we postulated the existence of a finite-sized mechanism that can switch state in an infinitesimally small time, which contradicts the laws of our world. — andrewk

That's precisely the argument being made.

There are some who claim that a supertask is possible; that if we continually half the time it takes to perform the subsequent step then, according to the sum of a geometric series, an infinite sequence of events can be completed in a finite amount of time.

Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible. Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility.

With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.

When it comes to the supertask of counting a "countably infinite set", by exponentially decreasing the time it takes to count, here's the problem I have with the scenario in the OP:

Tthe mathematical ability to count in a finite time is purely mathematical and, crucially, doesn't involve the concept of conscious experience.

Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. Let me explain:

You say he halves his rest period every step - but it's still implicit that each step has to be a *conscious experience*, a "choice". Which means, even though you can mathematicaly say he can count to infinity in 1 minute, I propose that he would be stuck consciously in that 1 minute for eternity, since that 1 minute includes an infinite series of choices and conscious experiences.

So if you send a person down such a staircase, in some mathematically perfect platonic realm where time is infinitely divisible, that person will be stuck in the eternity of that 1 minute.

Then your argument should be that supertasks are impossible, not that 60 seconds cannot elapse. — keystone

No, clearly I made the appropriate choice in deciding what to argue. The described "supertask" is incompatible with 60 seconds elapsing. We seem to agree on that now. The supertask is derived from the premises of your example, therefore the supertask is the valid conclusion to your premises. You have provided no propositions or premises whatsoever, to conclude that 60 seconds may actually elapse. This is derived from your prejudice about the the nature of time, and the conventions of measuring the passage of time. To base my argument on unstated premises, and prejudices, is to produce an invalid argument. Therefore I have no principles to validly argue that 60 seconds can pass, and no principles to argue logically that the described supertask is impossible. Your assertion that 60 seconds may pass is inconsistent with your stated premises, and is not supported by any premises.
It is derived merely from your prejudice.

The example is simply: 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 for 60 seconds, resetting to 0 at every tenth increment. — Michael

Clearly, what is implied by "and so on", contradicts "for 60 seconds". The two are inconsistent, incompatible. Therefore your example is self-contradicting and incoherent. To me, it is analogous to saying "the big blue computer is designed to produce the complete numerical expression of pi, and when it is finished we'll throw a big party. Do you see how "when it is finished" (analogous with "after 60 seconds") contradicts the described supertask "produce the complete numerical expression of pi" ( analogous with what is meant by "and son on")?

Clearly, what is implied by "and so on", contradicts "for 60 seconds". — Metaphysician Undercover

No it doesn't.

The "and so on" refers to repeating this formula:

Step 1 occurs after 30 seconds, step 2 occurs after a further 15 seconds, step 3 occurs after a further 7.5 seconds, and so on.

As per the sum of a geometric series this supertask takes 60 seconds.

If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be? — keystone

Presumable it would be at (the number of steps in the first staircase divided by 2). So? — Ludwig V

Actually, I've bethought myself and realized that the step numbers will only align if the number of steps is odd. If it is even, they won't be such a point. I still don't see that anything of interest follows.

If you'll allow me one more post.

This made me realize that if you can define the staircase down, you have defined the stair-case up. There is an intricacy about defining exactly what a step is, but let's leave that aside. Suppose we define a staircase down with 10 steps (floor level is 0). When I take the first step, there are 9 steps (10 - 1) left 2nd step (10-2)...9th (10-9) and 10th step (0) (floor level). Mutatis mutandis, that is the same in reverse 1st step up leave (10 - 1) and so on. So the staircase down defines the staircase up. No need for two staircases.

I could be wrong here, but I think that for a staircase of N steps, 1st step leaves (N-1) and so on.

Zeno contends that change is impossible, leading to stark implications depending on one's philosophical stance on time. Under presentism, this translates to an unchanging, static present—life as nothing more than a photograph. — keystone

That sounds like a Boltzmann Brain, a mere state from which all is fiction and nothing can be known. Under this sort of presentism, there is nothing but a mental state and no experience at all, so no Achilles, Tortoise, stairs, or whatever. Just a mental state with memories of unverifiable lies.
This is not the usual presentism where that state was caused by prior ones, and will cause subsequent states. I don't think what you describe can be validly categorized under the term 'presentism'.

