## Infinite Staircase Paradox

• 296
The story:

Icarus was walking through the woods when he stumbled upon a sign which read:

"One, two, three, four, five,
Each step named as down you dive.
Endless stairs stretch out of sight,
Walk them. Grasp the Infinite's might.
Your speed will double with each step as you soar
like an angel approaching a classical black hole's core."

He glanced down and noticed a dusty staircase plunging into darkness. Thinking that one step would be harmless, he descended and immediately grasped the profound gravity of his actions. He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period.

Despite the staircase being endless, he reached the bottom of it in just a minute. Looking around to ascertain his location, he was startled to find himself standing over a dead body. Clearly, the dead body was that of a man who had fallen from a great height. Horrified, he stepped back, intending to ascend the staircase, but it had vanished. Turning around, he found no steps in sight—how could there be, for what would they number?

A wave of anxiety overwhelmed him as he pieced together the events. Slowly, he looked down at the face of the corpse and recognized it as his own.

The infinite staircase was never real—an impossibility and an illusion from the very start. Or was it?

My Questions:

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? If we allow Hilbert's Hotel to exist in the abstract and possible realm, are we forced to accept the infinite staircase into the abstract and possible realm? Is this actually a paradox? What are your thoughts?
• 18
Infinity minus one equals infinity

Would the above qualify as a paradox, or just be silly in "the" non abstract and possible realm but fit into the abstract and possible realm? Or the reverse?
Can a paradox be conceived in the a&p realm?

Sorry if the above "thoughts" are confusing or ill expressed.
• 2.3k
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?

I think it's a quite nice paradox. It exemplifies that we can conceive of counting from 1 to infinity in a finite amount of time by halving the time it takes for stating each successive number, but that we can't perform the same process in reverse order, as it were, since there is no "last number" that we can step in first to retrace our steps. 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.
• 1.4k
I get shades of Zeno's paradox going on here, except Zeno get's there.

Despite the staircase being endless, he reached the bottom of it in just a minute.
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.
• 12.5k
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.

How does that work? He's traveling by steps. Each step takes a discernible amount of time which is a different time from the prior step. You say he reaches the bottom, yet there is not "last step". Clearly he doesn't do a bunch of last steps at the same time, so ambiguity is not the problem. 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?
• 12.5k
Despite the staircase being endless, he reached the bottom of it in just a minute.

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.

Yes, there is a "slight of hand" involved. The real solution is that the only "finite time" in the description is the starting time, and the formula for figuring the increments. And, according to the prescribed formula for figuring the increments, there can be no finish time. It's analogous to finding the end of pi, you just keep going. Despite the defined, and finite starting point, Icarus is currently covering an infinite number of steps in an infinitely short period of time, and by adhering strictly to the description we must conclude that the bottom will never be reached. Bye bye Icarus, enjoy the "black hole's core".
• 1.4k
Each step takes a discernible amount of time which is a different time from the prior step.
Exactly. Step n takes 60/2**n seconds. That's very much a nonzero duration for any n.

You say he reaches the bottom
After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60?

yet there is not "last step".
Just like there is no last natural number, yes. There is no last step to 'be' at.

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?
It's pretty clear from the mathematics. Where do you expect him to be then at 61 seconds if not 'past them all'?

according to the prescribed formula for figuring the increments, there can be no finish time
OK, so mathematics is not your forte. The sum of this infinite series is not 60 according to you.

The infinite staircase appears to only allow one to traverse it in one direction.
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.
• 207
Icarus reaches an infinitely distant location after one minute. This place can be named with a number with an infinite number of digits before the decimal point, e.g. ...444444 . You can add or subtract one to this number, but you can't get back to finiteness with a finite number of steps.
• 12.5k
After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60?

The specifications do not allow for a minute to pass, that's the problem. It's just like Zeno's Achilles and the tortoise paradox. What is specified by Zeno, disallows the possibility of Achilles passing the tortoise. Here, what is specified in the op by keystone, disallows the possibility of a minute passing.

So it's like you're saying, "after Achilles passes the tortoise...", when Achilles cannot pass the tortoise, because of what is specified, he must pass an infinite number of spaces first. Here, you are saying "after a minute..." when a minute cannot pass because of what is specified, Icarus must pass an infinite number of steps fist.

So it's just like the Achilles and the tortoise paradox, with a different conclusion. 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. But @keystone has made a invalid conclusion, and should have stuck to Zeno's formulation, to say that a minute cannot pass for Icarus because he always has another step to make first, and that step will be made in a shorter time than the last.
• 296
Infinity minus one equals infinity
Would the above qualify as a paradox

If that statement is logically unacceptable, then it could be considered a paradox. However, many people today might not see an issue with it, so you would need to provide further explanation to convincingly demonstrate its paradoxical nature.

Can a paradox be conceived in the a&p realm?

