There is no physical process. — fishfry

The scenario describes a fictional, physical process. The lesson is that the defined supertask (the fictional, physical process) is logically impossible, but this isn't apparrent when considering only the mathematical mapping.

The lesson is that the defined supertask (the fictional, physical process) is logically impossible, — Relativist

The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible?

the process of counting steps is not completable — Relativist

Are you suggesting that supertasks cannot be completed?

The process is:
1) Have fuzzy intuitions;
2) Study some math;
3) Develop far better intuitions. — fishfry

Agreed, but most importantly: (4) apply those intuitions to (the original) experiments.

It's only a finite number of steps back, even from infinity. — fishfry

I like where you're going with this. To navigate between the staircase and omega (and back), one must leap over infinite steps. This concept becomes more palatable if we consider that the steps become progressively smaller towards the bottom. However, let me try to rephrase your perspective: Icarus requires a finite number of strides to reach the bottom and a finite number to return to the top, thus avoiding any supertask. When Icarus adds 1/2, then 1/4, then 1/8, he gets bored and chooses to make a final leap. On his final leap, instead of adding an infinite series of smaller terms, he simply adds another 1/8 and reaches omega, where his calculator displays exactly 1. In this case, the infinity in the paradox describes the steps which he potentially could have traversed (and seen), not what he actually did (and saw). Since he never actually observed all steps, he is in no position to confirm that there were actually infinite steps...but there could have been...potentially. Paradox solved?

the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone

I'm asserting that an infinite process is necessarily never completed - by definition.

The lesson is that the defined supertask (the fictional, physical process) is logically impossible,
— Relativist

The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible? — fishfry

The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed.

I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist

Good. Then we're on the same page!

The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed. — Relativist

You've just described the ordinal $\omega + 1$, which has as one representation the sequence 1, 2, 3, 4, 5, ... , and another more familiar representation as 1/2, 3/4, 7/8, ..., 1.

These are perfectly rigorously defined and logically consistent mathematical objects (assuming ZF is consistent of course].

They're just limits. There is no mathematical mystery. People just get confused when you start making up fictional entities like switching circuits that change state in arbitrarily small amounts of time.

You're not (ahem) a .999... = 1 denier, are you? That's one of the standard crank arguments, that the process of adding the next 9 is "never completed." It's a fallacious argument. There is no temporal process of adding 9's. Rather, you have a mathematical function that assigns to each natural number the digit 9. That's a completed process (or function, more accurately) once you accept the axiom of infinity. When you interpret the string of 9's as a sum 9/10 + 9/100 + ..., the sum of the series is 1, by the definition of the limit of a convergent infinite series.

The "never ends" argument is simply mathematical ignorance. The fast-switch circuit is as realistic as Cinderella's coach that turns into a pumpkin at the stroke of midnight.

At 1/2 second before midnight it's a coach. At 1/4 second before midnight it's a coach. Dot dot dot. At midnight it's a pumpkin. How does that happen? It's a fairy tale. For some reason, philosophers recognize Cinderella as a fairy tale (scrub enough floors and you'll attract the devotion of a handsome prince with a foot fetish); yet these same philosophers take Thompson's lamp seriously. I can't account for this cognitive error.

No it doesn't. — Michael

The contradiction is very obvious. I'm surprised you persist in denial. The supertask will necessarily carry on forever, as the sum of the time increments approaches 60 seconds, without 60 seconds ever passing. Clearly this contradicts "60 seconds will pass".

I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state. — noAxioms

Exactly. Let’s deconstruct the argument:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible.
(2) We assume that the world functions according to presentist beliefs.
(3) Our experiences clearly indicate that motion is possible.

These three assertions cannot all be true simultaneously. It’s unlikely that anyone, including Zeno, would dispute (3). You find (1) to be unreasonable. However, consider the possibility that (2) is incorrect.

