It shows that no matter how fast you go you can never finish. — Michael
But is there any integer he never gets to? Nope. That's a point for finishing. — Srap Tasmaner
Suppose we do it differently: Give Zeus one second to recite "1", half a second to recite "2", a quarter to recite "3", and so on. Is there any natural number he hasn't recited after 2 seconds have passed? — Srap Tasmaner
Anyway, that's why I suggest that Planck units solve the paradox--space is not infinitely divisible.
Kant explains in the Critique of Pure Reason why it's hard for us to accept finite divisibility--it's outside of anything humans ever experience, so we can't wrap out heads around it. — NKBJ
This is Thomson's lamp paradox, except the question in this case will be "was the last number odd or even?" — Michael
It might be useful to consider a similar scenario. Zeus counts backwards to 1, getting slower as he counts. It took him 1 second to count from 2 to 1, half a second to count from 3 to 2, and so on. — Michael
I think these calculus solutions are just a bewitchment — Michael
I think we could play around with "first" as I have been with "last", but for many cases recursive specifications are exactly what we want, so I can just as well say that what you describe here is not a task at all. — Srap Tasmaner
So is there an argument for (1) — Srap Tasmaner
But this is just not true for infinite tasks. The number of natural numbers remaining to be recited is the same after reciting any finite number. — Srap Tasmaner
Counting up from 1 is a task but counting down to 1 isn't? Why is that? — Michael
I don't know. It seems a truism that if one has completed a series of consecutive tasks then some task was the final task. I know it's not much of an argument, but I honestly can't make sense of it being any other way. — Michael
In effect your claim is that "infinite task" is a contradiction. — Srap Tasmaner
Just in the sense that I don't know how to specify that task recursively. Is there a way? If not, is there some other way? — Srap Tasmaner
Can you specify the task of movement recursively? “Move to the first half-way point”? It’s a lot like counting the rational numbers between 0 and 1 in order — Michael
Moving by half the remaining distance can be specified recursively; doing the rationals between 0 and 1 in order cannot be if by "in order" you mean "smallest to largest". There's no smallest rational > 0 to be the first. — Srap Tasmaner
My point for the moment is that if anyone can do anything, then Zeus can recite all the natural numbers in 2 seconds. — Srap Tasmaner
Convergent series are necessarily inapplicable to supertasks. — Michael
Sometimes your objection is that a given task can't be finished, sometimes that it can't be started, sometimes, as here, that if it could be completed then something else you don't like could also be possible. — Srap Tasmaner
What turns out to be wrong with this family of arguments? It's not just about movement, for instance, but about there being any sort of change at all, about anyone, as I said before, ever doing anything. So what's going wrong here? — Srap Tasmaner
There's a fundamental difference between the number line example and movement. Disregarding the numbers between, it takes time to count any amount of numbers, but doing the same with distances the movement between two points would be instantaneous. Thus, if you don't stay at any location, the movement between two locations does not take infinite time. — BlueBanana
The first point would only be needed if it took time to travel to it. — BlueBanana
the movement between two points would be instantaneous — BlueBanana
Disregarding the numbers between — BlueBanana
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.