My point is that the stairs are countably infinite. Consequently, they COULD be counted, if we were traversing them. — Relativist

Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence?

I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction. — Relativist

That's true. What puzzles me is why they are not dismissed out of hand. Someone earlier described them as fairy stories, and the writers seem to be able to wave a hand and create impossibilities, which would be magic, so that description makes sense. But it seems to me more like an illusion and the problem is then to understand how that illusion works.

Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world — Relativist

Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world?
More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?

The counter, with it's supertask has one way of counting out time, by dividing seconds into shorter and shorter increments, while the stopwatch is designed to measure an endless procession of seconds. The two are incompatible. — Metaphysician Undercover

Yes. How come anyone can't see that? Since the difference is the difference between simple addition and division followed by addition, I think it is then possible to see how people can be misled into thinking they are compatible - even that they must be compatible.

there are no seconds unless measured out — Metaphysician Undercover

Yes there are. A second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom". This occurs even if we don't measure it.

Yet again you can't seem to get beyond our use of labels to understand that our labels refer to things that exist and do things even when we're not around.

My point is that the stairs are countably infinite. — Relativist

Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps.

[/quote][The physical process of descending stairs] fits this definition:
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."[/quote]Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask.
We seem to have lost @keystone, and the stairs thing was his. I prefer Zeno's scenario which doesn't seem to be plausibly physical so long as we take a classical continuous view of both time and space.

The article discusses the issue

It does, I'm quite aware. Just not in Zeno's argument.

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...

I pretty much quoted exactly Black's remarks just above. Yes, the task is not complete by this finite definition despite every step having been taken, and that final step must be taken for your counter to have a defined value after a minute.

The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete.

Again, the stairs is utterly abstract. There's no kinematics to it. Not so with the tortoise. I can pass the tortoise, thus completing (by the 'all steps' definition) the supertask.

The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.

How does the abstract mathematics not account for the physical ability of me passing the tortoise?

PSA:
The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.

I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all.
I cannot take a bite of an apple unless perhaps the bite taking is completed by the performance of the taking of a bite? Presumably the same bite??

Kindly translate. Perhaps it points out some error I'm making, but only if I can parse it. I can come back to your statement and respond more intelligently.

That's what I see going on with the posters who focus only on the mathematical series.

I'm trying to focus on the completion of all tasks and not on the measurement of a nonexistent value.

I agree we can't treat infinity as a number, and haven't suggested you should.

But I think you have. Your attempted counter (or the color change thing in the recent post) treats it as a number, and suggests taking its modulus relative to base 10 or 3. What is the lowest digit of the number of the final step? If there is no such number, then the output of your scenario is undefined, which is very differnt from the digit counter displaying a value of 'undefined', or an undefined lamp state somehow violating the law of excluded middle by being in some state between on and off.

But for the supertask to be meaningful, you have to identify where infinity fits in the kinetic task description. I'm saying it entails a never-ending sequence of tasks. Identifying the limit doesn't make this disappear.

Infinity means unbounded, which means there is a physical location and time interval.for any task n That's what makes it meaningful, and it only works if physicality is presumed not discreet.
Also, there is no violation of physics like faster-than-light movement as suggested by the OP.

I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction.

I can pass a tortoise without contradiction. That shows that at least one of three (two explicit, one implicit) premises are false. But it doesn't necessarily have to be the premise you just mentioned there, that supertasks are impossible.

You seem to be avoiding the contradiction by ignoring the incompleteness of the infinitely many kinematic steps. The presence of the contradiction implies supertasks are logically impossible (not merely physically impossible).

I'm ignoring it because those contradictions arise from a 4th premise (that there is a final step), one which I don't accept.

What puzzles me is why they are not dismissed out of hand. — Ludwig V

Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics?

A white box turns red when the Earth completes a half-orbit, turns blue when it completes another quarter-orbit, turns back to white when it completes another eighth-orbit, and so on.

What colour is the box when the Earth completes its orbit around the Sun? — Michael

Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'.

If someone would explain to me, in a way which makes sense, a better perspective, then I'd happily switch. — Metaphysician Undercover

Michael did very nicely with his first line in his reply.

Your reading comprehension skills are also off. I never suggested converting you to some opinion other than the one which you hold. I simply suggests that you seem incapable of understanding alternatives, to the point where you don't understand people who presume one of these alternatives.

Your most recent reply demonstrates this, as does the frustration evident in Michael's reply just above.
Yes, it can be explained in a way that makes sense, but apparently only to others.

Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'. — noAxioms

And so it is meaningless to claim that such a supertask can complete. The fact that we can sum an infinite series is a red herring.

Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence? — Ludwig V

Correct.

That's true. What puzzles me is why they are not dismissed out of hand. — Ludwig V

I think it's because they are interesting puzzles, and because they help teach certain concepts.

Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world? — Ludwig V

Yes- that's a better way to describe it.

More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?

The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable.

I think it's because they are interesting puzzles, and because they help teach certain concepts. — Relativist

I think they are interesting because they dangle the prospect of completing a task and persuade us to ignore the reality of the impossibility of the task.
It is rather like a lottery. A lottery ticket is sold by dangling the prospect of a big win without any noticeable effort; some people focus on that and ignore the probabilities. The way the proposition is presented makes a difference to the view that people tend to adopt.

Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics? — noAxioms

The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable. — Relativist

Maybe I've misunderstood what a supertask is. Are there not different kinds of cases?
In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed. I mean that we know that Achilles will pass the tortoise and even calculate when with simple arithmetic (no infinities required). But then the same problem, presented in a different way, seems to suggest that it cannot. So it's a question of how you choose to present the problem.
The staircase is different. It gives us a task (going down the infinite stairs) that cannot be completed, but links completion of the task to another process which also cannot be completed, but has a limit.

Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps. — noAxioms

We can assign those numbers as we take each step. That's counting, and it's perfectly meaningful.

Perhaps you mean there's no way to say we can meaningfully complete the counting of all the steps. That's true, but it seems to be contradicted by the fact that the infinite process completes before the 1 minute mark. .

Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask. — noAxioms

OK, but speed of light limitations put a physical limit on how fast the stairs can be descended, so that it eventually becomes physically impossible to descend a step in the prescribed period of time. The minimum size limitation also relates to a physical impossibility. But I'm making the stronger claim that it is logically impossible.

Relativist:"The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete."
Again, the stairs is utterly abstract. There's no kinematics to it. — noAxioms

The entire exercise is abstract, but the scenario is written in terms of the kinematic (not abstract) process of descending stairs: each step is a motion, taking place in a finite amount of time.

PSA:The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.

I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all. — noAxioms

Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1). The term (sequence) is not referring to the entire collection. The task is to reach the bottom of the stairs (as stated in the description in the first post of this thread). Perhaps you can already see that it's trivial: it's actually impossible to reach the bottom of the stairs, since there is no bottom to a staircase with infinitely many stairs.

In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed. I mean that we know that Achilles will pass the tortoise and even calculate when with simple arithmetic (no infinities required). — Ludwig V

It depends on how the race is framed. It CAN be described as a supertask, wherein Achilles runs to a series of destinations, each established by where the tortoise is located when he begins each leg of the race. In that case, Achilles never actually reaches the turtle, he just gets increasingly closer. If you frame it in terms of constant speeds by both, then it's not a supertask - it's a different kind of puzzle.

And so it is meaningless to claim that such a supertask can complete. — Michael

The lack of a defined number for the last task does not prevent completion (by the all-tasks definition), so I regard your statement as a non-sequitur.
It does prevent completion if completion is defined as the removal of the green ball in the bag of a dozen non-green balls (see green ball example above), so I agree with you there.

Maybe I've misunderstood what a supertask is. Are there not different kinds of cases? — Ludwig V

Several here have been defining completion effectively as measuring the value of the final task, and that instance I suppose differs from Zeno's that specifies no such requirement.

In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed.[/quote]Only because he posits a second premise incompatible with the first. 1) Supertasks are possible (by demonstration). 2) Supertasks are impossible, a second premise that isn't in any way justified.

I mean that we know that Achilles will pass the tortoise

Well, keystone suggested that Zeno denies this, and M-U suggests that time somehow stops due to the offense we've given it. Anyway, I agree with you, but it requires that implied premise that empirical evidence is valid.

But then the same problem, presented in a different way, seems to suggest that it cannot.

This suggests fallacious reasoning in the second presentation. Most of the fallacies I've seen posted seem to be based on the premise of there being a limiting step. It's why I like Bernadete's Paradox of the Gods (see post ~30) which explicitly leverages the lack of there being a limiting step, and drives that to a seemingly paradoxical result. That's a harder one to wave off.