Here's an alternative approach:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a block universe exists, change of the block is impossible.
(2) We assume that the world functions according to eternalists beliefs.
(3) Our singular, consistent historical experience gives us no reason to believe that the block universe is subject to change.

In this scenario, all three points could indeed hold true, suggesting that an eternalist viewpoint might be more suitable. However, quantum mechanics challenges point (3), necessitating a more nuanced argument. Despite this, an eternalist framework—albeit with some adjustments to incorporate the quantum aspects of our universe—appears to be the most rational choice.

Not sure of the difference. If I cut a string, I don't get points, I get shorter strings. — noAxioms

The cuts themselves are the points (think Dedekind cuts).

You can under some interpretations. — noAxioms

One can observe a superposition directly? Please share a link.

Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"

If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation. — noAxioms

What I aim to demonstrate is that there is a scenario where local motion is possible and continuous without involving supertasks. This occurs in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).

Necessary only if the first premise is to be accepted. — noAxioms

If the universe is discrete, then Zeno's paradoxes cannot occur as he described them. What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.

I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term. — Relativist

I did not get a mention for this post, does that happen sometimes? Maybe I just missed it.

As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal $\omega + 1$. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers.

However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step? — Relativist

You can model this situation with $\omega + 1$, as I've tried to explain a number of times.

After all, if we work in the close unit interval [0,1], the sequence 1/2, 3/4, 7/8, ... never ends, yet there's its limit right there at the right end of the interval. We "get there" through a limiting process. There is no last step if your steps are required to be discrete. But we can also take limits. Limits aren't steps, but that's a semantic quibble. We can adjoin 1 to 1/2, 3/4, 7/8, ... to form the $\omega + 1$ "extended sequence" if you want to call it that, 1/2, 3/4, 7/8, ..., $\omega$.

I don't know if this will help, but at least I can motivate the legitimacy of the ordinal concept by linking the wiki page on ordinal numbers.

But there's an easier way to think of it. We're just adjoining a formal symbol at the end of the natural numbers:

1, 2, 3, 4, ..., $\omega$. It's just a formal symbol, means nothing at all. But we can define it in such a way that it's the upper limit of 1, 2, 3, ... in exactly the same way that 1 is the upper limit of 1/2, 3/4, ...

Yes there is no "last step" but there is in fact a limit.

By definition, a limit is not reached, it is approached. — Relativist

That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.

In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.

It's not reached by a single step. Rather, it's reached by the limiting process itself.

Suppose Icarus writes the number of the step on a piece of paper with each step, erasing the previous number. What number will be on the paper at the end?

It cannot be finite. If it were n, why not n+1?

It cannot be infinite because no step has the number infinite.

Take the scenario here:

After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

We can sum the geometric series to determine that the limit is 60 seconds. The claim some make is that this then proves that this infinite sequence of events can be completed in 60 seconds.

However, then we ask: what colour is the square when this infinite sequence of events is completed? — Michael

This is just the lamp story with three states. The answer is that the sequence 0, 1, 2, 0, 1, 2, 0, 1, 2, ... has no natural completion or limit. So if we want to define its state "after" the natural numbers, we can say it's anything we want. I like the Cinderella analogy. The square turns into a pumpkin at midnight. That's no less realistic than the square story.

As per the setup, the square can only be red, white, or blue, and so the answer must be red, white, or blue. — Michael

Why? After all in the sequence 1/2, 3/4, 7/8, ... the "setup" is that each element of the sequence is a rational number strictly less than 1.

But the limit of the sequence is 1. This illustrates a general mathematical principle:

Taking limits does not necessarily preserve all properties of a sequence.

All of 1/2, 1/3, 1/4, 1/5, ... are strictly positive. But the limit of the sequence is 0, which is not positive. (I'm positive!)

You have to be very careful not to fall into the trap of assuming that a limit must preserve all the properties of the elements of the sequence that approaches it. You have made that mistake.

