Sadly, I am not good enough at maths and logic, so I can't post valid or interesting comments regarding this paradox. What I try to defend is that what keystone wrote is actually a paradox. — javi2541997

I think that if you're not good at maths and logic, I would think that you might not be in a good position to know if this is a valid paradox or just straightforward nonsense. If it is a paradox at all, it would only be in a mathematical sense. Surely not EVERYTHING contradictory is a paradox.

I think that if you're not good at maths and logic, I would think that you might not be in a good position to know if this is a valid paradox or just straightforward nonsense — flannel jesus

Okay. Let's leave it at that. Fair enough. I will not continue with posting on this thread.

I'm flattered that you replied. It was just a bit of fun and I expected to be totally ignored or possibly reprimanded. But since you have replied.....

Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1. — keystone

That's a complicated remark, because the numbers assigned are assigned in a specific context. If the staircase existed in the way that a physical staircase exists, the steps can easily be re-numbered in the new context (wanting to go up, rather than down). In that context, the first step up is numbered, even though it would not be numbered 1 in the context of going down. I think I recognized the problem when I said:-

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. — Ludwig V

My conclusion in the light of what you say is that the staircase up is not the same as the staircase down.

What I was wrong about was how the staircase is "created". The person going down does not create the staircase. We (the readers) (or, if you prefer, you, the writer) create the staircase. But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. So my advice should not be given to the person going down but to you and your readers.

Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. — keystone

I wouldn't say that. If you read Aristotle's Physics, you'll see that he describes the principal definition for "time" being used at his time, as a sort of number, used for measurement. That, he distinguished from a secondary sense, as something measured. So the secondary sense might be consistent with what you say, but it's not the primary definition for "time" at that time.

No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. — keystone

But the end is not reached. According to what is stipulated Icarus always has more steps before a minutes passes. Here's what I wrote in the other thread:

"For Icarus a minute cannot pass because he always has steps to cover first, just like Achilles cannot pass the tortoise for the same reason. Maybe if we call the staircase a line, and the steps are "points" it would make more sense to you. No matter where Icarus stands on the prescribed line, he has to cover an infinite number of points before a minute can pass. And to traverse each point requires a non-zero amount of time. Therefore no matter where Icarus is on the line (stairs), there will always be time left before a minute passes. A minute cannot pass, and Icarus' journey cannot end."

You only claim that the end is reached, because you assume that a minute passes. But that assumption is not provided for, as explained. Therefore, the valid conclusion is that Icarus never reaches the bottom, just like Zeno concluded that Achilles never surpasses the tortoise. The required premises about space and time, to conclude otherwise, are not provided.

I suggested that movement was discrete, not that space was discrete. — Michael

This is a very good point, to keep in mind when we get a good understanding of, and move beyond, the appearance of paradoxes like Zeno's. What is indicated is that motion is discrete. To account for the reality of this, we need to model either space or time as discrete, but not necessarily both. So when we employ the concept of "space-time" we deny ourselves the capacity of separating time from space, and considering the possibility that one is continuous, and the other is discrete.

To me, what works best for understanding the nature of reality is to allow for a continuous time, with a discrete space. This means that the so-called "quantum jump" is a feature of space. As time passes, the things which we know as having spatial (material) existence change in discrete jumps. Those discrete jumps limit our empirical capacity to understand the true nature of time, because no change can be observed to occur during the time which passes in between such jumps. However, since time would be conceived as continuous, while spatial motion is discrete, we can still conceive of time as passing in between such jumps.

Those premises allow us to understand the "immaterial realm of Forms", as the activity which is occurring between the spatial jumps. The immaterial realm of Forms is what determines how the spatial world will be, at each moment of ("observable") time. The human being as a free willing agent, has causal power within the immaterial realm of Forms, to influence what will be spatially present at any moment as time passes. Notice the requirement of real points in time. These are the points when the observable (spatial) world is materialized as discrete "jumps" during the passage of time.

The issue we have is that if there is no smallest unit of time then the counter is metaphysically possible, but this entails a paradox as the answer to what the counter shows after 60 seconds is undefined yet the counter will show something after 60 seconds. Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael

I don't think this is correct, I think Andrewk is correct. The counter is not programed to reach 60 seconds, that is outside its described capacity. Here's what you said:

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. — Michael

.
See, the counter is not programmed to show anything after 60 seconds. "There is no answer", because if the counter were to do as prescribed, it would never get to 60 seconds. There is no such things as 60 seconds for the counter, it can never get there if it does what its supposed to do. The assumption that 60 seconds will pass, is the mistaken conclusion of the OP, because this requires a premise about time which is not provided in the example. That premise being that time will pass at some rate which will surpass the actor in the example. That's what creates the appearance of paradox, if you allow yourself to be influenced by the contradictory idea, that time will surpass the actor.