In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and future

There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning.

So there is still motion and change over time under eternalism, and the 'paradox', as worded, works under either since no reference to the present is made. Hence my suggestion that the topic has nothing to do with whether or not one posits a preferred moment in time.

Which view do you think is more reasonable?

Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement.

Consider whether it is easier to draw a one-dimensional line by assembling zero-dimensional points consecutively or to cut a string (akin to dividing a line into segments).

That sounds like Zeno's arrow thing, a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. Under the discreet interpretation, there are a finite number of points making up a finite length line segment. Under the continuous interpretation, no finite number of points can make up a line segment, but a line segment can still be defined as (informally) all points from here to there.
About the only practical difference is that for two non-identical points, they can be said to be adjacent only in the discreet view.

Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time.

But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this.

Instead, modern Zeno would suggest that the entire timeline already exists as a block universe

Irrelevant, per above. The block universe can still be interpreted as discreet or not, just like the presentist view. The difference between the two has nothing to do with any of the scenarios Zeno is describing.

However, there's a twist: abstract strings, like time, are infinitely divisible. No matter how many cuts we make (one after another), we never actually reduce the string to mere points.

You do if it is discreet. A physical string is very much discreet, but that is neither space nor time. Zeno seems to favor the continuous model since all his paradoxes seem to presume it. E.g: "That which is in locomotion must arrive at the half-way stage before it arrives at the goal", a statement that simply isn't true under a discreet view.

the eternalist perspective reframes the impossibility of supertasks from an unacceptable notion—that motion itself is impossible

Nonsense. It says no such thing. It is only a difference in the ontology of events.

that observing every instant in history is impossible.

This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular.

If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. — keystone

Non sequitur. It presumes the length of the staircase is a number, which is contradictory.


Everybody here seems to be attempting to introduce a premise that there is a number that represents the number of steps, despite the immediate contradiction with the premise to which it leads.

Presumable it would be at (the number of steps in the first staircase divided by 2) — Ludwig V

Case in point.

But the last step down is not defined, which means it can't be reached. — Ludwig V

Doesn't follow, since clearly I can overtake the tortoise in a universe that is continuous.
I can also do the reverse (the dichotomy version), which is the equivalent of counting down from infinite steps.

So instead of the assertions, show formally how this contradictory conclusion follows from the premise. Nobody has done that.

The paradox does not require the physical possibility of such a counter. It simply asks us to consider the outcome if we assume the metaphysical possibility of the counter. If the outcome is paradoxical then the counter is metaphysically impossible, and so we must ask which of the premises is necessarily false. — Michael

Point taken. This thread is the first time I hear of "supertasks". What I can't agree to immediately is that

I would suggest that the premise that is necessarily false is that time is infinitely divisible.

It is metaphysically necessary that there is a limit to how fast something can change (even for some proposed deity that is capable of counting at superhuman speeds). — Michael

If we agree that time is infinitely divisible, it seems to follow that an infinite task may be completed in a finite amount of time, just like there are infinitely many numbers between 1 and 2, Cantor nonwithstanding. What is being argued is simply the metaphysical possibility of infinity — I don't see anything metaphysically necessary or impossible about a certain speed threshold.
If we admit infinity is metaphysically possible, the counter is metaphysically possible too, and it counts to infinity. We may dislike that conclusion, but aesthetic appeal is not an argument but a motivation to seek one, which is not rejecting a premise outright.

Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible — Michael

Except there have been plausible solutions given to Thomson's Lamp. Which is more of a problem than it is a paradox.

For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description.

Except there have been plausible solutions given to Thomson's Lamp. — Lionino

I wonder if there's such a solution to my variation.

If we agree that time is infinitely divisible, it seems to follow that an infinite task may be completed in a finite amount of time — Lionino

And so conversely, if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible.