Let me draw an analogy. Historically, our understanding of the world was believed to be tangible and possible. We thought we grasped the truth, whether through Newtonian mechanics or general relativity. Then, some thinkers pointed out inconsistencies in these prevailing views that defied explanation. What was once deemed tangible and possible turned out to be tangible and impossible. As a result, the model of the world was revised, and the new model was then assumed to be tangible and possible. Over the years, this process repeats, gradually bringing us closer to the truth.
• 296
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.

What's your take on this? Do you believe he never finishes descending the stairs? If that's the case, then where would he be after one minute has passed?
• 296
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.

How is it possible for him to ascend the stairs if there isn't a first step? Or do you think that he might not be able to fully descend the stairs?
• 296
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.

Do you truly believe that Achilles is unable to surpass the tortoise? Do you think that Icarus's deeds influence the passage of time? Is there a concrete analogy in which your actions alter how time progresses for me?
• 12.5k
Do you truly believe that Achilles is unable to surpass the tortoise?

By what is stipulated, yes, Achilles cannot surpass the tortoise. But, the stipulations are not a true representation and that is why there is a problem. The issue is with the way that motion is described. We employ a conception of time and space as a continuity which is infinitely divisible. However, if motion actually consists of discrete changes like a "quantum jump" for example, then the representation of a continuous existence, is false.

Do you think that Icarus's deeds influence the passage of time?

No, I do not think that Icarus's deeds influence the real passing of time. However, the passing of time is a subject in your example, and it is determined by the premises of the example. So. according to the premises of the example, half as much time is covered each time Icarus makes a step, as compared to the previous step, and Icarus can keep on stepping forever. Therefore the passage of time is defined in relations to Icarus's deeds in the example. That is what is stipulated in the example, That the passage of time is relative to Icarus's steps. Whether it is a truthful representation of the passage of time is irrelevant at this point.
• 1.4k
How is it possible for him to ascend the stairs if there isn't a first step?
This is nicely illustrated by Zeno's 'dichotomy paradox'. Per wiki:
"Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on."

Each 'step' of the path from here to there must be preceded by a prior step. The fallacious conclusion is that no journey can be taken anywhere since there can be no first step when they're set up as you have done. Likewise, Zeno fallaciously concludes that Achilles cannot overtake the Tortoise due to the journey being divided into infinite steps. Both are a non-sequitur fallacy since it simply does not follow that the goal cannot be reached just because there exists a way to slice it into an unbounded quantity of segments.

The specifications do not allow for a minute to pass,
By what is stipulated, yes, Achilles cannot surpass the tortoise.
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.

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.

That's a far better wording, and less fallacious than the way Zeno is reported to have worded it.
• 1.4k
Is there another source for this paradox? Or did you just invent this yourself?
• 2.3k

Is this not a question of special relativity? It seems paradoxic until we apply physics.
• 1.4k
where does relativity come in?
• 2.3k

He's accelerating exponentially along a linear trajectory (the infinite staircase). So he's approaching the speed of light. Hence relativity becomes an integral factor. One that hasn't been addressed in this "paradoxic" hypothetical.
• 1.4k
I don't think the intention was for physics to be a problem. It's probably supposed to be a purely mathematical problem, it's too fantastical for physics to be a concern.
• 2.3k

Ah okay. Fair. Then where is the reaching the bottom in under 1 minute coming from? 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.

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.
• 1.4k
Then where is the reaching the bottom in under 1 minute coming from?

He just made it up, it doesn't come from anywhere. That's why I'm questioning if it's really a paradox, that's why I want another source for it
• 12.5k
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.

It's not a non-sequitur, the conclusion follows logically from the way that Achilles' movement is described. From the description there is always further distance for Achilles to move before he overtakes the tortoise. Therefore he cannot overtake the tortoise. The issue is not that the argument is invalid, it is that the argument is unsound. The description of motion employed provides false premises.

Time not being allowed to pass was never a specification in the OP.

In the OP, it is not the case that time is not allowed to pass, but the premises imply that a minute cannot pass for Icarus, who always has to take more steps before a minute can pass. Just like in Zeno's paradox, the premises which describe how Icarus moves down the stairs are faulty.

So, in the OP, the false premise is the description of acceleration. Acceleration from rest is described as continuous and open ended (infinite). But this is false, acceleration does not happen like this in reality. Imagine if the OP was expressed in the following way. Someone states that the universe is infinite in size. Then the person states that a rocket accelerates from being at rest on earth at a rate of acceleration which will take it to the edge of the universe before a minute passes. Then the person concludes that after a minute passes the rocket is at the edge of the universe. Do you see the incoherency? That's what's going on in the OP. The premises are arranged so that there cannot be an end to the staircase, just like there cannot be an end to pi. Then it concludes, that after a minute has passed, the end has been reached.

So the OP makes a non-sequitur by concluding that the end is reached. 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.

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.