However, as per the setup it will never stay on any particular colour; it will always turn red some time after white, turn blue some time after red, and turn white some time after blue, and so the answer cannot be red, white, or blue. This is a contradiction. — Michael

It's not a contradiction. It's the straightforward observation that the sequence 0, 1, 2, 0, 1, 2, ... has no sensible limit. So if you wish to define a final state, you can make it anything you like. I choose pumpkin.

Remember, Cinderella's coach is a coach at 1 second before midnight; at 1/2 second before midniht; at 1/4 second before midnight; and so forth. Yet at the stroke of midnight, the coach turns into a pumpkin.

That story makes exactly as much sense as Thompson's lamp. Except that with Cinderella, we introduced a discontinuity. Where as with the lamp, and with your three-state lamp, there is no possible way to define the limiting state in such a way as to preserve continuity.

The conclusion, then, is that an infinite sequence of events cannot be completed, — Michael

An infinite sequence of events can have a limit. I assume you agree that 1/2, 3/4, ... has the limit 1. We can think of 1 as the "completion" of the sequence. It's reached not by a "final step," but rathe by the limiting process itself.

and the fact that we can sum the geometric series is a red herring. — Michael

No, it's the heart of the matter. .999... = 1 even though there's no "last 9." The limiting process is real. It's important. It exists.

To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible. — Michael

That is flat out false and does not follow at all.

We can "complete" the sequence .9, .99, .999, .9999, ... with the number 1, which is reached via a limiting process.

we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: — Michael

Sorry, what? You don't believe that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? You don't believe in calculus? You are arguing a finitist or ultrafinitist position? What do you mean?

Of course if you mean real world events, I quite agree. But your three-state lamp is not a real world event, it violates several laws of classical and quantum physics, just as Thompson's two-state lamp does.

Suppose Icarus writes the number of the step on a piece of paper with each step, erasing the previous number. What number will be on the paper at the end? — SolarWind

42. Can you argue otherwise? The final state is not defined. It can be anything we like.

@fishfry:

"Not defined" does not mean that you are free to choose the result.

Which solution has n = n+1?

Certainly not 42.

"Not defined" does not mean that you are free to choose the result. — SolarWind

Yes it does. If I define the first three elements of a sequence, like 3, 12, 84, what number comes next? Mathematically, it can be any number at all.

Likewise if I define a function at 1, 2, 3, 4, ..., and I want to also define it at $\omega$, a symbolic point AFTER all the natural numbers, I can define it to be anything I want. Like 42. Or like Cinderella's coach, which is a fine, beautiful coach at 1 second before midnight, 1/2 second before midnight, 1/4 second before midnight, etc., yet turns into a pumpkin at the stroke of midnight.

Which solution has n = n+1? — SolarWind

Not the same kind of undefined. Here, it can't be defined. But the state at the bottom of the stairs can be anything at all.

So there's a distinction between something that can't be defined because it's impossible, and something that simply hasn't yet been defined, and that can then be defined as anything at all.

Certainly not 42. — SolarWind

Don't see why not. When you first heard the Cinderella story, did you make the same objection to the coach turning into a pumpkin at the stroke of midnight?

As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal ω+1

+
1
. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers. — fishfry

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.

By definition, a limit is not reached, it is approached.
— Relativist

That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.

In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.

It's not reached by a single step. Rather, it's reached by the limiting process itself. — fishfry

The limit of the series is "reached" only in the sense that we can reach a mathematical answer. The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. Deriving the limit just means we've identified where the sequential process leads. In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? 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.

if a physical process ends, there has to be a final step. — Relativist

This is equivalent to asserting that 'infinity' is the largest integer. Does nobody else see that making such an assertion is going to lead to contradiction? It doesn't mean that there cannot be an unbounded thing.

I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist

This depends on one's definition of completing a process. The SEP article on supertasks has this to say about it:
"But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires.

I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed.

Good. Then we're on the same page! — keystone

And a different page than me.