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? — keystone

Icarus was a Greek. To the Greek, and especially the Stoics, time is an infinite circle, and they (many of the Stoics) thought that everything repeats after a full cosmic cycle. Such that all paths hitherto, and all paths henceforth have and will be a reality. Even if reality doesn't work that way, the emotional effect of believing in such a thing allows one to overcome hardships by accepting them as part of the necessary path they are currently on, and that anyway of overcoming that obstacle is correct as all paths henceforth will eventually play out. It's quite similar to several quantum theories.

What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age. — keystone

There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. In the case of the staircase, there actually is no last step - so it was correct to say the staircase was "endless".That would be analogous to saying the largest natural number can be reached by counting. This same objection has been raised in regard to the Zeno walk (see this SEP article).

We can consider the steps to be implicitly numbered - they map to the natural numbers. Traversal is one step at a time, moving from step n to step n+1. Every such n is a member of the set of natural numbers, but the supertask obviously never runs out of these. The contradiction is introduced by the stipulation that the end (of something endless) is reached by this process.

One reason the thought experiment can be misleading is that we're accustomed to treating infinite sets as mathematical objects. So we can consider the set of natural numbers and discuss it's cardinality (aleph-0). The set of supertask steps (step 0 to step 1, step 1 to step 2...) is also an infinite set with cardinality aleph-0 so it maps 1:1 to the set of natural numbers. The mapping is "complete" because it's defined for each member of the sets, but a supertask is a consecutive PROCESS, not a formulaic mapping identifying the correspondence. So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS.

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.

Step 1 occurs after 30 seconds, step 2 occurs after a further 15 seconds, step 3 occurs after a further 7.5 seconds, and so on. — Michael

I see that 30 and 15 and 7.5 sums up to 52.5 seconds. I also see that as it progresses the sum approaches 60. But I do not see how it could ever get to 60. Show me how you believe that "and so on" could indicate a sum of 60 is achieved please.

I wonder if there's such a solution to my variation. — Michael

What digit does the counter show after 60 seconds?

If time is infinitely divisible, the counter would go up to infinity. Not a conclusion that many of us may like, but there doesn't seem to be anything logically absurd with it.

I am generally in agreement with fishfry that

the story's made up. In freshman calculus, the sum of that series is 1. But freshman calculus is just another made up story too. Just a highly useful one. There are no summable infinite series in the physical world. No physical computer can calculate the sum. — fishfry

If it is in a made-up universe where such counters are possible, and time is infinitely divisible, the counter should count to infinity after 30s.

Let's say even, the counter counts 1 at 15 seconds, 2 at 22,5, 3 at 26,25 and so on. It seems it would converge to infinity at time 30s. However what would the counter show at 60 seconds? Are we talking about aleph-0 and aleph 1 and so on?

if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible — Michael

Of course, but it seems that supertasks have not proven that the issue is truly the nature of time instead of their phrasing or some other thing.

It is indeed strange, if supertasks are impossible, it is because of the nature of time. So concluding something about the nature of time from thought experiments seems to put the horses behind the chariot or maybe to be analogous to ontological arguments, where we conclude something about the world by relying our own, perhaps mistaken, human intuitions.

If time is infinitely divisible, the counter would go up to infinity. Not a conclusion that many of us may like, but there doesn't seem to be anything logically absurd with it. — Lionino

I disagree. It's absurd because the counter progresses through natural numbers, and can never reach a final one. Infinity isn't a natural number. In the context of a temporal counting process, infinity = an unending process, not something that is reached (and not a number).

If it is in a made-up universe where such counters are possible, and time is infinitely divisible, the counter should count to infinity after 30s. — Lionino

I don't follow. In calculus, the sum of an infinite series is defined. According to that definition, the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... is 1.

But even if we reject that in the physical universe, or we reject it for other reasons (we're ultrafinitists, we're cranks, we hate math, we hate infinity, we simply don't care, etc) it's still perfectly clear that no matter how many finite number of terms you take, the sum is always less than 1. So I don't see how you can justify claiming that the sum should be infinity, even in a world where you reject the modern theory of convergent infinite series.