I don't think that this is representative of the OP at all. You have changed the divisibility of time in the OP to a divisibility of space in your interpretation. Then, instead of dealing with the problem of acceleration, which the OP is concerned with, you have to employ "magic barriers" to make sense of the steps. There are no such magic barriers employed by the OP, only steps, and each step is made in half the time of the prior step. So clearly, the OP deals with the issue of infinite acceleration.

I don't think the intention was for physics to be a problem.

If a person does not take into account what is physically possible in this type of thought experiment, then one can make up false premises however one wants, and create the illusion of a "mathematical problem" when no such problem actually exists. The real problem is that the premises are false (physically impossible), and by employing the false premises the illusion of a mathematical problem is created.
• 932

I don't understand the rules of this game.

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.

But it would be a bad idea for him to ask whether the stairs up were the same stairs as the stairs down, or whether the staircase exists. At best, those questions would scramble his mind, possibly to the point where he might get so distracted as to forget to keep moving. At worst, the staircase might disappear beneath his feet.

He should allow at least twice as long to climb up as he took to get down. But he can expect to complete his climb in the same amount of time as he took to descend.
• 1k
Let S denote the set of stairs, let N denote the standard natural numbers and let N* denote the nonstandard numbers. We can model the cardinality of S, which is equivalent to the height of the top of the staircase, by using a non-standard natural number h* from N*. Lets assume

i) There does not exist an injection N --> S
ii) There exists a surjection I ---> S, where I is a subset of N.

Condition i) represents the hypothesis that we do not know how many stairs there are, or equivalently that we cannot know the height of the top stair due to assuming that we will never reach the bottom of the staircase.

Condition ii) represents the physically plausible situation that although we cannot count the stairs, there cannot be more stairs than some finite but unboundedly large subset of the natural numbers.

In other words, we are assuming that S is subcountable.

Let s(n) denote the n'th stair that is visited when descending. Using this order of descent on S, we have a total function S --> N* describing the height of each stair as a non-standard natural number, namely

s (0) => h*
s(1) => h* - 1
s(2) => h* - 2
..

which when written directly in terms of the indices denoting the order-of-descent is a function f

f : N --> N* :=
f (n) = h* - n*

This function describes an infinite descent in N*, and is paradoxical because

1) Every nonstandard natural number e* that is in N* corresponds to some standard number e in N, and vice-versa.

2) We have defined an infinitely descending chain of non-standard natural numbers in N*.

The paradox is resolved due to the fact that the order-of-descent we are using when descending the "infintie staircase" from the top has no recursively definable relationship in terms of the order of ascension when climbing the staircase from the bottom; although Peano's axioms rule out the existence of non-wellfounded subsets for recursively enumerable subsets of the natural numbers, our subset isn't recursively enumerable in terms of those axioms, and is therefore an external subset that cannot be talked about by Peano's axioms.
• 2k
Violation! The fact that the stair has a bottom shows we are dealing with Hegel's "bad infinity."

Anyhow, Aristotle claims that we cannot have an actual infinity, only a potential one. However, Hegel famously rebuts this claim with the Essence chapter of the Greater Logic, a text that is infinitely dense and impenetrable.
• 18
Perhaps, when translating from mathematical to non-maths usage, infinity acquires an extra qualification i.e. potentiality, which is required to make any use of "infinity" useful as a quantifier. Otherwise, saying anything other than numbers can or can't be infinite leads to issues of illogic.
Maybe, Aristotle was mistakenly "transcribed" from his words or personal writings as "only potential infinity" instead of "infinity of potentiality(ies).
• 14.4k
I think there’s a simpler way to phrase this problem.

After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.

What digit does the counter show after 60 seconds?

If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time.
• 12.5k

That's beautiful, simple and eloquent.
• 14.4k
Thanks, although it's actually a variation of Thomson's lamp.
• 1.4k
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.
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'.

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.
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.

What's not defined in either case is a last or first step respectively, just like there is no highest integer.

From the description there is always further distance for Achilles to move before he overtakes the tortoise.
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:
"In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead". The 2nd bolded part is the non-sequitur, and the first bolded part follows from the 2nd. None of it makes the assertion you claim. The non-sequitur makes the argument invalid. There are ways (such as with the light switch) that make it seem more paradoxical.

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.
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.

So, in the OP, the false premise is the description of acceleration.
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.

there cannot be an end to pi.
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.

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.
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.

I don't think that this is representative of the OP at all.
No, it is more the reverse of Michael's digit counter, just like Zeno's dichotomy scenario is the Achilles/tortoise thing in reverse.

What digit does the counter show after 60 seconds?
You have changed the divisibility of time in the OP to a divisibility of space in your interpretation. — Metaphysician Undercover
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.

it's actually a variation of Thomson's lamp.
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.
Side note: Would be awful nice if the site put numbers on the posts.

Is there another source for this paradox? Or did you just invent this yourself?
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.

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.
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.
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