(Some)... compound expressions suffer the fate I attribute to 'completed infinite sequence of tasks' and 'thinking robot'. What seems most notable about such compounds is the fact that one component (e.g., 'infinite sequence') draws the conditions connected with its applicability from an area so disparate from that associated with the other components that the criteria normally employed fail to apply. We have what appears to be a conceptual mismatch. — Benacerref on Supertasks

I'm not sure whether that doesn't amount to a contradiction or whether it is an entirely distinct issue. But it seems like that if that's the case, one doesn't get as far as a contradiction.

But this implies that one has to consider what sense can be attributed to, for example, "complete" in the context of its use. It is already clear, isn't it, that the meanings of various words have to change to be applied in the context of "infinity". Just because we understand what it is to complete a finite sequence, it doesn't follow that we understand what it is to complete an infinite sequence. "Countably infinite sequence", for example, doesn't mean what you might think it means. Each and every term in the sequence can be counted, but you couldn't complete a count of all of them. In finite cases, counting each one ends up counting all. "Larger" and "smaller", if I've understood right, have to be re-defined as well.

For me, the most persuasive example is that I want to be able to say that as Achilles passes the winning post, he is completing Zeno's sequence. Or, at least, I think it is reasonable to say it. That task, of course is very different from Thompson's lamp because the sequence on/off,... doesn't have a limit, so I don't see that working in that case.

I hope you meant that actions taken outside the system are neither consistent nor inconsistent with the rules. Could we not express this by saying that the rules don't apply, or that it is not clear how to apply the rules, in the new context? — Ludwig V

No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules.

I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of a lamp and button. And I don't see Thomson as getting tripped up either. — TonesInDeepFreeze

Believe @Michael has different assumptions as Thomson, but I haven't gone back to check Thomson's original formulation.

That was a complete description. — Michael

Not so. What lamp? Describe the full setup.

There are no hidden assumptions. — Michael

Not possible, else Benacerraf's objections to Thomson's formulation would apply.

P1 is implicit in Thomson's argument. Using the principle of charity you should infer it. As neither you nor Benacerraf have done so I have had to make it explicit. — Michael

My charity ran out long ago regarding this subject. The lamp is a solved problem.

As a comparison, consider the following:

The lamp is off at 10:00. The button is pushed 10100100 times between 10:00 and 10:01. Is the lamp on or off at 10:02? — Michael

Perfectly clear that you have stated nothing about 10:02. For all we know it turns into a pumpkin.

Any reasonable person should infer that nothing else happens between 10:01 and 10:02. — Michael

Ah, hidden assumptions. Argument by "any reasonable person." You're wrong. Nothing was said about 10:02.

Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — Michael

Benacerraf himself anticipated the spaghetti, saying the lamp might vanish in a puff of smoke.

There is no Supreme Button Pusher arbitrarily willing the lamp to be on or turning it into a pumpkin. There is only us pushing the button once, twice, or an infinite number of times, where pushing it when the lamp is off turns the lamp on and pushing it when the lamp is on turns the lamp off. — Michael

Us. You and me? Human beings pushing a button in arbitrarily small intervals of time.

Nonsense. You're just typing in nonsense. Us pushing the button?

Can't you see why I'm demanding that you write out, in one place, your entire description of the problem. That way you would be able to catch yourself making stuff up as you go.

This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips. — Metaphysician Undercover

You're agreeing with me. Play the lamp game twice. Sometimes the terminal state is on, other times it's off, other times the lamp turns into a plate of spaghetti, or "vanishes in puff of smoke" as Benacerraf says.

I think the problem is precisely that there is nothing to constrain the lamp and we want to find something. In theory, we could stipulate either - or Cinderella's coach. But we mostly think in the context of "If it were real, then..." Fiction doesn't work unless you are willing to do that. It's about whether you choose to play the game and how to apply the rules of the game. — Ludwig V

Which is why I endlessly challenge @Michael to make his version explicit once and for all. And why he won't.

This seems to be more in tune with common sense, for what it's worth. The question is, why? I think it is because of the dressing up of the abstract structure. We assume the lamp has existed before the sequence and will continue to exist after it. — Ludwig V

What justifies such an assumption with regard to an entirely fictional lamp, coach, or pumpkin?

So the fact that the sequence does not define it does not close the question and we want to move from the possible to the actual. But it is not clear how to do that - and we don't want to simply stipulate it. Perhaps that's because defining the limit of the convergent sequence as 1 - or 0, which have a role in defining the sequence in the first place, — Ludwig V

Neither 0 nor 1 is the limit of the sequence of alternating 0's and 1's.

invites us to think in the context of the natural numbers (or actual lamps), whereas defining ω as the limit of the natural numbers does not. — Ludwig V

$\omega$ can be defined such that it is the limit of the sequence of the natural numbers.