Let's say even, the counter counts 1 at 15 seconds, 2 at 22,5, 3 at 26,25 and so on. It seems it would converge to infinity at time 30s. — Lionino

Yes, agreed, if we allow the value $\infty$ as in the extended real numbers.

However what would the counter show at 60 seconds? Are we talking about aleph-0 and aleph 1 and so on? — Lionino

We'll never get past $\aleph_0$ by adding more finite numbers. Likewise $\aleph_0 + \aleph_0 + \aleph_0 + \dots$ where there are $\aleph_0$ terms, is still $\aleph_0$.

According to that definition, the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... is 1. — fishfry

Yeah, the series has infinite terms. Michael's example flips it, and the counter counts how many elements there are in the series, not the sum of the terms, which is the passing of time.

So I don't see how you can justify claiming that the sum should be infinity — fishfry

I am not, check his problem: https://thephilosophyforum.com/discussion/comment/898574

You'll never get past ℵ0 by adding more finite numbers. Likewise ℵ0<br/>+ℵ0+ℵ0+...
. — fishfry

See https://thephilosophyforum.com/discussion/comment/898574

Speaking of extended real numbers, is there any useful application of it?

See https://thephilosophyforum.com/discussion/comment/898574 — Lionino

Ok I hadn't seen that before. Whatever shows at the end (if that even makes sense) it's certainly finite, since you're adding up finitely many finite numbers then resetting to 0. So the final total, if you can define such a thing, is 0 or some positive finite number. I don't believe such a series has a well-defined sum, since the sequence doesn't converge.

That is, the sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ...

That sequence doesn't converge.

@michael then says, "If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time."

I don't see how that follows at all. No mathematical thought experiment can determine the nature of reality. We can use math to model Euclidean geometry and non-Euclidean geometry, but math can never tell is which is true of the physical world. You can use math to model and approximate, but it is never metaphysically conclusive.

But even taken on its own terms, I don't follow the reasoning. How does observing that the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ... doesn't converge, imply anything at all about the nature of time?

By the way the Thompson's lamp sequence is 1, 0, 1, 0, 1, 0, ... and that doesn't converge either.

Speaking of extended real numbers, is there any useful application of it? — Lionino

It's used in calculus to talk about "limits at infinity" and "infinite limits." For example, the limit as x goes to infinity of 1/x is 0; and the limit of 1/x as x goes to 0 is infinity. You need a formal definition of infinity in order to make those statements rigorous.

Those usages are convenient but not necessary. We could talk about the limit of 1/x as "x gets arbitrarily large," but instead we just say, "as x goes to infinity," and everyone understands the meaning.

The extended reals are also used in measure theory (a generalization of length, area, volume, etc) so that, for example, we can sensibly say that the real line has length infinity.

STAIRCASE PARADOX

Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... ω
. Any step back takes you to a number that is only finitely many steps from the beginning. You don't need infinitely long legs. In fact your legs can be arbitrarily small. Any step backward (or up the stairs) necessarily jumps over all but finitely elements of the sequence. — fishfry

I see your point, and I appreciate your analogy with the [0,1] interval. However, you need to clarify what happens in the narrative. The purpose of this narrative is to ensure that one cannot simply retreat behind formalisms. This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. If you're suggesting he doesn’t have infinitely long legs, then perhaps he possesses infinitely powerful legs that enable him to leap over infinite steps. This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug.

I've bethought myself and realized that the step numbers will only align if the number of steps is odd. If it is even, they won't be such a point. — Ludwig V

This brings us to another paradox - Thomson's Lamp - in that the last step can neither be even nor odd.

So the staircase down defines the staircase up. — Ludwig V

Now explain how your algorithm works for infinite stairs.

So why don't you just link me to the reading materials that would lead me to believe that the supertask you described in your op is possible to complete? That specific supertask, not supertasks in general. Let's not beat around the bush, let's get right to it. — flannel jesus

Instead, please present any supertask you consider viable, and I will demonstrate its connection to Icarus descending the staircase. For instance, do you agree that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... equals exactly 1?

Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. — flannel jesus

I'm unclear on whether you're disputing the existence of supertasks or merely the ability of humans to perform them. Do you believe it's conceivable for anyone physical or abstract, perhaps even a divine being like God, to accomplish a supertask?

Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility. With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed. — Michael

Reading your posts gives me a sense of calm. :D

ou have provided no propositions or premises whatsoever, to conclude that 60 seconds may actually elapse. — Metaphysician Undercover

I said he "reached the bottom of it in just a minute." Thus, the premises address both the completion of the supertask and the passing of a minute. It seems you are challenging the incorrect premise.

There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. — Relativist

I would contend that all of the infinity paradoxes clearly illustrate contradictions inherent in the concept of actual infinity. Furthermore, I would argue that every definition of real numbers inherently suggests that supertasks are completable.

So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS. — Relativist

We can also map the steps to the elapsed time (1 → 0.5, 2 → 0.75, 3 → 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps?

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process. — Relativist

Why not?

ZENO'S PARADOX

I don't think what you describe can be validly categorized under the term 'presentism'. — noAxioms

You're correct that presentists don't explicitly hold this belief. However, what Zeno's Paradoxes demonstrate is that if their ideas are taken to their logical conclusion, this belief is implicitly suggested.

There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning. — noAxioms

Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. So if you're saying that the experience of the present has nothing to do with Zeno's Paradox, then I agree with you. But there is a very significant difference between a photo and a movie reel.

Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement. — noAxioms

Reconciling general relativity with presentism is quite challenging. Therefore, if empirical evidence influences your thinking, eternalism might be a more suitable perspective to adopt. Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.

a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. — noAxioms

You're approaching this with a whole-from-parts mindset, where you aim to construct everything from smaller components. Thus, you believe the only options are to assemble a continuous line from infinite points or from discrete line segments. Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.

While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said “My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.”

But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this. — noAxioms

However, you're working under the assumption that a timeline consists only of discrete points in time. You cannot directly observe a particle in a superposition state, but this doesn't mean that superposition states are merely fictional. I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points) and the unobserved a wavefunction (comparable to a line) that lies between them.

The block universe can still be interpreted as discreet or not, just like the presentist view. — noAxioms

I believe you are discussing whether time is discrete or continuous. In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted). I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.

You do if it is discreet. A physical string is very much discreet — noAxioms

I explicitly wrote abstract string.

Nonsense. It says no such thing. — noAxioms

Perhaps it's not my place to speak for others, but let’s say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.

This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. — noAxioms

Additionally, none of the paradoxes explicitly rule out this as a possible solution.

Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular. — noAxioms

If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see. They're blacked out. In fact, I would argue that we can only ever observe countably many frames so in fact, most of the frames remain unobserved (in a superposition of sorts). This allows the story to advance and avoids singularities.

And so conversely, if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible. — Michael

This only applies if you adhere to a whole-from-parts construction approach. As I mentioned in my discussion with NoAxioms, a seldom considered alternative is that the universe is constructed parts-from-whole. I really hope you will engage with me on this possibility.

THOMSON'S LAMP

Depends on if the calculator is required to follow the mathematical theory of convergent infinite series. If yes, 1, If no, then it can be anything at all. — fishfry

In this scenario, the calculator isn't equipped to perform calculus; it's a basic model tasked with adding each term of the infinite series. While mathematical theory predicts that at 60 seconds, it will display 1, it's true that the narrative does not specify what should appear at that moment. I am even welcoming of the idea that it turns into a black hole at 60 seconds. Nevertheless, isn't it concerning to you that there's a discrepancy between mathematical theory and your intuition? I completely agree that freshman calculus is invaluable, and I'm not suggesting that infinite series or any aspect of calculus are without merit. I use aspects of it everyday. Instead, I propose a new interpretation of what these infinite series represent. The story of the calculator isn't really about what it displays at 60 seconds; it's about the approach to 60 seconds. Likewise, I suggest that infinite series don't actually sum up to a specific number, but rather they outline a continuous, unbounded process. We don't have to assert that there's a least upper bound to this process.

