Yes it does, this distance from Achilles to the Turtle is finite, even after accounting for the movement of the Turtle. The only thing Zeno's paradox does is create an infinite number of points on a finite number line. — Jeremiah
...if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)
Now the resolution to Zeno’s Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long—only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.
but this doesn't actually matter, because the rationals can be re-arranged into a list — Srap Tasmaner
Except to keep this analogous to movement the counting has to be ordered. We don't jump to the half-way point and then back to some earlier point. — Michael
I'd have to brush up on this to answer properly, but my instinct is that that's an interpretation problem, essentially a matter of labeling. There's the standard interpretation, associated with the number line, of what order numbers are in, but they don't have to be. That may not look like much of an answer. — Srap Tasmaner
Of course if space is granular, then our task is finite, yes? — Srap Tasmaner
Given an infinite amount of time I could hop to all the rationals between my starting point and any destination (inclusive) — Srap Tasmaner
You are just talking about a continuous number line, which is actually a requirement for the FTC. The curve must be continuous from a to b. — Jeremiah
This would actually make all motion impossible — Jeremiah
I don't see what this has to do with mathematics. — Michael
Map/territory confusion. — Baden
The realm is physics. If the mathematical models (maps) cause paradoxes so much the worse for their application in this instance. — Baden
If it took me 1 second for each hop, it would take me countably many seconds to do all the rationals, the same number of seconds it would take to hop to all the natural numbers in order. — Srap Tasmaner
A supertask has countably many steps completed in a finite amount of time. — Srap Tasmaner
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