C1 is a premise.
— TonesInDeepFreeze

It’s not, it’s a valid inference from the premises. — Michael

The premises don't not specify that the button is ever pushed.

The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit.

w can be defined such that it is the limit of the sequence of the natural numbers. — fishfry

Of course, w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers.

But, just to be clear, we still need to prove that there exists a set such that every natural number is a member of that set, since that set is the domain of the aforementioned sequence.

But, just to be clear, we still need to prove that there exists a set such that every natural number is a member of that set, since that set is the domain of the aforementioned sequence. — TonesInDeepFreeze

wut? axiom of infinity. what's wrong with you tonight? you just blasted out six mentions of me, one more mindless than the next.

(edit) Some of them were compliments. Those weren't mindless!

Pending completion

wut? axiom of infinity. what's wrong with you tonight? — fishfry

The axiom of infinity is how we prove that there is a set that has every natural number as a member.

From the axiom of infinity, we derive that there is a unique least inductive set.

Then we define: w = the least inductive set.

Then we prove that all the natural numbers are members of w.

My point is that we have to be careful in thinking of making this definition:

w = the limit of the sequence of all the natural numbers

since the domain of that sequence is the set of natural numbers, which would already have to have been defined, and so we would have already defined w.

w = the ordinal limit of the sequence of all the natural numbers*

is a theorem, but it would be tricky were it a definition.

* Given a reasonable definition of 'limit of sequence of ordinals'.

/

This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him.

ω can be defined such that it is the limit of the sequence of the natural numbers. — fishfry

Quite so. Except I thought that it had actually been done.

Neither 0 nor 1 is the limit of the sequence of alternating 0's and 1's. — fishfry

Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ωth entry?

w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers. — TonesInDeepFreeze

Yes. My only point was that it is not a natural number, whereas 1 and 0 are. Hence, although both are limits of their respective sequences, as 1 or 0 also are, 1 and 0 are used in other ways in other contexts. This makes no difference to their role in this context and does not affect their role in other contexts, but does affect what we might call their meaning. ω is not used in any other context - so far as I know.

What justifies such an assumption with regard to an entirely fictional lamp, coach, or pumpkin? — fishfry

I agree that we can agree not to ask questions about the lamp outside the context of Thompson's story. But I'm not sure that an assumption really requires a justification. But, for the sake of argument, if I'm telling you a story about a real ball and the shenanigans the prince got up to, you would make that assumption. So if I'm pretending to myself that Cinderella's ball actually happened, I will make the same assumption. This is one reason why I prefer to stick to the abstract structure and shed the dressing up.

My charity ran out long ago regarding this subject. The lamp is a solved problem. — fishfry

Can I ask what your solution is? Just out of interest.

No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules. — Metaphysician Undercover

But actions which are outside of the rules are not contrary to the rules, so they are consistent with the rules. However, on thinking about it, I think my answer it that it depends on the rule. Sometimes the rule means that actions that are not permitted are forbidden and sometimes the rule means actions that are not forbidden are permitted. And sometimes neither.

Any reasonable person should infer that nothing else happens between 10:01 and 10:02. Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — Michael

Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. (If it could, it would not be an infinite sequence). The limit has to be something that is not an element of the sequence. It has to be, to put it this way, in a category different from the elements of the sequence. (I'm trying to think of a self-limiting activity, but my imagination fails me. Perhaps later.)

Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. — Ludwig V

That's precisely the problem. Both of these things are true:

1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.

Therefore we must accept that the supertask cannot be performed.

And even if we were to grant an alternate account that allows for the lamp to spontaneously and without cause be on, doing so does not answer Thomson's question. He wants to know what the performance of the supertask causes to happen to the lamp. Having some subsequent, independent, spontaneous, acausal event after the performance of the supertask does not tell us what the performance of the supertask causes to happen to the lamp. It's a red herring.

Perfectly clear that you have stated nothing about 10:02. For all we know it turns into a pumpkin. — fishfry

As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, and there cannot be a god or wizard or gremlin magically turning the lamp into a plate of spaghetti.

And then as per P2, P3, and P4, pushing the button can never cause the lamp to vanish in a puff of smoke.

So the lamp can never turn into a pumpkin. It can never turn into a plate of spaghetti. It can never vanish in a puff of smoke. It can only ever be either off or on.

Before we even consider a supertask, do you at least understand that if the button is pushed to turn the lamp on (and then not pushed again) then the lamp stays on?