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible. — keystone

Zeno's argument is that X is possible, and another that X is not possible.
I see no mention of presentism in his arguments. I cannot follow your arguments here if you don't show how he presumes any such thing, or why it matters. Motion is defined under either view, and the argument can be made in either view of time. By modus ponens, at least one of Zeno's premises must be false unless empirical evidence is entirely dismissed as invalid.

I think you are under the impression that motion is not meaningful under eternalism, and that this somehow absolves Zeno's conclusion, but all his arguments still apply, and are still self contradictory.

The cuts themselves are the points (think Dedekind cuts).

OK, so now we have point cuts separating shorter strings, each with nonzero extension.

One can observe a superposition directly? Please share a link.

Any interpretation that denies wave function collapse has everything in superposition at all times. One simply finds ones self in superposition with the observed state. So I observe both the dead and the live cat, presuming that "I" dong the observing is the same person as the person a moment ago with the closed box.

No, I'm not don't personally accept MWI, but the simplicity of it is elegance itself.

in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).

Moton is change of postion over time. The block universe very much has that for any moving object. The worldline of that object is a different spatial locations at different times. All of Zeno's arguments still apply, and are still contradictory.

Yet again, the only difference between the view is the positing of the preferred moment, which is irrelevant to the subject at hand. Both are effectively block view, but presentism assigns different (at least four kinds of) ontological states to different events based on its relation to the preferred moment, and eternalism assigns identical ontological states to all events.

The kind of motion you are referencing (the changing of the block (over what??)) is not suggested by either view, nor by Zeno.

If the universe is discrete, then Zeno's paradoxes cannot occur as he described them

The first premise would be demonstrably false. The second premise (that supertasks are impossible) would be moot, but arguably true then.

What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.

You seem to do this by reducing the universe to a point (your 'photo'), which is not something that is continuous. A point in time at least, which is the same as denial of time at all.

Sorry, what? You don't believe that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? You don't believe in calculus? You are arguing a finitist or ultrafinitist position? What do you mean?

Of course if you mean real world events, I quite agree. But your three-state lamp is not a real world event, it violates several laws of classical and quantum physics, just as Thompson's two-state lamp does. — fishfry

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.

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). 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.

So if you wish to define a final state, you can make it anything you like. I choose pumpkin. — fishfry

If you want to say that supertasks are possible 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.

The contradiction is very obvious. I'm surprised you persist in denial. The supertask will necessarily carry on forever, as the sum of the time increments approaches 60 seconds, without 60 seconds ever passing. Clearly this contradicts "60 seconds will pass". — Metaphysician Undercover

An ordinary stopwatch is started.

After 30 seconds a white box turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

When the stopwatch reaches 60 seconds, what colour is the box?

Your claim that the box changing colour entails that the stopwatch will never reach 60 seconds makes no sense. The stopwatch is just an ordinary stopwatch that counts ordinary time as it ordinarily would and is unaffected by anything the box does.

An ordinary stopwatch is started.

After 30 seconds a white box turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

When the stopwatch reaches 60 seconds, what colour is the box? — Michael

You have not avoided the contradiction, only obscured it. Increments of time must be measured, the are the product of a measuring device. The measuring device in this example is an ordinary stopwatch. What is implied by "and so on" is increments of time which the stopwatch cannot measure. There lies your contradiction. The stop watch is the designated device which measures time, but you talk about increments of time which cannot be measured by it, therefore contradiction.

In your other example, the counter was the device that measured the passage of time, and it was designed to measure all those increments (do the supertask). But that measuring device denied the possibility of 60 seconds of time passing, making the measurement of a separate timepiece which would measure 60 seconds, contradictory.

Therefore we can conclude that the contradiction lies between the two very distinct descriptions of how time is measured. One way is the supertask of the counter. This is obviously a theoretical way of measuring time. The other way is a description of how time is actually measured in practise. There is contradiction between these two ways of describing the measurement of time. As you yourself indicate, the practical way gets the nod, as the real way, because it is supported by empirical evidence, and the supertask way, since it contradicts the practical way, is designated as impossible. As philosophers though, we are trained to be skeptical of sense evidence, having been educated in the ways that the senses commonly deceive us. So the philosopher knows that there is more to this problem than what meets the eye. It's not simply a matter of dismissing the supertask, and accepting the conventional way of measuring time, as the true way to measure time.