The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. — Michael

This is the mistaken assumption. There is no limit prescribed. The premises set up an infinite process, which means "no limit". The issue exposed now, is that training in mathematics (calculus specifically) inclines one to see the scenario as a limit at 60 seconds, when the example simply does not state it that way. Someone who knew no complicated mathematics, only simple arithmetic, would work through the prescribed process, adding up the periods of time, and then realize that the sum gets closer and closer to 60 seconds, without at all thinking that 60 seconds is "a limit" here.

This imaginary "limit" is added by the mathematical way of looking at the scenario. It is "the intent" in producing the example. Start with the limit, and set something up which approaches the limit but does not reach it. The limit though, is not part of the example, it was only employed by the mind which produced the example, as a guiding principle which does not enter into, or become part of the example. In other words, we need to read the premises exactly as they are written, and there is no mention of 60 seconds as a limit. It's only when you take the short cut, don't read, but jump to the end, the intent, that you think of 60 seconds as a limit.

60 seconds will pass in the universe. The counter is just one thing that exists in the universe and it changes according to the prescribed rules.

So given the prescribed rules, when the universe is 60 seconds older, what digit will the counter show?

60 seconds will pass in the universe. The counter is just one thing that exists in the universe and it changes according to the prescribed rules.

So given the prescribed rules, when the universe is 60 seconds older, what digit will the counter show? — Michael

Yes, that is the point. Your expressed conceptualization "60 seconds will pass in the universe" is not consistent with the conceptualization prescribed by the OP. But this conceptualization "60 seconds will pass in the universe" is not part of the example. So your introduction of it is not valid. We could call it an equivocation fallacy. We have "time passing" as prescribed in the example, and "time passing" in the universe. The two are not consistent. To introduce the latter into the example, is to equivocate.

The obvious point is that we can describe a scenario which is logically possible, but physically impossible. When working with this scenario, we need to bare in mind, the fact that it is physically impossible, and adhere rigidly to the logic of the scenario, only. If one's mind gets influenced by other principles, such as what is physically possible in the universe, and what is going on in the physical universe, this will surely create confusion.

I don't understand what you are saying.

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.

What digit does the counter show after 60 seconds?

Your suggestion that the above entails that 60 seconds won't pass makes no sense.

Infinite geometric progression with first term 30 and ratio 0.5.
The sum is $\frac{30}{1-0.5}$. The counter is the number of members of the GP, to reach 60 seconds the counter must go to infinity. But in reality the counter, no matter how powerful, will just break the closer we approach the 60s mark.

As t→60, probability of the counter breaking goes up.

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. 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).

Bernadete's Paradox of the Gods: — Michael

Ah, thank you for that. I sort of remembered the story but not the name/author.
It seems far more paradoxical than Zeno's thing since motion is prevented despite the lack of any actual barrier.

It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.

But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).
But it doesn't follow that the journey thus cannot start, since clearly it can. By such a method, one can count from negative infinity to zero. You just need to not take some minimum time to do a given count.

The solution, similar to my proposed solution above, is that movement is not infinitely divisible

Mathematically it is, and mathematics seems to have no problem with it. Yes, I believe certain axioms must be accepted, but I'm no expert there.
As for physics, the assertion that motion is infinitely divisible seems to be a counterfactual assertion, not necessarily false, but unjustifiable. Such is the nature of quantum mechanics. But a journey can begin in either case, whether or not motion is continuous. Zeno does not illustrate otherwise in my assessment.
Zeno did his thing well before QM made us all question our classical notions of motion, so we can for the sake of argument make classical assumptions for this topic. If there's a limit to divisibility, then the problem goes away since there are finite steps.

If movement is continuous then an object in motion passes through every marker in sequential order, but there is no first marker, so this is a contradiction. — Michael

I don't find that to be a contradiction.

The false premise for Zeno is that each distance, and each time period will always be divisible. — Metaphysician Undercover

OK, if you deny the continuous nature of both space and time, then the number of iterations is finite, and the argument falls apart. My arguments presume a more mathematical interpretation: the continuous nature of both. If space is discreet, Achilles passes the tortoise after finite iterations. There would be a last one, after which the tortoise is passed. The conclusion of the inability to overtake doesn't follow because the premise upon which it is based becomes false.