That's the problem with all these puzzles. — fishfry

Your argument that the paradox is nonphysical is a red herring. This narrative takes place in the abstract realm, and unless you can pinpoint a contradiction within that context, we should consider it as abstract and possible and acknowledge its validity. Perhaps you lean towards theoretical perspectives, but it's important not to undermine the significance of thought experiments. They have arguably been among the most influential types of experiments conducted by humans.

We can also map the steps to the elapsed time (1 → 0.5, 2 → 0.75, 3 → 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps? — keystone

Indeed, the stipulated elapse of a minute implies all the steps would have been traversed, but that implication is contradicted by the fact that the process of counting steps is not completable. The presence of this contradiction implies there's something wrong with the scenario.

Here's what's wrong: a mapping reflects a logical relationship, not an activity. The activity is a stepwise process: step n+1 is counted AFTER step n; the logical relation is present atemporally - it's an entailment of the way the scenario is defined.

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.
— Relativist
Why not?

Same as above: it's a logical relation (atemporal) that does not account for the stepwise process that unfolds in sequence (temporally).

I see your point, and I appreciate your analogy with the [0,1] interval. However, you need to clarify what happens in the narrative. The purpose of this narrative is to ensure that one cannot simply retreat behind formalisms. — keystone

The formalisms are wonderfully clarifying of one's formerly fuzzy intuitions.

For example the idea of stepping back from the bottom. It's only a finite number of steps back, even from infinity. Absolutely nobody has that intuition at first. Once one has studied the ordinal numbers, it's literally a theorem that it's always only finitely many steps back from a transfinite ordinal. And once you understand why that is, you now have a better intuition.

The process is:

1) Have fuzzy intuitions;

2) Study some math;

3) Develop far better intuitions.

Some may think of that as "retreating behind formalisms." I think of it as developing better intuitions about the real numbers, infinite processes, and so forth.

This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. — keystone

You can not use the word "reality" in this context. In reality there is no such staircase. This is an abstract conceptual thought experiment. It has a mathematical answer. If you are at 1 and you take even the tiniest step backward, you necessarily jump over all but finitely many elements of any sequence that approaches 1. That's a better intuition than the pre-mathematical intuition.

You don't find it counterintuitive that moving slightly to the left of 1 jumps over all but finitely many elements of any sequence approaching 1 from the left, correct? That's clear to you I assume.

Well, that's the staircase. It's a better intuition, informed by a precise formalism.

The best I can do to meet you halfway here is to agree that I have had some mathematical training, and that I have had "improved" intuitions beaten into me by professors at some of our finest universities. But in truth, studying math clarifies all the fuzzy intuitions about the real numbers, infinite sequences and series, infinite processes, and so forth. The Thompson lamp is a sequence of alternating 0's and 1's and it's not defined at infinity. So the final state is anything you care to define. You can't make the series continuous no matter how you complete it.

If you're suggesting he doesn’t have infinitely long legs, then perhaps he possesses infinitely powerful legs that enable him to leap over infinite steps. — keystone

As the example of [0,1] shows, even the tiniest imaginable legs, the weakest possible legs, necessarily jump over infinitely many elements of any infinite sequence approaching 1 from the left.

There is no such thing as an infinite staircase, so this physical analogy confuses more than it enlightens.

I often make this point about Hilbert's hotel. You go online and people will argue that there's no such hotel, and how can there be room in an infinite hotel, and so forth. Many people end up more confused than they were before. In fact any infinite set may be placed into bijective correspondence with one of its proper subsets, and that's the "formalism" that clarifies the vague and unrealistic story. Hilbert, by the way, only mentioned the hotel once in his life, at a public lecture, and never wrote or spoke about it again. It's been blown up way out of its negligible importance, to the point where many people think it's a mathematical argument. It's not. /end rant

This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug. — keystone

He has a magic carpet.

