The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists and doesn't exist? Does this make sense? — keystone
He reaches the bottom of something with no bottom. It taking a minute is fine, but there being a bottom is contradictory. Hence I think resolution. Just as there is no first step to take back up, there is no last step to reach, even if it is all reached in a minute.Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone
He reaches the bottom of something with no bottom. It taking a minute is fine, but there being a bottom is contradictory. Hence I think resolution. Just as there is no first step to take back up, there is no last step to reach, even if it is all reached in a minute. — noAxioms
Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone
But also, there is a slight of hand that occurs when we are encouraged to imagine Icarus's position immediately after he's finished traversing the infinitely long staircase in the original direction. If he would have traversed the staircase in Zeno like fashion, as specified, although he would have stepped on all the steps in a finite amount of time, there would be no definite position along the staircase that he was at immediately before he had arrived at his destination. — Pierre-Normand
Exactly. Step n takes 60/2**n seconds. That's very much a nonzero duration for any n.Each step takes a discernible amount of time which is a different time from the prior step. — Metaphysician Undercover
After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60?You say he reaches the bottom
Just like there is no last natural number, yes. There is no last step to 'be' at.yet there is not "last step".
It's pretty clear from the mathematics. Where do you expect him to be then at 61 seconds if not 'past them all'?How do you think it is possible that he got finished with all the steps, in the described order, yet there was no last step?
OK, so mathematics is not your forte. The sum of this infinite series is not 60 according to you.according to the prescribed formula for figuring the increments, there can be no finish time — Metaphysician Undercover
Your poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds.The infinite staircase appears to only allow one to traverse it in one direction. — keystone
After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60? — noAxioms
Infinity minus one equals infinity
Would the above qualify as a paradox — kazan
Can a paradox be conceived in the a&p realm? — kazan
If he would have traversed the staircase in Zeno like fashion, as specified, although he would have stepped on all the steps in a finite amount of time, there would be no definite position along the staircase that he was at immediately before he had arrived at his destination. — Pierre-Normand
Your poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds. — noAxioms
Here, instead of concluding that a minute cannot pass, as Zeno concluded that Achilles cannot pass the tortoise, keystone changes things up to say that after a minute has passed the infinite number of steps has been reached. — Metaphysician Undercover
Do you truly believe that Achilles is unable to surpass the tortoise? — keystone
Do you think that Icarus's deeds influence the passage of time? — keystone
This is nicely illustrated by Zeno's 'dichotomy paradox'. Per wiki:How is it possible for him to ascend the stairs if there isn't a first step? — keystone
The specifications do not allow for a minute to pass, — Metaphysician Undercover
What do you mean stipulated? That Achilles cannot overtake is a non-sequitur. It simply doesn't follow from there being a way to divide the journey into infinite segments. This isn't a stipulation, it is merely a fallacious conclusion. Time not being allowed to pass was never a specification in the OP. Of course the lack of the stairs back up was actually a specification, and I find that contradictory.By what is stipulated, yes, Achilles cannot surpass the tortoise. — Metaphysician Undercover
Then where is the reaching the bottom in under 1 minute coming from? — Benj96
What do you mean stipulated? That Achilles cannot overtake is a non-sequitur. It simply doesn't follow from there being a way to divide the journey into infinite segments. — noAxioms
Time not being allowed to pass was never a specification in the OP. — noAxioms
The dichotomy thing was better illustrated by something that actually seems to be a paradox.
You are at location x < 0. The goal is to traverse the space between x=0 and x=1.
Thing is, a magic barrier appears at x=1/2 if you are at x <= 1/2, but x > 1/4.
A second barrier appears at x=1/4 if you are at x <= 1/4, but x > 1/8.
And so on. Each barrier appears only if you're past the prior one.
Furthermore, for fun, the last barrier is red. The prior one blue, then green, then red again. Three colors in rotation, all the way up the line.
Per the dichotomy thing (and Keystone's stairs), there can be no first barrier. So you walk up to x=0 and are stopped, despite there not being anything there to stop you. I mean, if there's a barrier, you'd see it and know its color, which is like suggesting a remainder if you divide infinity by three.
So paradoxically, you are prevented from advancing despite a total lack of a first barrier. You can see the goal. But you can't move. — noAxioms
I don't think the intention was for physics to be a problem. — flannel jesus
The mathematics is clear. The sum of the infinite series 1/2 + 1/4 + 1/8 ... is 1, not more, not less. Nobody has claimed 'under a minute'.Surely even if halfing the time with every step, a minute will still eventually be exceeded somewhere along the infinite steps and before this so called "finite bottom" to an infinite staircase?!? Doesn't make sense mathematically either. — Benj96
Well, the counterexamples have shown otherwise. I can subdivide the trip from 0 to 1 the other way around, with the smallest steps coming first, thus showing that it can be physically traversed in either direction.The most interesting thing I found about this is the unidirectional counting. You can count from 1 toward infinity but you can't begin counting from infinity toward 1. — Benj96
This is not true. Perhaps you are reading a different account of the story than I did, which is the one on wiki, which says simply:From the description there is always further distance for Achilles to move before he overtakes the tortoise. — Metaphysician Undercover
Same non-sequitur. It is not true that Icarus always has more steps to take, only that he does while still on a step, but the time to complete all the remaining steps always fits in the time remaining in his minute.In the OP [...] the premises imply that a minute cannot pass for Icarus, who always has to take more steps before a minute can pass.
Sort of. I agree It has no basis in physical reality like Zeno's examples do. The OP poetry is only mathematical in nature and isn't meaningfully translated into physics. No amount of physical acceleration can traverse an infinite physical distance in finite coordinate time.So, in the OP, the false premise is the description of acceleration.
Then it concludes, that after a minute has passed, the end has been reached.[/quote]No. It concludes that all of the steps have been traversed. It does not assert that there is a last one. In this suggestion, the OP at least does not commit the fallacy that Zeno does.there cannot be an end to pi.
OK, which premise then is false in the Zeno case? The statement is really short. One premise that I see: "the pursuer must first reach the point whence the pursued started", which seems pretty true to me.Zeno on the other hand, concludes that Achilles cannot overtake the tortoise, which is the valid conclusion. And the absurd conclusion reveals the falsity of the premises.
No, it is more the reverse of Michael's digit counter, just like Zeno's dichotomy scenario is the Achilles/tortoise thing in reverse.I don't think that this is representative of the OP at all.
What digit does the counter show after 60 seconds? — Michael
Yes, my example is more on par with Zeno dividing space than the OP dividing time. It has the same problem as Michael's counter: Measuring something where the thing being measured is singular, which makes the whole thing invalid.You have changed the divisibility of time in the OP to a divisibility of space in your interpretation. — Metaphysician Undercover
I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called.it's actually a variation of Thomson's lamp. — Michael
It was unclear if this was addressed to the OP, or to me since this question was asked immediately after I posted the thing about the barriers. Anyway, not mine, but I can't find a link.Is there another source for this paradox? Or did you just invent this yourself? — flannel jesus
It is indeed unexplained why the guy, after taking the first step, is somehow compelled to continue his journey after 30 seconds and not just turn around. Mathematically it has some meaning, but it never has physical meaning, as several have pointed out.However, I recommend that Icarus stops looking for the last step down and starts looking for the first step up. He should find that as easily as he found the first step down. — Ludwig V
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