Your assumption of discrete space is interesting, given that space (and everything else) is abstract to you, and thus any abstract space can be halved.

In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. — keystone

Presentism is still presentism even if time is continuous. You seem to describe a discreet view there, which runs into problems.
I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. Eternalism doesn't resolve the problems with any of them.
I was unaware of Zeno's 'eternalist' leaning. Yes, the term didn't exist back then (not until perhaps the 11th century)

n this comprehensive perspective, motion is impossible. — keystone

Block view also defines motion as change in position over time, and thus motion is very much meaningful under the view.

rip from 0 to 1-I don't get it. — keystone

All these are trips from beginning to end. Zeno's initial state (0) to the point where the tortoise is passed (1). In your OP, 0 is time zero, and 1 is time 1-minute.

Yes, that is the point. Your expressed conceptualization "60 seconds will pass in the universe" is not consistent with the conceptualization prescribed by the OP. But this conceptualization — Metaphysician Undercover

This seems to contradict yourlelf. You say time is discreet, in which case the number of digit changes is finite, and there is an answer. You also seem to deny that the sum of the converging series is not 1, or that time somehow is obligated to stop, which is the same thing.

Michael: The output of the counter is undefined. I can think of no better answer than that.

But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).

But it doesn't follow that the journey thus cannot start, since clearly it can. — noAxioms

It does follow that the journey cannot start. Therefore given that the journey can start then the premise that there is no first division is false. It's a proof by contradiction.

As such there is some first division and so movement is discrete.

By such a method, one can count from negative infinity to zero. — noAxioms

Given that each division is some ${1\over{n}}m$ then such a movement is akin to counting all the real numbers from 0 to 1 in ascending order. Such a count cannot start because there is no first real number to count after 0.

Let's first remember the fact that the limit of a sequence isn't defined to be a value in the sequence.

Re : The Cauchy Limit of a Sequence

"When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others"

A converging sequence might eventually settle on value equal to its limit, but even then the two concepts are not the same. So it doesn't matter whether we are talking about Thompson's Lamp, or merely a constant sequence of 1s. In either case, a limit, if it exists, doesn't refer to any position on the sequence, rather it refers to a winning strategy in a type of two-player game that is played upon the "board" of the converging infinite sequence concerned.

So it make no literal sense to consider the value of an unfinishable sequence at a point of infinity, so the meaning of a "point at infinity" with respect to such a sequence can at best be interpreted to mean an arbitrary position on the sequence that isn't within a computable finite distance from the first position. In the newspeak of Non Standard Analysis, such a position can be denoted by a non-standard hyper-natural number, meaning an ordinary natural number, but which due to finite limitations of time and space cannot be located on the standard natural number line.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). Whereas solutions to Zeno's Paradox tend to start by accepting (iii) but reject the assumption that motion can be analysed in terms of a countably dense linear order of positions, either by denying (i) (namely the assumption that the sequence of positions is infinite, which amounts to a denial of motion) or by denying (ii) (namely the assumption that motion can be used to count positions, for example because the motion and position of an arrow aren't simultaneously compatible attributes).

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). — sime

I didn't think he proposed a solution. Rather, it was an example to show that supertasks are impossible.

It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

didn't think he proposed a solution. Rather, it was an example to show that it is impossible to complete a supertask. — Michael

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence.

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence. — sime

True, but that's only part of the issue.

If after 30 seconds he's flipped the switch once and if after a further 15 seconds he's flipped the switch a second time and if after a further 7.5 seconds he's flipped the switch a third time, and so on, then it would suggest that a supertask can be completed in 60 seconds.

So if a supertask can't been completed in 60 seconds then the time between each flip cannot continually decrease. At some point no further division is metaphysically possible.

Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone

There is a contradiction in the stated scenario: there's an END to the ENDLESS staircase. Better to ask where he is after a minute.

Assess progress after each step he takes by noting the number of steps yet to be taken: there are always infinitely more to take. So at no point does he actually make progress - even after traversing infinitely many steps because that relation holds at all points along the way.

The answer to all those paradoxes is that you haven't defined what happens at the limit.
— fishfry

I think this is a misrepresentation. The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael

No misrepresentation. And why must something happen at the limit? Take this mathematical example.