Your argument that the paradox is nonphysical is a red herring. This narrative takes place in the abstract realm, and unless you can pinpoint a contradiction within that context, we should consider it as abstract and possible and acknowledge its validity. — keystone

It's perfectly valid. I already agreed that it's perfectly natural to consider that you are present at the bottom of the staircase after one minute, because that's the choice that makes the sequence continuous. I already said this. There is no paradox. As an abstract thought experiment it works out perfectly well.

In fact I don't even understand what the supposed paradox is. I do not believe your original exposition was sufficiently clear on this point.

The staircase is nicer than the Thompson lamp, which can not be made continuous by any choice of final state, since 0,1,0,1... does not converge.

Perhaps you lean towards theoretical perspectives, but it's important not to undermine the significance of thought experiments. They have arguably been among the most influential types of experiments conducted by humans. — keystone

I haven't undermined the staircase story, I've agreed that it's perfectly sensible to assume the walker is present at the bottom, since that preserves the continuity of the sequence.

That's in contrast with the lamp, in which there is no possible final state that makes any more sense than any other.

I see that 30 and 15 and 7.5 sums up to 52.5 seconds. I also see that as it progresses the sum approaches 60. But I do not see how it could ever get to 60. — Metaphysician Undercover

Because 60 seconds will pass. I don't understand the problem you're having. The passage of time does not depend on what the counter is doing.

So to make this simpler; I am watching a stopwatch whilst the counter is counting according to the prescribed rules. When the stopwatch reaches 60 I look at the counter. What digit does it show?

If time is infinitely divisible, the counter would go up to infinity. — Lionino

The counter resets to 0 after 9. It will only ever show the digits 0-9.

No mathematical thought experiment can determine the nature of reality. — fishfry

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction.

You can argue that reality allows for the possibility of contradictions if you want, but most of us would say that it is reasonable to assert that it doesn't.

So to make this simpler; I am watching a stopwatch whilst the counter is counting according to the prescribed rules. When the stopwatch reaches 60 I look at the counter. What digit does it show? — Michael

I imagine the counter would be spinning at a near-infinite speed by that stage, making it very difficult to read.

The counter stops after 60 seconds.

Because 60 seconds will pass. I don't understand the problem you're having. The passage of time does not depend on what the counter is doing. — Michael

You just reaffirmed the same contradictory statements. It's impossible, by way of contradiction, that the counter can do the assigned task, and 60 seconds can pass. I see the contradiction as very clear and obvious, so I do not understand why you can't see it as contradictory.

The counter, by the prescribed specifications, is designed so that 60 seconds cannot pass until the counter counts every logically possible fraction of a second. Since logical possibility is defined by convention, and the convention allows for an infinite number of possible divisions, the counter, by the prescribed specifications, cannot finish the task. Therefore 60 seconds cannot pass. Your insistence that it can is blatant contradiction.

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction.

You can argue that reality allows for the possibility of contradictions if you want, but most of us would say that it is reasonable to assert that it doesn't. — Michael

Why the double standard? When speaking to fishfry you readily acknowledge the contradiction. When speaking to me, you insist that the counter can perform the supertask and 60 seconds of time can also pass, as if there is no contradiction involved.

I'll repeat what I said to andrewk above:

There are some who claim that a supertask is possible; that if we continually half the time it takes to perform the subsequent step then, according to the sum of a geometric series, an infinite sequence of events can be completed in a finite amount of time.

Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible. Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility.

With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.

You seem to take issue with that first paragraph, but your reasoning against it doesn't make any sense. Unless the universe ceases to exist then 60 seconds is going to pass. The passage of time does not depend on the counter.

With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed. — Michael

I think that's a sensible direction.

But does that imply necessarily that time and or space in our universe must be discrete and not continuous?

But does that imply necessarily that time and or space in our universe must be discrete and not continuous? — flannel jesus

If continuous space and/or time entail that supertasks are possible and if supertasks are not possible then space and/or time are not continuous.

I can't tell if that's a "yes" or more of a "I don't know"