As I said before, the supertask way needs to be proven to be wrong, rather than simply dismissed because it contradicts the way of current practise. The empirical evidence of the stop watch is nothing other than current practise, convention, so it is in fact manufactured evidence. If we always accepted the current practise as the best, or true way, then we'd never improve ourselves. This is why I say simply accepting it and dismissing the supertask, is prejudice, and nothing else. The fact that we have not devised a supertask machine, and so we use other ways to measure time, does not mean that it is impossible that the supertask way is the real "true" way to measure time, and our current practise is actually giving us a false measurement. Therefore the supertask must be proven to be impossible.

Here's an example of an attempt at a similar type of proof. In ancient Greece, there was a principle accepted by many, that the orbits of the sun, planets, etc., were eternal circular motions, as a sort of divine activity. Being a prescribed activity which continues for an infinite duration of time, the eternal circular motion is a supertask. Now, Aristotle in his "On the Heavens" (De Caelo) showed how eternal circular motion is a logically valid and consistent principle, a real logical possibility, just like the supertask counter is. However, he then went on to explain how anything which moves in such a spatial pattern must be a material body. He then described "matter" as the principle of generation and corruption, and determined that a material body must have been generated in the past, and will be destroyed in the future. In this way he provided the principles required, to prove logically, that (the supertask) eternal circular motion is actually logically impossible. This proved that the heavenly bodies were not eternal, and not divine. Then the principle which he employed, "matter", became the keystone for understanding the nature of the physical reality because it provided the principle for associating change and becoming on the earth, with change and becoming in the heavens.

Increments of time must be measured — Metaphysician Undercover

No they mustn’t.

if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer. — noAxioms

Wrong. The statement (the completion of a consecutive series of physical steps entails a final step) is necessarily true. When we consider this statement in conjunction with a statement about the series being "complete" (in terms of convergence) we introduce a contradiction. This is the point! These statements cannot both be true, but both are entailed by the scenario.

But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires. — noAxioms

The SEP article says:
"Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. One might only doubt whether or not the standard topology of the real numbers provides the appropriate notion of convergence in this supertask. "
Indeed, I'm denying that the topology of the real numbers applies to the execution of the supertask itself - although I agree it applies to the series.

As I noted above, a physical, step-counting process that completes must entail a final step. Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction. That seems a cop-out. The point of the thought experiment is to highlight the contradiction.

I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed. — noAxioms

I agree with this, but this simply ignores the implication of the physical process of step-counting. For the scenario to be coherent, BOTH view of completeness have to be true. But they aren't - so the scenario is actually incoherent.

the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone

Yes- and that's because the role of infinity in the task. The task entails a sequence of events, so the infinity can only mean an infinite chain of events - one after another without end.

No they mustn’t. — Michael

That's fundamentally incorrect. If you truly believe that an increment of time exists without being measured, tell me how I can find a naturally existing, already individuated increment of time.

if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer.
— noAxioms
Wrong. The statement applies universally to the physical process of descending stairs. — Relativist

The physical process of descending stairs is not a supertask. I couldn't think of a way to make it a supertask, even by making each step smaller. A supertask has no final (or first, respectively) step, so by counterexample, the assertion "there has to be a final step." is incorrect.

A contradiction is introduced when this statement ("a completed step counting entails a final step)

I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent.
The series (say the time needed to complete all tasks) converges. The count does not.

Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete

The SEP article says:
"... From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity"

I notice the SEP article correctly doesn't claim that the last step is taken.

As I noted above, a physical, step-counting process that completes must entail a final step.