We work in the space (0,1), the open unit interval of real numbers. It excludes the endpoints.

We consider the sequence 1/2, 3/4, 7/8, 15/16, 31/32, ...

Clearly this sequence has the limit 1 ... except that 1 is not in our space. So this sequence has no limit. Such a sequence is called a Cauchy sequence. It's a sequence that should "morally" converge, whether its limit happens to be in the set of interest or not.

Say the Thompson lamp is turned on at 1/2, off at 3/4, on at 7/8, and so forth.

Why on earth must there be a behavior defined at the limit? In this case there is no limit because 1 is not in our set.

But now do the same thing, but in the closed unit interval [0,1], which does include its endpoints.

In this case the limit, 1, is defined and exists. But still, the behavior of the lamp is not defined at 1.

That's the point. There's no paradox. You've simply neglected to tell me what the lamp does at 1, and you're pretending this is a mystery. It's not a mystery. You simply didn't defined the lamp's state at 1.

Does this example better explain that the the "paradox" is simply that you're arguing over what is the state of the lamp, when the state of the lamp is undefined?

How about if we defined the state of the lamp as turning into a swordfish at 1. Then that's the answer. It's on, off, on, off, ... at each point of the sequence, and a swordfish in the limit. There is no contradiction and no mystery.

Also note that the sequence 1/2, 3/4, 7/8, ... is order-isomorphic to the sequence 1, 2, 3, 4, ...

And if we include the limit, then 1/2, 3/4, 7/8, ..., 1 is order-isomorphic to 1, 2, 3, 4, ... $\omega$ as I described earlier. From the standpoint of order theory, they have the same order.

People who have trouble imagining that we could reach a limit after counting the natural numbers, would have no trouble agreeing that 1 is the limit of 1/2, 3/4, 7/8, ... But those two situations are identical with respect to their order properties. Now we never "reach" a limit, which is another phrasing that confuses people. We don't reach the limit, but the limit exists.

Why on earth must there be a behavior defined at the limit? — fishfry

By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off.

That's the point. There's no paradox. You've simply neglected to tell me what the lamp does at 1, and you're pretending this is a mystery. It's not a mystery. You simply didn't defined the lamp's state at 1. — fishfry

We're being asked what the lamp "does at 1", so you saying that we must be told what the lamp "does at 1" makes no sense.

Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible.

The paradox is resolved by recognising that the premise is flawed.

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

I guess I misunderstood your example. It is obviously not consistent with the OP. That little part where you say "and so on for 60 seconds" is unclear. The OP lays out the conditions in steps, but your "and so on" tells me very little.

What digit does the counter show after 60 seconds? — Michael

The problem set-up, which gives the axioms of the system we are working with, does not provide enough information to decide.
Hence the statement, for any numeral n, that:
"The counter shows digit n at 60 seconds"
has the same status in this logical system as the Continuum Hypothesis has in ZFC set theory. That is, we can adopt the statement as an axiom, and the system remains consistent (ie no contradictions arise). And we can assert its opposite ("the numeral showing at 60 seconds is not n") and the system remains consistent.

I think where people tie themselves in knots on this is that they feel they should be able to use things we know about our world to reason their way to an answer. But they can't, because we threw away that possibility when 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. So, having thrown away everything we know about our world, all we have available to use for our deductions are the axioms given in the set-up, and those axioms are consistent with the answer being any one of 0, 1, 2, ... or 9.

So I assert the answer is 2, without proof (because proof is impossible), with complete confidence that nobody can prove the assertion wrong.

It does follow that the journey cannot start. — Michael

This seems to be an assertion, not a logical consequence of the premise. In fact it leads to a contradiction of the premise, hence demonstrating that the journey being able to start very much does follow from the premise, unless you can also drive that to contradiction, in which case the premise has been shown to be false.

Therefore given that the journey can start then the premise that there is no first division is false.

I swear you changed this. You had something that logically followed from your assertion. The conclusion that movement is discreet contradicts Zeno's premise that "That which is in locomotion must arrive at the half-way stage before it arrives at the goal". So by contradiction, the journey not being able to start doesn't follow from the premise.

If movement is discreet, there is a finite number of steps. The first (smallest possible) step cannot be divided, and the inability to do so violates Zeno's premise.
As for Bernadete's Gods, there would be a first God and there would actually be a barrier preventing motion. No paradox at least. That is decent evidence that Zeno's premise is false. But suppose space and time is continuous. Then the journey can start without contradiction, unless you can find one. You say above that it implies a first division, but nobody has suggested that such a journey must begin with a first segment. It only needs to take finite time (1 unit in this case). The simple example is me going from here to there, which you apparently assert is impossible if space/time is continuous. A bold assertion.

