Counting down by halving to 0

Question

Say that you are counting down from 10. You say how long is left after half the amount of time you said how long was left (Like 10, 5, 2.5, 1.25, 0.125, etc.). Because when you halve repeatedly you can never get down to 0, wouldn't you have to say how long is left infinitely many times before you can reach 0 seconds?

Update:
8 months ago, Vsauce uploaded a video about a similar problem to this, which he called Supertasks: Watch it at https://www.youtube.com/watch?v=ffUnNaQTfZE.

Literally, the answer is yes, you need to say infinitely many times before your reach 0 seconds. But what is the real question you want to ask? Are you uncomfortable that you need to say infinitely many times in a finite time period or you sense some contradiction somewhere hidden in this case? — achille hui, Oct 24 '13 at 09:15
That was a good video. But definitely not worth bumping this old question for that. It would have made a far better comment than an edit. — Asaf Karagila, Sep 06 '16 at 09:00

score 2 · Accepted Answer · answered Oct 24 '13 at 09:05

You haven't given enough of an explanation. You've only explained what you're doing during the times $t$ in the interval $[0,10)$, where $t$ measures the seconds since you started counting. However, you do say something infinitely many times within that interval.

You haven't said anything about what you're doing for the remaining time. Normally, given that the problem is phrased in terms of the "real world", we'd invoke continuity, and conclude that you would say "0 seconds left" when $t=10$. However, this breaks physical assumptions -- e.g. the speed at which your mouth is moving increases without bound, and the sound waves from your speech would eventually be strong enough to shatter the Earth.

So the usual implicit conventions we make for such problems really don't apply: you have to describe what happens at (and possibly after) $t=10$ before we can know if you will ever reach $0$.

That sounds like a nice "What If" for the xkcd what if blog... — Asaf Karagila, Oct 24 '13 at 09:07

score 1 · Answer 2 · answered Oct 24 '13 at 08:53

1

Yes, you would, but as time decreases, you would have to say it very very fast...

answered Oct 24 '13 at 08:53

malin

1,201

but then how would you reach 0? – Zach Barham Oct 24 '13 at 10:00

score 1 · Answer 3 · answered Oct 24 '13 at 08:57

You will have to speak very fast because you'll have to squeeze in infinitely many words into these $10$ seconds. But you will still only wait $10$ seconds.

If the clock stops while you're saying the amount of time that has left, then you will never reach $0$, but you will get so very close to it. By definition this would be a Sisyphean task.

score 0 · Answer 4 · answered Oct 24 '13 at 09:11

0

You should read about Zeno's paradoxes of motion.

answered Oct 24 '13 at 09:11

dfeuer

9,069

Counting down by halving to 0

4 Answers4