The staircase ... gives us a task (going down the infinite stairs) that cannot be completed

Just not physically. Mathematically it can, but then the story mentions 'the bottom' which implies something final that 'no more stairs' does not. So it lacks rigor.


The Littlewood-Ross Paradox illustrates an interesting way to treat infinities that highlights the dangers of treating infinities as numbers. From the SEP supertask page:
"We have a jar and a countably infinite pile of balls, numbered 1, 2, 3, 4, …. First we drop balls 1–10 into the jar, then remove ball 1. (This adds a total of nine balls to the jar.) Then we drop balls 11–20 in the jar, and remove ball 2. (This brings the total up to eighteen.) Suppose that we continue in this way ad infinitum, and that we do so with ever-increasing speed, so that we will have used up our entire infinite pile of balls in finite time. How many balls will be in the jar when this supertask is over?"

The answer is, as is argued by just about everybody: Zero. At any finite step n, there are n*9 balls in the jar. But after the supertask is complete, there is no final step Z with a state of Z*9 balls. In fact, every ball is numbered, and we know when it went in and when it went out. There is no exceptions to this, so the jar is empty at the end. Totally not intuitive, but not necessarily contradictory. Arguments against it have been attempted.

But I'm making the stronger claim that it is logically impossible. — Relativist

I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory.

PSA

Has always meant 'prostate specific antigen' to me. I get my PSA checked at least once a year.

Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1)..

OK, 'act' is a step (go half the remaining way to the goal). 'task' is a goal (pass the tortoise).
It makes sense now, thanks.

Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. I cannot buy into that PSA statement. It is just a rewording of the 'do the last step' definition of completion, a definition which only works for tasks requiring a finite number of steps.

Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. — noAxioms

Yes, the PSA entails taking a final step. We agree infinity is not a number, so there is no final step.

I cannot buy into that PSA statement.

Show the PSA is false.

But I'm making the stronger claim that it is logically impossible.
— Relativist
I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory. — noAxioms

Why? The claim is indeed justified by the necessity of a final step for completion. Simply denying a final step is necessary doesn't make it so - you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.

There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility. — Michael

Of course it's not a physical possibility.

If by metaphysical you really mean physical, then it's not a metaphysical possibliity.

But clearly we humans have the ability to conceptualize infinite sets and infinite processes, and we can even formalize the idea and get freshman calculus students to get a passing notion of the idea. If metaphysics includes abstract concepts created by humans, then infinite sets and mathematically infinitary processes are definitely part of metaphysics.

But it depends on what you mean by metaphysics. There is no doubt in my mind whatsoever that infinite sets, infinite sequences, and the theory of convergent infinite series have mathematical existence. Whether you include that in your metaphysics is up to you, but the mathematical existence of convergent infinite series is beyond dispute.

Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time). — Michael

You are now talking about a physical process. Of course we can not write out infinitely many terms of the series. That has nothing to do with the mathematical truth expressed as 1/2 + 1/4 + ... = 1.

Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity. — Michael

Nobody is writing anything down and this is not a physical process and you are entirely wasting your time trying to convince me that we can't physically write down an infinite series because I already know that.

If you want to say that supertasks are possible — Michael

In reality? In the physical world? No, I deny them entirely. It's tiresome to argue against your representation of positions I don't hold.

but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it. — Michael

I never claimed any such thing. I have no idea why you think I claimed any such thing. Supertasks can be defined abstractly, as in limiting processes. They are not physically instantiable as far as we currently know.

As far as the "final state," think of it as a function on the ordered set

{1/2, 3/4, 7/8, ..., 1}.

We can define a function any way we like. We can assign 1 to 1/2, 0 to 3/4, 1 to 7/8, and so forth.

We are then entirely free to define the value of the function at 1. We can call it pumpkin if we simply declare pumpkin to be an element of our output set.

There is no requirement that the value of a function at any point is required to be any particular thing. Functions are pretty much arbitrary. Just like Cinderella's coach. A coach at 1/2 second before midnight. A coach at 1/4 second before midnight. Dot dot dot. And then a pumpkin at the stroke of midnight.

It's a perfectly legal function. It just doesn't happen to be continuous. But it's perfectly legal to define a function that's a coach at each of infinitely many elements of a sequence, and then a pumpkin at the final limit point.

Mathematically it's just a function $f : \{\omega + 1\} \to \{ \text{coach, pumpkin}\}$

where $\omega + 1$ is just the order type of the set {1/2, 3/4, 7/8, ..., 1}.

Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval. — Relativist

I agree it's about a right-open interval. We have 1/2, 3/4, .. in (0,1). We can adjoin 1 to work in (0,1].

The limit of the series is "reached" only in the sense that we can reach a mathematical answer. — Relativist

But I'm not talking about anything else! This is purely a mathematical problem! There is no lamp that switches in arbitrarily small intervals of time. Adding time to this problem confuses the issue. It makes people think there's a physical component to the problem when there isn't. It's purely mathematical.

The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. — Relativist

There isn't any physical process to speak of. The lamp is fictional. Purely mathematical, a function on {1/2, 3/4, ...} with its completion defined in {1/2, 3/4, ..., 1}

Deriving the limit just means we've identified where the sequential process leads. — Relativist

It may "lead" somewhere but there's no law that constrains the final state. It may be discontinuous, like Cinderella's coach that's a coach at 1/2, 1/4, 1/8, ... seconds before midnight, then becomes a coach at midnight. That's why it's perfectly possible that the lamp becomes a pumpkin after 1 second.

In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? — Relativist

Lost me there, limit of what is infinity? If you put a symbolic "point at infinity" after the natural numbers and you define a function on the augmented set 1, 2, 3, 4, ..., $\infty$, you can define a function on the augmented set whose value at infinity is anything you like.

The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else. — Relativist

Kind of lost me here. The process 1, 2, 3, ... never ends, but we can still stick a symbolic point at infinity. Just like we can add the point 1 to the set {1/2, 3/4, 7/8, ...} to make {1/2, 3/4, 7/8, ..., 1}

Nobody is questioning the fact that 1/2 + 1/4 + ... = 1.

This is an example of a supertask:

I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.

According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.

Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.

It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote.

Show the PSA is false. — Relativist

The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal.
Either PSA is wrong or the premise is. In neither case is PSA valid for a supertask.

Simply denying a final step is necessary doesn't make it so

Simply asserting that such a step is necessary doesn't make it so, especially when it being the case directly violates the initial premise. That violation does very much demonstrate not only the lack of necessity of a final step, but the impossibility of it, given the premise.

you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.

I don't know how the task being 'kinetic' changes the argument. You can phrase it as a n inertial object overtaking a slower one in frictionless space.

Issues that I see: The problem is 1 dimensional as phrased: The position of Achilles is given only in x. To overtake the tortoise, he'd have to collide with it, so he has to be off to the side,. If he's off to the side, there's at least two axes x and y. If they're in 2D+ space, then which of the two is in front is dependent on the chosen orientation of the axes. If you hold the orientation stable throughout the exercise, then the scenario still holds as described.

None of that seems to have any relevance to your reqirement of a rephrasing around 'kinetic'. Why does that word somehow invalidate the premise?

Yes there are. A second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom". This occurs even if we don't measure it. — Michael

Nice try Michael, but "ground state" is an ideal which does not occur in nature. It's like a "blackbody" and things like that, ideals used for theory, which do not ever truly exist. Like "60 seconds" in the OP it is approached as a limit, but never truly achieved.

Besides, what you've provided does nothing to resolve the problem I explained. It's just the same as your example of "a year". The activity is what occurs, time passes. What you gave is the definition of "a second", that means that a second is the duration of time which is equivalent to that activity. A second is not that activity itself. So the activity occurs, and there are no seconds unless someone determines the duration of time required for the specified number of oscillations. That's why beginning and end points are required which constitute a measurement, just like I explained to you in the case of "a year".

Yet again you can't seem to get beyond our use of labels to understand that our labels refer to things that exist and do things even when we're not around. — Michael

I have no problem accepting that things exist even when we're not around, and I gave you examples of such things, dogs and cats. However, what you are not getting, is that some labels refer to things created by human beings, these are called "ideals". So, we have two categories of labeled things, natural things which exist even when we're not around, and things created by minds, such as ideals, which are dependent on minds. And, you refuse to distinguish between the natural things, and the artificial ideals. So you insist that because it is a labeled thing, it must exist even when were not around. That's the folly of Platonism. Do you not understand that "the second" is a mathematic object of ideal time, just like "the circle" is a mathematical object of ideal space?