Do you at least accept these?

C4. If the button is only ever pushed at 11:00 then the lamp is on at 12:00
C5. If the button is only ever pushed at 11:00 and 11:30 then the lamp is off at 12:00

Can't you see why I'm demanding that you write out, in one place, your entire description of the problem. That way you would be able to catch yourself making stuff up as you go. — fishfry

I did so. It's here.

The premises don't not specify that the button is ever pushed. — TonesInDeepFreeze

If the button is never pushed then as per P1 and P4 the lamp will forever be off, consistent with C1.

The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit. — TonesInDeepFreeze

Yes, "off" means "not on". The lamp's bulb is either emitting light or not emitting light.

See also the first part of my response to fishfry above.

1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.
Therefore we must accept that the supertask cannot be performed. — Michael

Not quite. If the last stage of the supertask was on, it is not on spontaneously and without cause.

The problem is that whether the supertasks can be performed is not really the issue. The issue is about how to perform a thought experiment - how much of reality you can import into the story. The state of the lamp, and even its existence, is not defined after the limit. So we can fill in the blanks. You prefer common sense to fantasy, but the story is fantasy, so common sense is not necessarily appropriate.

If you are accepting that the button can be pushed an infinite number of times in a finite time, you have already abandoned causality. I could add a premiss that the lamp cannot switch from one state to another in less than a nano-second, and prove that supertasks cannot be performed. Would that convince anyone? I tried that a long time ago and was put right in short order. You earlier brusquely told me that the reason you didn't run the program was that the computer couldn't process the information fast enough.

If the last stage of the supertask was on... — Ludwig V

A supertask has no last stage. Again to quote Thomson, "I did not ever turn it on without at once turning it off [and] I did in the first place turn it on, and thereafter I never turned it off without at once turning it on."

Therefore, if the lamp is on after having performed the supertask then the lamp being on has nothing to do with me having pushed the button to turn it on. The lamp being on would be spontaneous and without cause, which just isn't possible given our premises.

The problem is that whether the supertasks can be performed is not really the issue. — Ludwig V

Yes, it is. Thomson's argument attempts to prove that supertasks cannot be performed.

I'm sorry. There is a serious typo in The first sentence of my reply to you. It should have read:-

Not quite. If the last stage of the supertask was odd, it is not on spontaneously and without cause. — Ludwig V

I hope it makes better sense now.

I refer to the last stage only because the question presupposes it. That presupposition is false, of course, which is why there is no answer to the question.

So I don't see the point of your argument here. It's about something else.

So I don't see the point of your argument here — Ludwig V

Benacerref claimed that the supertask being performed and then the lamp being on is not a contradiction. I am trying to prove that it is (or rather that Thomson already proved this).

The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask – precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off.

The pseudocode I provided before shows this. Its logic does not allow for echo isLampOn to either be determined to output true or false or to arbitrarily output true or false. Therefore, we must accept that it is impossible in principle for while (true) { ... } to ever complete.

And so we must accept that it is impossible in principle for a supertask to be performed.

The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask – precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off. — Michael

Thank you. That is much clearer.

If you want to include a wider, more commonsensical context, you could think that a lamp does not spring in to existence at the beginning, or disappear in a puff of smoke after the limit (12:00 or 2:00 or whatever it is). Nor does time stop. But in that case, you can confidently say that its status cannot be determined, with the implication that you need to wait to find out what its status is.

But once you've gone down that road, there are other things you might need to bring in, such as the time it takes for the lamp to transition from one state to another. Then the scenario falls apart - the experiment cannot be conducted.

Benacerraf claimed that the supertask being performed and then the lamp being on is not a contradiction. — Michael

Nor is it. He talks about two instances of the game, and either outcome would be consistent - on its own. But they contradict each other and that's the problem. I don't rate that "refutation" any more than you do.

But, to be fair, he does grant that Thomson's demolition of the arguments for supertasks are valid. It's just his argument against that he takes exception to. It's interesting, though, that neither he nor Thomson considers the other solution - including the limit in the series.

The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency. — SEP on Supertasks

An interesting indeterminate comment. But I think that the impossibility of the final cycle before the limit does put paid to it. It's all about what "complete" means in the context of infinity. Benacerraf, it I've read him right, allows that Achilles can be said to complete infinitely many tasks in a finite time, but argues (rightly) that Thomson's lamp is a different task and suggests to me that he is inclined not to allow that conclusion in that case.

Quite so. Except I thought that it had actually been done. — Ludwig V

Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea.

There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., $\omega$ such that $\omega$ is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ...

