Am Montag, 2. Oktober 2017 05:22:32 UTC+2 schrieb Quadibloc: > On Sunday, October 1, 2017 at 1:22:43 AM UTC-6, 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 > > ?? > > Then following all of these infinitely many commands won?t get you to point > > 1. If you reached point 1 you have disobeyed those commands, because every > > single of those infinitely many commands tells you to get closer to 1 but > > NOT reach 1. > > You would be correct if Zeno's paradoxes were correct. But they're not. > Achilles can and does overtake the tortoise every day. > > 0.9999... does *NOT* mean actually doing those infinitely many steps. There > is never time to do that many commands. Instead, it means the place that > doing them would take you, if you _could_ do them. > > Yes, doing any _finite_ number of those commands would not get you to 1. You > would have to disobey them to get that far.
Even doing an _infinite_ number of those commands wouldn't get you to 1.
1. 0.99 + 0 2. 0.99 + 0 3. 0.99 + 0 ...
Neither a finite number of the steps on the list above will get you to 1 nor an infinite number of the steps on the list above will get you to 1. We can tell that - no matter if we can do all the steps or not. For each particular line is true that we don't reach 1. And this is true for this list also:
> But you *can't* do an infinite number of commands. Period. > > So that isn't the criterion you use to figure out what 0.9999... actually > is. > > Is 0.9999... not equal to 1? In order for it _not_ to be equal to 1, it > would have to be less than 1 by some finite number.
Why that? 0.999... cannot be located at point 1 of the number line. Why do you think that means that it must be short of 1 then? It cannot be located at a point < 1 either.
> But pick any such > number, and by doing a sufficiently large finite number of commands, you can > get closer to 1 than that. > > So 1 is indeed the only thing it can be equal to, even though that looks > funny. But that's just a problem with the decimal system of writing numbers > - it doesn't perfectly match the real numbers it refers to - not with the > numbers themselves. It doesn't mean infinitesimals have to be added to the > real number line. > > John Savard