Your reading comprehension skills are also off. I never suggested converting you to some opinion other than the one which you hold. I simply suggests that you seem incapable of understanding alternatives, to the point where you don't understand people who presume one of these alternatives. — noAxioms

I think you misunderstand. I understand the alternatives very well, so well in fact that I can comprehend the contradictions which inhere within some of these alternatives. So I see the need to reject them. The people who assert, and insist on some of these alternatives do so without proper understanding themselves. They accept and assert by the force of convention, which is simple prejudice, and they refuse to acknowledge the contradictions within, assuming that convention cannot be faulty.

I don't know what to tell you. 60 seconds can pass without anyone measuring it. If you can't accept this then we can't continue.

The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal. — noAxioms

Or the PSA is correct, and the goal can't be met.

Relativist: "Simply denying a final step is necessary doesn't make it so."
Simply asserting that such a step is necessary doesn't make it so — noAxioms

I'm not merely asserting it. You have to agree that a final step is necessary for completion when there are finitely many steps. Why would it matter if the number of steps is infinite?

Most importantly: What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer?

Relativist: "you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed."
I don't know how the task being 'kinetic' changes the argument. — noAxioms

Here's how: the infinity is manifested as a never-ending kinetic process.

Points on a number line exist concurrently (in effect). Steps in a kinetic process do not: they occur sequentially, separated by durations of time.

I'm going to defer commenting on the Achilles/tortoise problem. It just clouds the issue with the stairway supertask.

It depends on how the race is framed. It CAN be described as a supertask, wherein Achilles runs to a series of destinations, each established by where the tortoise is located when he begins each leg of the race. In that case, Achilles never actually reaches the turtle, he just gets increasingly closer. If you frame it in terms of constant speeds by both, then it's not a supertask - it's a different kind of puzzle. — Relativist

I'm not sure it is even a puzzle if it is framed in terms of constant speeds by both. Let's say Achilles gives the tortoise a head start of 100 units of length, that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds. (This was suggested to me by a friend.) It seems OK to me, but perhaps I'm wrong to think that it will generalize.
But I don't really want to pursue it. I just wanted to point out that a puzzle can be the result of framing the question in the "wrong" way. Everyone seems intent on resolving the staircase problem on its own terms, which seems to me a mistake.

Or the PSA is correct, and the goal can't be met. — Relativist

I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask.

Why would it matter if the number of steps is infinite? — Relativist

Because a contradiction results from making that additional assertion. In the example given, it is a very direct contradiction.

What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer. — Relativist

If the process continues forever, by definition it isn't a supertask. It's a different process than the one being discussed.
I don't have an answer because I don't understand what 'kinematic' adds to the issue.

Points on a number line exist concurrently (in effect). — Relativist

I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept.

Steps in a kinetic process do not: they occur sequentially, separated by durations of time. — Relativist

OK. I buy that. But this works mathematically as well, so 'kinetic' doesn't add anything. I can draw the worldlines of Achilles and the tortoise on some medium and all you get is two lines that cross at some point. The axes on the plot are x and t, so in this mathematical representation, the steps do not occur simultaneously, but are separate durations of time. What did 'kinetic' add to that?
I'm trying to understand your point about how the word somehow is relevant.

the Achilles/tortoise problem ... just clouds the issue with the stairway supertask. — Relativist

OK, this has been about the stairway. There is no objective kinematics about that since it involves a space-like worldline, so the steps are not unambiguously ordered in time. The ordering of the steps becomes ambiguous due to relativity of simultaneity, and it becomes meaningless to use the word 'sequential' in this context.

Hence my always referencing the tortoise example since it hasn't any physical ambiguities like that. There are still frame dependent fact, so for instance in another frame, it is the tortoise trying to overtake Achilles, both of whom are facing backwards.

Those are my thoughts on 'kinematics'.

Or the PSA is correct, and the goal can't be met.
— Relativist
I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask. — noAxioms

No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation.

Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.

You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal. Labeling the process a "supertask" is handwaving, not proof. Show your logic.

If the process continues forever, by definition it isn't a supertask. — noAxioms

Fair enough, I misstated it. The process does not continue forever, however there is no end to the process.

Let's compare the supertask to a scenario in which the time interval between each step is a constant (e.g. 1 second). You'll agree that this process does not complete, right? But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.

Points on a number line exist concurrently (in effect).
— Relativist
I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept. — noAxioms

My point was that the kinematic stair-stepping process has a temporal element that is not reflected in a number line.

The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process. The goal is therefore unreachable by the kinematic process.

No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation. — Relativist

1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)

You don't find this contradictory?

Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.