Agree. But the only attempted step counting processes are examples like the lamp or Michael's digit counter, and those examples are not physical. The Achilles example can be physical, but it isn't counting anything.

Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction.

There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step?

I agree with this, but this simply ignores the implication of the physical process of step-counting.

Kind of like I ignore the green ball in the bag, yes.

For the scenario to be coherent, BOTH view of completeness have to be true.

I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number.

No they mustn’t. — Michael

:up:

Once again, M-U cannot comprehend a view outside his own idealistic assumptions.

If you truly believe that an increment of time exists without being measured, tell me how I can find a naturally existing, already individuated increment of time. — Metaphysician Undercover

I don't know what you mean by "finding a naturally existing, already individuated increment of time", but it is a fact that 60 seconds of time can pass without anyone looking at a clock or a stopwatch. Billions of years passed before humanity evolved, and this isn't some retroactive fact that only obtained when humanity started studying the past.

I don't know whether you're arguing for some kind of antirealism or if you're failing to understand a use-mention distinction.

Regardless, the arguments I am making here are directed towards the realist who believes that supertasks are possible.

Once again, M-U cannot comprehend a view outside his own idealistic assumptions. — noAxioms

If someone would explain to me, in a way which makes sense, a better perspective, then I'd happily switch. Simple assertions like "It must", and "no it's not" just do not suffice for helping a poor lost soul such as myself, comprehend another view.

but it is a fact that 60 seconds of time can pass without anyone looking at a clock or a stopwatch. — Michael

Simple assertions do not help me to understand what you are trying to say. "60 seconds" is the reading we get of the clock. It's just a generic symbol, like "dog" or "cat". But I can show you many things which would be called "a dog", and things called "a cat". Now, if you think that there are some things called "60 seconds", other than the reading taken from a measuring device, then show them to me.

Consider Wittgenstein's example of "the standard metre in Paris". "One metre" is a measurement, and there are many items which can be measured to be a metre. The standard metre is the paradigm, the official example of that convention. But there are no objects in the world which "one metre" refers to, not even the standard metre, as this is the paradigm, it is not "one metre" itself. Likewise, we can measured a multitude of different times as "60 seconds", and there is a paradigm, or standard which is the oscilation of the cesium atom, but there is nothing in the world which is referred to by "60 seconds"

the second, defined as about 9 billion oscillations of the caesium atom. — Wikipedia

https://en.wikipedia.org/wiki/Unit_of_time#:~:text=The%20base%20unit%20of%20time,oscillations%20of%20the%20caesium%20atom.

Billions of years passed before humanity evolved, and this isn't some retroactive fact that only obtained when humanity started studying the past. — Michael

A ''year" is nothing but a human convention, a standard of measurement, just like "a metre". Therefore there was no years prior to humanity. What you talk about here is a projective measurement backward in time. There was something occurring prior to humanity, which we commonly call "the passing of time", but what is referred to as the passing of time did not consist of years, as "years" is the product of the measurement, just like "metres" is the product of the measurement. To claim that standards of measurement existed prior to humanity, which actually invented them, is Pythagorean idealism, Platonism. Then these standards become eternal principles, such that God was measuring the passage of time prior to humanity, to determine the passage of "years". But this Platonism is demonstrably wrong, because it excludes the possibility of error.

That the passing of time is something completely different than a succession of years is very evident from the fact that it can be measured as a succession of a vast multitude of different increments, years, days, minutes, seconds, picoseconds, nanoseconds. Each of these increments serves as a measurement standard, (just like metre, centimetre, kilometre, foot, inch, mile), but not one of them is the thing which is measured. And, the little inconsistencies between them which show up in conversions, where we have to adjust the clock so that they keep up with each other, shows that none of them is actually real or true. Notice the wiki quote, a second is "about nine billion oscillations".

