Of course in my supertask I removed marble 1 at the first step. The consequence of this is that there is no n such that the marble with n written on has not been removed. Suppose instead I removed marble 2 at the first step, marble 3 at the second and so on. Then after the minute marble 1 remains in the pot, but no other. In this way I can get up to 9 marbles left over in the pot. Can anyone think of a procedure for removing marbles so that there is a finite number of marbles which is greater than 9 left in the pot after 1 minute?
Originally posted by PatNovak The OP does not fit neatly into the idea of a sum,
That is because it isn't, its a sequence.
However, it is clearly not a sequence, because we know that the glass hits the floor, and if it were a sequence it would never hit the floor, so a sequence can be discounted immediately. The glass nevertheless must traverse every member of the sequence. The so called 'paradox' is set up by convincing you that the glass must traverse every member of the sequence, and then essentially pointing out that it must then get to the destination. But the sequence is infinite and has no 'last member' so the question is how the glass achieves the completion of an infinite series.
But the question is still very much mathematical.
Originally posted by DeepThought No, the series is the complete sum with an infinite number of terms, the sequence is a sequence of partial sums which the series is the limit of.
Sorry, I should have said sequence.
Zeno's paradox is an example of a supertask, and there is a nice page on Wikipedia about them. One that I like is the following paradox. Take an empty pot and start putting marbles in. At each step for every 10 marbles you put in remove 1. After completing the first step (instantaneously), do the second step after 30 seconds, the third after 15 seconds and so on. How many marbles are in the pot after 1 minute? The intuitive answer is an infinite number. However, suppose the marbles were numbered, on the first step marbles 1 - 10 are added and marble 1 is removed. On the second step marbles 11 - 20 are added and marble 2 is removed. After 1 minute there is no marble that has not been removed and therefore the pot must be empty. Of course if instead we had removed marble 10, 20, 30 etc. the pot would contain an infinite number of marbles. Which just goes to show you have to be careful how you go about a supertask. Thanks for the great example. It demonstrates the dangers of playing infinities off against each other.
Originally posted by PatNovak Merely your assertion, an assertion which you yourself have disproven by agreeing that the sequence can never reach 1.
The OP, sets up a paradox by explaining that the glass must traverse an infinite sequence prior to reaching the ground. That is not my assertion, it is right there in the OP. And yes, the sequence does not include the ground.
It seems paradoxical because our minds have difficulty with infinities. But with mathematics, and viewing the situation in different ways, the paradox can be resolved.
Originally posted by shavixmir Alright. At the risk of sounding stupid: What?
Why does an infinite amount of time [b]have to cancel out an infinite number of halves (or distances traveled)?[/b]
It doesn't. In the OP, a finite amount of time, in an infinite number of pieces, correspond to an infinite number of distances (which have a finite total length). But they correspond (cancel out) because the problem is set up that way, not because all infinities cancel each other (they don't).
Originally posted by shavixmir Ahhhh... No. Still have no idea of what you are saying.
Are you saying that (OP): an infinite number of halves can't be used, because time is finite and distances have to be finite as well?
Imagine a piece of pie that you divide into half. Now you keep adding half of the remaining piece to the pie. So you start with your half a pie, and add a quarter. Now you have three quarters of a pie. Add half of a quarter, and you get seven eights (you can visualize this quite easily with a drawing). At any point in the process, you will have some pie left, but keep doing it infinite times and you will end up with a full pie. So the full pie (a finite thing) can, at least mathematically speaking, be divided into an infinite number of pieces. In this paradox, one takes a finite distance and time and divides them simultaneously into infinite pieces.
Originally posted by shavixmir Ahhhh... No. Still have no idea of what you are saying.
Are you saying that (OP): an infinite number of halves can't be used, because time is finite and distances have to be finite as well?
No, I am saying that both the total time and total distance involved are finite. The number of pieces of each are infinite.
If physical time and distance are infinitely divisible, then there is no problem. However it is conceptually a problem if you think of the situation as being a sequence of steps with the ground as the last step. In that case you end up with an infinite number of steps, and then one more, which just doesn't make conceptual sense.
Something drops half a distance
24 Oct '14 17:55
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