This shows that no specific halfway step reaches the goal, which is the same as saying that the goal cannot be reached in a finite number of steps.

It seems that every post seems to attempt finite logic on an unbounded situation. If you accept that motion is possible, there is a flaw in at least one of the premises.
— Relativist

You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal.

Yea, I do, don't I? I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. Attack them if you will. The do require some axioms that are not obvious, so there's a good place to start. Nevertheless, I can do more than just handwave, by several unrelated methods.

Demonstration that immediate contradictions arise from denying either of the premises or presuming your conclusion 3 is also more than just handwaving. For instance, given the usual scenario, where is Achilles at time t=1? If he's not at the goal then, then where else is he?

There are those that deny an object falling past the event horizon of a black hole by suggesting that 'time stops' in a somewhat similar manner that some posting here have suggested. But that's just an abstract coordinate effect (and the leveraging of finite logic). Change the coordinate system to one that isn't singular at the point of contention and the object falls in, no problem. Similarly, Achilles is stuck in an abstract sense due to a deliberate choice of coordinate system that is singular at the goal. The impediment is entirely abstract and not physical at all.

Per modus ponens, empirical observation shows that motion is possible, as is the overtaking of a slower object. One need not accept that empirical evidence (keystone attempted this avenue), but I choose to start with acceptance of empirical evidence. There are a few places where it is inappropriate to do so, and this isn't one of them.

Also, no impediment to the reaching of the goal has been identified, so in a similar way, your stance (what is your stance? Supertasks are nonexistent, even given continuous assumptions?) is also achieved by handwaving when it is not just being flat out contradictory. You do seem to heavily rely on definitions that come only from finite logic. A definition that is being leverage outside its range of applicability is

The process does not continue forever, however there is no end to the process.

There is a temporal end to it, a final moment if not a final step.

But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.

There is a bijection yes. It does not imply that both or neither completes.
This reminds me of some of the discussion behind Gabriel's horn, and attempting to suggest that its infinite area implies that it has infinite volume.

Yes, your example here very much illustrates how a deliberate abstraction can be made to be singular at any chosen point, in this case tying infinite time to a finite duration. Yes, this works even in uncountable infinities: There is a bijection between the space from 0 to 1 and the space from 1 on up, by the simple relation of y = 1/x. This in no way implies that 1 meter cannot exist.

The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process.

The 'process' can go beyond the end of the line despite it ending before the goal. This is sort of a different issue since you're putting an uncountable set of points between 0 and 1. Why not just 1/2, 1/4, ...

The goal is therefore unreachable by the kinematic process.

Disagree. The kinematic process isn't restricted to only points on the number line.

This is an example of a supertask:

I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.

According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.

Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.

It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote. — Michael

You're continuing to argue against a position I don't hold. Why are you doing this? There's no interesting conversation to be had. Supertasks are not consistent with known physics. We're agreed on that.

I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. Known physics changes all the time, sometimes radically.

You're continuing to argue against a position I don't hold. Why are you doing this? — fishfry

Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks.

I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. — fishfry

I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility.

No physical law can allow for an infinite sequence of events to be completed. The very concept of an infinite sequence of events being completed leads to a contradiction. To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense.

If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity.

Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks. — Michael

Ok.

I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility. — Michael

Now that's something I disagree with. But I don't care about supertasks much so it's better if I don't engage.

No physical law can allow for an infinite sequence of events to be completed. — Michael

This is an open question. Of course no physical law currently known allows for supertasks, but you can't say what we will regard as physical law in another couple of centuries.

The very concept of an infinite sequence of events being completed leads to a contradiction. — Michael

You keep repeating that, but you have no evidence or argument.

To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense. — Michael

Do you deny infinite mathematical sets?

If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity. — Michael

Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number.

I already agree with you that there are no infinite collections of physical objects according to currently accepted theories of physics. But you can't claim that there will never be any such theory.

And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity.

Do you deny infinite mathematical sets? — fishfry

No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them.

Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number. — fishfry

That's not relevant to the claim I'm making.

I'm saying that if I have finished counting the members of some set then some member must be the final member I counted.

And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity. — fishfry

I don't deny the possibility of something not ending. The issue is that supertasks entail that there is an end to infinity, which is nonsense.

You keep repeating that, but you have no evidence or argument. — fishfry

Thomson's lamp, my box changing colour, the example of writing out each natural number, etc. I've offered plenty. Your attempt to rebut them by reference to mathematical limits fails to address the issue.