I don't know whether you're arguing for some kind of antirealism or if you're failing to understand a use-mention distinction. — Michael

Perhaps if you took the time to explain to me how you understand this use-mention distinction, and how it is applicable in this context, that might be helpful to me understanding your position, which at the time seems completely ridiculous. I really do not understand how you can believe that the product of a measurement "one second" can exist without the act which produces it. Again, the best course of action for you to help me understand, would be to show me these things, called "a second", or explain to me how I might find one without performing the act of measuring which is what, I believe, actually creates them.

Regardless, the arguments I am making here are directed towards the realist who believes that supertasks are possible. — Michael

As I explained yesterday, we must consider that any such task (supertasks) are possible until proven otherwise. This is because they are logically possible, and the only thing which makes them appear to be impossible is that they are inconsistent (in contradiction with) the conventional way of doing things. But the conventional way is not necessarily the best way, it is only the way which is supported by empirical evidence, which has been proven to be unreliable and misleading. Therefore we cannot dismiss the supertasks as impossible until we have sound logic which disproves them. Simply asserting that supertasks are impossible just displays an empiricist prejudice. Then you support your prejudice with Platonism.

That we coin the term “X” to refer to some Y isn’t that Y depends on us referring to it using the term “X”. This is where you fail to make a use-mention distinction.

If we take the term “1 year” as an example, the Earth orbiting the Sun does not depend on us measuring it. It just orbits it, independently of us.

So to rephrase my example:

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?

Your claims so far are akin to claiming that the Earth will never complete its orbit around the Sun, which just makes no sense. The box does not have the power to influence the Earth's velocity or the Sun's gravitational pull.

I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent. — noAxioms

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

The series (say the time needed to complete all tasks) converges. The count does not.

Yes, the sequence of defined temporal points (1/2, 1/4, 1/8...) is a series, but the mathematics that identifies the limit does not take into account the kinematics of the task. Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world - regardless of whether or not you wished to consider it.

The physical process of descending stairs is not a supertask. — noAxioms

It fits this definition:
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."

Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete. — noAxioms

The goal of removing all the marbles will therefore never be met if there are at least 2 green marbles, and it will rarely met even if there is only 1. How does this relate to a supertask that allegedly completes?

I notice the SEP article correctly doesn't claim that the last step is taken. — noAxioms

The article discusses the issue:

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...
... there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. ... The two meanings for the word “complete” happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.

The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete. The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.

Here's a paper in which a philosopher proves it to be impossible to complete infinitely many tasks in a finite time based on the "Principle of Sequential Acts":

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.

The author argues that those who argue the task completes implicitly deny the PSA, without considering it, and therefore not refuting it. That's what I see going on with the posters who focus only on the mathematical series.

Relativist: "Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction."
There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step? — noAxioms

Yes, it's a cop-out because it ignores the kinematic process. Stating this in terms of the PSA gives you something specific to address, if you want to not cop out.

If your sole purpose was to discuss the math associated with the limit of a series, you'd have been better off avoiding putting it in terms of a supertask.

Relativist: "For the scenario to be coherent, BOTH view of completeness have to be true."
I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number. — noAxioms

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

I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction. You seem to be embracing the completeness of the mathematical series, then concluding that there can't be a last step because that would entail a contradiction. So look at it this way:
1) the completeness of the series does not demonstrate an analogous supertask is possible.
2) If there is no last step (or if the process is not consistent with the PSA), then the kinetic process (which is a supertask) is logically impossible.

If we take the term “1 year” as an example, the Earth orbiting the Sun does not depend on us measuring it. It just orbits it, independently of us. — Michael

OK, the earth goes around the sun indefinitely, even if there were no humans on earth. But there is no "years", nor is there any individual "orbits" separated out without someone, or a device to make the judgement of beginning and end.

That is the problem with your example. Time passes, we agree on that, but there are no seconds unless measured out. Now, both the counter doing the supertask, and the ordinary stopwatch are designed to measure the passage of time. 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.

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

Same problem, the device is not designed to reach the end of an orbit. It will keep on changing colours faster and faster without ever getting to the end of an orbit. I assume it would probably burn up though, from going too fast.