A topological structure is an abstraction of expressing closeness with open intervals, as in the real numbers. The point is that $\omega$ is the limit of 0, 1, 2, 3, ... in exactly the same sense as "abstracted freshman calculus," if you think of it that way.

Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ωth entry? — Ludwig V

No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. Note that it has two subsequences, 0, 0, 0, ... and 1, 1, 1, ,,, that each have respective limits of 0 and 1.

Now it's a theorem that if a sequence converges, all of its subsequences must converge to the same limit. Makes sense, right? A convergent sequence "squishes down" to near the limit.

So a sequence like 0,1,0,1 ... that has two subsequences with different limits, proves that the sequence can not have a limit.

Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. Like

1/2, 3/4, 7/8, ...; 47

In this case 47 is still the value of the "extended sequence" function at $\omega$. I call it the terminal state.

I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke."

As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, — Michael

Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer.

Regarding the rest of it, I'm lamped out, so I will not debate your ideas further. We have heard each other's talking points multiple times at this point. At least I've got a big time philosopher on my side. How cool is that, right? To actually have professional vindication for a personal idea. I've gotten more than my money's worth from this conversation.

If you feel like answering whether you put the same constraint on Cinderella's coach, I'd be intereted to know. Can't respond anymore to the rest of it. When I get to the point that I haven't typed any words on the subject that I haven't typed before, that's how I know I"m done with that topic.

Thanks for the chat plus any Cinderella comments.

Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer. — fishfry

I don't understand your question.

Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off.

We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible.

Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea. — fishfry

I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though.
It's called a performative speech act. Do you know about them? Very roughly, the saying of certain words is the doing. The classic example is promising. A particularly important - and complicated - variety of speech act is a definition. Particularly interesting cases are the definition of rules. (Well, definitions are always regarded as rules, but there are cases that are a bit tricky.)

The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. In addition to the rule, there is a distinct action - applying the rule. That is where, I think, all the difficulties about infinity arise. We understand how to apply the rule in finite situations. But not in infinite situations. Think of applying "countable" or "limit" to "+1". The concept has to be refined for that context, which, we could say, was not covered (envisaged) for the original concept. (By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?)

There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., ω such that ω is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ... — fishfry

Oh, yes, I get it. I think.
Forgive me for my obstinacy, but let me try to explain why I keep going on about it. I regard it as an adapted and extended use of the concept in a new context. (But there are other ways of describing this situation which may be more appropriate.) My difficulties arise from another use of the "1" when we define the converging sequence between 0 and 1. It seems that there must be a connection between the two uses and that this may mean that the sense of "limit" here is different from the sense of ω in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context.

No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. — fishfry

I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42.

Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. In this case 47 is still the value of the "extended sequence" function at ω. I call it the terminal state. — fishfry

So we say that all limited infinite sequences converge on their limits. Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit.

I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke." — fishfry

I have completist tendencies. I try to resist them, but often fail.

The issue is about how to perform a thought experiment - how much of reality you can import into the story. — Ludwig V

That's exactly right. And as I told Michael, way back, in the beginning, if the rules allow for zero importation from the "real world", (which is distinct from the possible world of the supertask), then the allotted amount of time known as the limit, can never pass. And the supertask dilemma never gets off the ground. When Michael insists that the duration must pass, "reality" is needlessly being imported into the thought experiment. We have two incompatible possible worlds.

That the allotted amount of time must pass, if true, is enough evidence to reject the supertask as impossible. However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time.

That's exactly right. — Metaphysician Undercover

I'm glad you agree. And you are right to go on to consider choices we could make.

However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time. — Metaphysician Undercover

That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. Actually, if the task is suspended before it is concluded for any reason, no conclusion can be drawn either way. So I would think that we have to say that the passing of time is a presupposition of the problem. So I wouldn't use this case as an argument against the infinite divisibility of time (or space, in the case of Achilles). (Actually, following our earlier argument, I'm inclined to see that as a mathematical or conceptual proposition, rather than a fact about the real ("physical") world.)

There's a principle here, that we are willing to import any presuppositions of reality (common sense reality) that are needed to make the argument work, in the sense of drawing a conclusion. But that is limited to what I call presuppositions.

There is another presupposition. There is a presupposition that real people are reading the story and arguing about it - and making choices about how much reality to import.

We can, of course, import whatever we want, in one sense. The issue might then arise whether the new version of the story is the same story or a new one.

It's complicated.

That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. — Ludwig V

In the thought experiment, the allotted amount of time cannot pass. The switch must complete an infinite (endless) cycle of on/off before the allotted time can pass. The endless cycle cannot ever be finished, so there will always be more switching to do, and the allotted amount of time will never pass. This is just like Zeno's thought experiment, Achilles will never pass the tortoise, because there will always be more distance to cover first..

Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. That is reality So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider.

We'd think that the rational human being ought to choose "reality" over the ideas of the thought experiment, then we'd reject the nonsense. But this "reality" is concerned with "what will happen", and Hume's problem of induction applies. How do we know that there will be a tomorrow? Because there always has been in the past. How do we know there will be a next hour? Because there always has been. How do we know that there will be a next moment? Because there always has been. However, "because there always has been" does not provide proof that there will continue to be into the future.

Therefore "reality" concerning "what will happen" is lacking in certainty, due to the problem of induction. And, theoretically, a system which prevents the allotted amount of time from passing, through a mechanism similar to the one of the thought experiment could possibly be arranged. Imagine that what is really represented is a continuous slowing down of "our time". Imagine that the mechanism is in a different time frame, so that in the different time frame, the switching on/off is at a constant speed. From our perspective, the switching appears to get faster and faster, but what is really happening is that our time is passing slower and slower in comparison to the other time frame. As it slows more and more and more, it approaches a complete stop, without ever reaching that complete stop, so that every tiny fraction of a second which goes by in our frame, is extremely long in the other. Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount.

Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider. — Metaphysician Undercover

OK. I'm with you that far. Comment:-
That's exactly what Zeno and Thomson want us to do. I guess the complications come in when we want to resist their conclusion, without resorting to "It's just a silly story". We could dismiss lots of perfectly respectable fiction on that basis, of course. But no-one worries about the implausibility/impossibility of the events in "Start Trek" or "Star Wars" or "The Hitchhiker's Guide..". That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is.

However, "because there always has been" does not provide proof that there will continue to be into the future. — Metaphysician Undercover

Yes. What you are doing is applying the actual context (reality) of the story, but instead of drawing on "common sense", drawing on philosophy. That seems to be not unfair, given that Zeno drew a rather radical philosophical conclusion in direct contradiction with "common sense". (He doesn't even have the grace to compromise by dismissing change as an illusion.) Thomson is different because all he wants to conclude is that supertasks are impossible. That's one thing I've never grasped - If supertasks were possible, what philosophical conclusions would follow?
So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point.

Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount. — Metaphysician Undercover

Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue.
For what it's worth, I thought for a while, that one could argue that Achilles always has plenty of time to get to the finishing line, because he passes each stage more and more quickly. But crossing the finishing line requires him to pass the last stage, and there isn't one. That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. (I do realize that there's some difficulty understanding that distinction, but I'll assume it for the moment, if I may.) I found something very like that point in Benacerraf's article.

If we have made a continuous uninterrupted journey from A to B we can be said to have covered all the stretches described in the first premise; that is, our motion can be analyzed as covering in turn AA', A'A", etc. (his italics) — Benacerraf on Supertasks p. 766

That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is. — Ludwig V

I agree, and the conclusion which needs to be drawn is that there is inconsistency between what we observe as the reality of space and time, and the way that we portray space and time through the application of mathematics.

So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point. — Ludwig V

I brought all that in, to say, as I've been saying since the beginning of the thread, that the rejection of the fiction, because it is inconsistent with "reality", isn't that simple. What we know as "reality" has the problem of lacking in certainty. This is what Plato demonstrates in how to deal with Zeno's problems. We cannot simply refer to what we know as "reality", because this is based on sense perception, and the bodily senses are proven to mislead the mind. This is why that sort of "paradox" persists, we dismiss them with reference to "reality" and they go away, that is until skepticism about "reality" reappears.

The issue here is that we really know very little about the nature of the passing of time.

Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue. — Ludwig V

The point, was not to resolve the problem, but to demonstrate how the "unreal" situation described could actually be real. So I was showing how the lamp could actually switch on and off infinitely, in the described manner, such that the allotted amount of time would not ever pass, and how it is possible that Achilles might never pass the tortoise. It is an example of time slowing down, and approaching a complete stop. Instead of the action of the lamp switching speeding up, think of the passage of time as slowing down, so it appears like the action is speeding up. Then the point which marks the limit, midnight or whatever never comes

That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. — Ludwig V

I agree with this, but I'd describe it as how we apply mathematics to space and time.

I don't understand your question. — Michael

Ok.

Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off. — Michael

Pending either of us having anything new to say, I am out of this conversation.

We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible. — Michael

"We" does not include me. I regard Thomson's lamp as a solved problem. When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. Pushing a button in an arbitrarily small time interval to activate a circuit that likewise switches in an arbitrarily small time interval is already a spontaneous, uncaused event. That's why I commend to you the parable of Cinderella's coach.