1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)

You don't find this contradictory? — noAxioms

Of course it is, but the the contradiction can be resolved by denying either one of two premises. You chose to deny the PSA, and I responded that the PSA could be true - we'd merely have to reject the other relevant premise - that the goal is reached. You have not made an argument that shows it is more reasonable to deny the PSA than to deny the reaching of the goal. I don't think it make sense to deny that a completed task entails a final step.

Demonstration that immediate contradictions arise from denying either of the premises or presuming your conclusion 3 is also more than just handwaving. — noAxioms

Sure. You have to agree the PSA is true for finite tasks. Is there something different about infinite tasks? It doesn't seem so: consider the process: stepping increasingly closer to temporal point in time 1, but the process never actually reaches it. So the goal is unreachable by the process.

I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. — noAxioms

No need. I understand that the math shows that the series reaches a point of convergence at time 1. However: the kinematic process never actually reaches time 1. That's why the series doesn't adequately account for the kinematic process -and why I've stressed we need to examine the process, not just do the math on the mathematical series.

no impediment to the reaching of the goal has been identified, — noAxioms

On the contrary, there's a logical impediment to reaching the goal through the process: the process does not reach time 1.

You do seem to heavily rely on definitions that come only from finite logic — noAxioms

I'm actually basing my claims on real analysis, which analyzes the characteristics of real numbers - including the associated infinities.

There is a temporal end to it, a final moment if not a final step. — noAxioms

That makes no sense. The process does not have a final moment. because there are infinitely many moments prior to time 1. There is no end to the series of kinematic steps, in spite of the fact that the mathematical series converges.

Relativist: "But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes."

There is a bijection yes. It does not imply that both or neither completes. — noAxioms

Why not?

Relativist: "The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process."

The 'process' can go beyond the end of the line despite it ending before the goal. — noAxioms

No it can't - that is logically impossible. The process entails taking steps with increasing shorter durations: 1/2 second, 1/4, 1/8,.... The process can only approach 1, it can never reach it.

. The kinematic process isn't restricted to only points on the number line.

No! Each new step is half the duration of the last step, and this halving process has no end.

No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them. — Michael

Ok. Clearly this is a matter of semantics.

Mathematically, if I have a set of events $\{e_1, e_2, e_3, \dots \}$, there's no problem whatsoever.

You seem to assign some meaning to the word "event" that I don't understand. Must an event be physical? In probability theory we have events that need not be physical, such as the probability of choosing a random real number between 0 and 1/3 from the unit interval. That's an event with no physical meaning at all.

An infinite sequence of events from the set I defined above would be $e_1, e_2, e_3, \dots$. No muss no fuss. That's an infinite sequence of events.

Perhaps you can tell me what an event is, bearing in mind that event is a technical term in probability theory that does not imply physicality.

https://en.wikipedia.org/wiki/Event_(probability_theory)

That's not relevant to the claim I'm making. — Michael

The claim you're making is not one I'm disputing.

I'm saying that if I have finished counting the members of some set then some member must be the final member I counted. — Michael

I disagree. Counting means to place the elements of some set in order-bijective correspondence with the natural numbers, or in a more general context, with some ordinal.

By that definition, we can easily count the natural numbers. The identity map will do.

You seem to think counting is a physical process. That's fine for most contexts, but it's not the only meaning of counting.

For example we have the famous countable/uncountable distinction between infinite sets. A set is countable if it can be placed into bijection with the natural numbers. The natural numbers, the integers, the rational numbers, and the algebraic numbers are all famous examples of countable sets that are infinite.

If you mean to say that we can't physically count the natural numbers, of course I agree. I personally could not get past 13 or 14 or so without losing interest. We could use a supercomputer, but even that has finite capacity. We could use the entire observable universe, but that contains only $10^{78}$ atoms. So sure, physical counting is constrained by resources.

But who's saying otherwise? Perhaps you can explain that to them, since I have never said anything remotely like that.

However: the kinematic process never actually reaches time 1 — Relativist

That's no surprise. It is an imaginary contrivance impossible to physically fabricate. Just a thought.

That's true, but that just makes it physically impossible. I think it's stronger: logically impossible.

↪jgill That's true, but that just makes it physically impossible. I think it's stronger: logically impossible. — Relativist

@Michael keeps making the same claim, and I do not understand the argument.

I agree that it's impossible to do infinitely many physical things in finite time according to present physics.

I do not see what the logical impossibility is.