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.
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. 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.