Given that each division is some 1/n then such a movement is akin to counting all the real numbers from 0 to 1 in ascending ordering. Such a count cannot start because there is no first number to count after 0. — Michael

No, the reals are not countable. The example we've been using is. There is no final count of steps in Zeno's dichotomy, so there is no demonstrated requirement of a 'first step' or any kind of final count of steps. Insistence otherwise seems to be leading to contradictions.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible — sime

Applying this to Zeno's cases, or to the OP: All three seem to be true. I disagree that only two can be.

OK, the OP has infinite length to deal with, but finite time to do it, which is just a different way of expressing the same mathematics, totally discarding physics.
Zeno doesn't violate physics. If space/time is continuous, then the number of steps is countably infinite, and it is exhausted in finite time, as illustrated by my ability to move and/or to overtake something slower. Zeno makes no mention of 'point at infinity'. The OP kind of does ('bottom of it'), but also doesn't since there's no 'bottom step' apparent from the post-1-minute state. The poetry obfuscates what's actually going on, so I mostly am in denial of Zeno's conclusions.

If I overtake the tortoise, I have also reached the 'bottom of the supertask', so I don't find the wording necessarily contradictory.


What am I missing? A formal proof that it leads to contradiction would be nice, but all I seem to get is assertions.

By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael

The lamp violates the laws of physics, so it's not a real lamp. It's only a metaphor for a mathematical puzzle. Why can't it turn into a pumpkin at midnight, like Cinderella's coach? What rule of the puzzle constrains a light, which can not physically exist, to be on or off at a time when its state is not defined?

Consider this mathematical variant.

In [0,1], the closed unit interval, we start with the sequence 1/2, 3/4, 7/8, ...

We have a function f on the sequence such that f(1/2) = 1; f(3/4) = 0, f(7/8) = 1, and so forth, alternating between 0 and 1.

What is the value of the function at 1? Well clearly, that value is not defined. It could be 0, 1, 47, or the Mormon Tabernacle Choir.

Until you tell me what is f(1), you're playing a silly game to ask me what it should be. It can be anything you like.

We're being asked what the lamp "does at 1", so you saying that we must be told what the lamp "does at 1" makes no sense. — Michael

It makes perfect sense, once you replace the lamp with a function on a sequence. The lamp is a red herring. No circuit could switch that fast. It's not a real lamp and there is no reason for it to be in any particular state where that state has not been specified. It's not a real lamp so it is not limited to be on or off.

Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? — Michael

Is f(1) = 1 or 0 or 47?

If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible. — Michael

It's not a real lamp.

The paradox is resolved by recognising that the premise is flawed. — Michael

The premises of the mathematical version are perfectly sensible, as is the answer I gave.

STAIRCASE PARADOX

It's always only a finite number of steps from infinity back to zero — fishfry

This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution.

You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus

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.

But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. — Ludwig V

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?

But the end is not reached. — Metaphysician Undercover

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

ZENO'S PARADOX

I suggested that movement was discrete, not that space was discrete — Michael

Consider linear motion. If you plot position against time, are you suggesting that the resulting curve, when examined closely, appears stairstepped rather than smooth? If that's the case, what would be the width of these incremental steps? This presents the same issue, as I could always plot a more accurate curve of motion using even smaller incremental steps.

I wouldn't say that. — Metaphysician Undercover

This response does not adequately address my reinterpretation of Zeno's ideas.

I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. — noAxioms

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. In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and future—akin to a film strip. Which view do you think is more reasonable? Of course, this raises profound questions, such as why we experience time's flow, but that discussion is for another thread.

Let me reframe Zeno's argument in different terms more relatable to a modern audience. 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). Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time. Instead, modern Zeno would suggest that the entire timeline already exists as a block universe, and our experience is merely about observing different parts of it, similar to making cuts in a string. 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. Each cut still leaves a segment of string, however minuscule. This introduces new challenges (for which there are answers) but as it relates to the discussion at hand, the eternalist perspective reframes the impossibility of supertasks from an unacceptable notion—that motion itself is impossible—to a more acceptable one—that observing every instant in history is impossible. This essentially echoes Aristotle's proposal, but it is only in the quantum era that such a solution becomes truly acceptable.

