On Sunday, October 1, 2017 at 8:22:32 PM UTC-7, Quadibloc wrote: > 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.
That finite quantity is the infinitely small or the very first finite quantity that is first after zero. Defined as One divided by infinity. .9 repeating is less than one by the infinitely small. Therefor they share a sameness to one another.
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