Originally posted by AThousandYoungThe time intervals become infinitesimally small so we do reach t=0. Whether you can calculate some limit with L'Hopital's rule is irrelevant to that fact.
Achilles reaches the tortoise because even as the number of "catching up events" is infinite, the length of each of those events in distance (or time because velocity is constant) becomes infinitesimally small. You end up multiplying the number of "catching up events" times the distance per event which gives you a fraction in which both the numerator ...[text shortened]... ng infinitely fast, putting in 2 and taking out 1 each time. Thus you get infinite balls.
Originally posted by AThousandYoungI think that if you say it's impossible to do an infinite number of operations in a finite amount of time then I have no answer for you. I can think of cases where such things do not have a defined answer. I definitely can see where mtthw is coming from when he says the problem has no defined solution. For example, imagine you just have a white ball and keep switching it with a black ball in every iteration. At t=0, which ball is in the bag? This has no defined answer. But that doesn't mean that any such sequence of operations has no solution. What I think we can say is that the answer "infinite" leads to a contradiction and so it cannot be the right answer.
Hmm, let me think some more. I'll try to write it out in a formal, clear way. In short it comes down to - ball swapper speeds up, Achilles does not.
Originally posted by PalynkaWhat contradiction does an answer of "infinite" create that an infinite rate of ball swapping per unit time does not?
I think that if you say it's impossible to do an infinite number of operations in a finite amount of time then I have no answer for you. I can think of cases where such things do not have a defined answer. I definitely can see where mtthw is coming from when he says the problem has no defined solution. For example, imagine you just have a white ball and keep ...[text shortened]... is that the answer "infinite" leads to a contradiction and so it cannot be the right answer.
Originally posted by AThousandYoungThe contradiction I'm thinking of is the inability to pick a ball from a bag with infinite balls and read its number without that number corresponding to an iteration number and corresponding t when that same ball was removed. How does that relate to an "infinite rate of ball swapping per unit time"?
What contradiction does an answer of "infinite" create that an infinite rate of ball swapping per unit time does not?
Originally posted by PalynkaThere's no such thing as a bag with infinite balls; not even as a mathematical construct. Even as a mathematical construct, there can only be bags in which the number of balls approaches infinity as some other variable approaches something else. All math that seemingly involves infinites really only involves limits as variables approach infinity.
The contradiction I'm thinking of is the inability to pick a ball from a bag with infinite balls and read its number without that number corresponding to an iteration number and corresponding t when that same ball was removed. How does that relate to an "infinite rate of ball swapping per unit time"?
Originally posted by AThousandYoungSo you think the set of natural numbers don't exist even as mathematical construct? Or the set of reals? Or the set of points in the interval [0,1]? And ergo all functions defined on domains on these sets (or any set with infinite cardinality) are also "unexisting" mathematical constructs?
There's no such thing as a bag with infinite balls; not even as a mathematical construct. Even as a mathematical construct, there can only be bags in which the number of balls approaches infinity as some other variable approaches something else. All math that seemingly involves infinites really only involves limits as variables approach infinity.
Wow. I think you'll find your views require jettisoning much more than you realize.
An old and well known problem apparently.
http://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox
This is the answer I feel most comfortable with (from the above link):
if infinitely many operations have to take place (sequentially) before [t=0], then [t=0] is a point in time that can never be reached. On the other hand, to ask us how many balls will be left at [t=0] is to assume that [t=0] will be reached. Hence there is a contradiction implicit in the very statement of the problem, and this contradiction is the assumption that one can somehow 'complete' an infinite number of steps. This is the solution favored by mathematician and philosopher Jean Paul Van Bendegem.
Originally posted by AThousandYoungI don't think that's the same as you were saying. I have more sympathy to your opinion than to that one (at least in the way it's described). The description of Bendegem's opinion seems like going back to Zeno to me. Infinity is never reached so Achilles never reaches the tortoise.
An old and well known problem apparently.
http://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox
This is the answer I feel most comfortable with (from the above link):
if infinitely many operations have to take place (sequentially) before [t=0], then [t=0] is a point in time that can never be reached. On the other hand, to ask us ho eps. This is the solution favored by mathematician and philosopher Jean Paul Van Bendegem.
Note that in that quote nothing is said about convergence (which you can read as impact of each operation as the number of steps increases, which I thought you implied by mentioning the "size" of steps).
Can we just renumber the balls?
Let's call the first ball added "1", and the second "-1", then we'll remove "-1".
For the second step, add "2" and "-2", and remove "-2".
Step 3 involves adding "3" and "-3", and removing "-3".
Repeat for all successive steps.
Now the set of balls added is {...-3, -2, -1, 1, 2, 3...},
and the set of balls removed is {...-3, -2, -1}.
Therefore what remains is {1, 2, 3...}, and so there are infinitely many balls in the bag.
It's the same "set" argument with a different numbering system, and you can hardly expect me to believe that the amount of balls remaining actually depends on the way the balls are numbered.
EDIT: Meh, I just read the other thread. Somehow I feel like this argument would just meet with the same fate as the one proposed there.
Originally posted by PalynkaNo, Achilles does not have to perform infinite iterations of some action sequentially.
I don't think that's the same as you were saying. I have more sympathy to your opinion than to that one (at least in the way it's described). The description of Bendegem's opinion seems like going back to Zeno to me. Infinity is never reached so Achilles never reaches the tortoise.
Note that in that quote nothing is said about convergence (which you can r ...[text shortened]... number of steps increases, which I thought you implied by mentioning the "size" of steps).