For only two of the three following premises can be true of a sequence: i) The length of the sequence is infinite. ii) The sequence is countable iii) The sequence is exhaustible — sime

The issue arises if Achilles toggles Thomson's Lamp with each stride, leading to a contradiction: his feet suggest that the sequence is exhaustible, but his hand indicates it is not.

SINGLE DIGIT COUNTER PARADOX

Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael

First, instead of using decimal, let's switch to binary, where the counter can only be 0 or 1. You suggest that quantum mechanics resolves this by introducing indivisible units, perhaps akin to Planck time. Looking to QM for inspiration is a good idea. However, the idea of Planck time doesn't hold up because in the abstract realm, we can always conceptualize a smaller increment. I propose that the correct solution is that at 60 seconds, the counter is in an unobserved state where its status fundamentally remains unknown. It could be either 0 or 1, so let's say it's in a state of (0 or 1). If we wish to steal technical terms from QM, we might refer to this state as being in superposition.

The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael

What if the undefined state is fundamentally unobservable? This raises the question similar to "If a tree falls in a forest and no one is around to hear it, does it make a sound?" The limitations I'm suggesting on observation should not be surprising to a generation that has grown up in the era of quantum mechanics.

It is metaphysically necessary that there is a limit to how fast something can change — Michael

Yet, it's impossible to determine what this limit might be. Would you argue that there is a limit to the slope of a line?

THOMSON'S LAMP

Why on earth must there be a behavior defined at the limit? — fishfry

Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? After all, the narrative doesn't specify what his calculator must indicate at 60 seconds. It seems to me that you're contesting the very idea which you support - that infinite series can have definitive sums.

By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael

Yeah, that law needs updated. I propose "for every proposition, either this proposition or its negation can be measured to be true." This introduces the possibility of a third, unmeasured state—when we're not observing, the lamp could either be on or off, placing it in a state of being (on or off).

STAIRCASE PARADOX

The "ground", thus defined, is a point that cannot be reached from the stairs, being infinitely far below it. Similarly, you cannot reach the stairs from that point, as every stair is infinitely far above it. That's why the man on the "ground" can't see any stairs as described in the OP story. They are all too far away above him. By making such a definition, we are essentially dividing our thought-experiment-world into two parts, neither of which can reach the other. — andrewk

You are effectively arguing that supertasks cannot be completed since if he cannot reach the ground, he must still be on the stairs.

This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution. — keystone

I gave the example of the first transfinite ordinal $\omega$. Any step you talk backward from it lands you on a natural number, from which it's only finitely many steps back from zero.

This is a perfectly well known mathematical fact. See Asaf Karagila's answer here. It's always only finitely many steps back from any ordinal, even uncountable ones.

https://math.stackexchange.com/questions/3980267/infinite-strictly-decreasing-sequence-of-ordinals

Let me use the same example I gave earlier. In the closed unit interval [0,1], consider the infinite sequence 1/2, 3/4, 7/8, 15/16, ..., which has the limit 1.

Suppose we start at 1 and take a tiny tiny tiny step to the left, as small as we like, as long as we land on an element of our sequence. Then you can see that no matter how small a step you tak, you will land on some element of the sequence that is only finitely many steps away from the beginning of the sequence at 1/2. Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... $\omega$. 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.

Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? — keystone

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.

That's the problem with all these puzzles. You take a situation that's mathematically straightforward, and you add in lights that flicker faster than the laws of physics would allow, and calculators to operate faster than the laws of physics would allow, and you try to reason sensibly on partial information. If a light can flicker faster than the laws of physics allow, what else can it do? What are the laws of physics in this made up universe?

Remember, Cinderella's coach turns into a pumpkin at the stroke of midnight. And for all we know, so does your hypothetical calculator. Because first, we already know that it doesn't follow the known laws of physics. And second, you've only told us what it does at each natural number step 1, 2, 3, ... You haven't told us what it does in the limit. So I say it turns into a pumpkin.

Can you prove me wrong? No, because 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.

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?

Mathematically it has some meaning, but it never has physical meaning, as several have pointed out. — noAxioms

Yes. With a real staircase would exist in both contexts and independently of both of them. Then the first step down is the last step up and the last step down is the first step up. But the last step down is not defined, which means it can't be reached. That's why the game is fascinating and frustrating at the same time, even though it is what I would call, arbitrary.

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 — keystone

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.