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.