer to, but which do it without calling on not-yet-defined > concepts. > > (In my opinion, it is in the definitions that one most > often see the brilliance of a mathematician. Good definitions > are not at all trivial to devise.) > > This is what we do with the definition of the value of an > infinite decimal expansion. > (It is by that definition that 0.999... = 1) > That definition uses addition and it uses infinite sets, but > it does not use infinite repetitions of addition operations. > We start, before our definition, with addition and infinite > sets, so this is a good definition, one that does not merely > trade one thing needing explanation for another thing needing > explanation. > > To return to my question: what do you, netzweltler, mean by > "infinitely many 9s following"? It might be the same as > what I mean (above). If it is, then we agree.
I say there are infinitely many nines following the '.' What I mean by "infinitely many" here is that there is a map, one-to-one but not onto, from those after-dot decimal places to those after-dot decimal places. And '9' is in every place.