$0, 1, 0, ..., 1\\0, 1, 0, ..., 0$

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off.

As a comparison, even though imaginary numbers have a use in mathematics it is more than just physically impossible for me to have $\sqrt{\text{-}1}$ apples in my fridge; it is metaphysically impossible.

No pretend physics can allow for me to have an imaginary number of apples in my fridge and no pretend physics can allow for the above two mathematical sequences to model the state of a lamp over time.

With Thomson's lamp, these are our premises:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t₀
P5. The button is pushed at least once between t₀ and t₁
P6. The lamp is either on or off at t₁

And these are our conclusions:

C1. If the lamp is on at t₁ then the button was pushed to turn it on, prior to which it was off
C2. If the lamp is off at t₁ then the button was pushed to turn it off, prior to which it was on
C3. The button is pushed $n \in \mathbb{N}_1$ times between t₀ and t₁

These conclusions prove that a supertask is not performed.

Saying infinity is a concept doesn't actually tell us anything about it. I don't even know what it means to say infinity is a concept. Is there anything that isn't a concept? — fishfry

Yes, of course you are right. It is no more help than saying that a table is an object. I was trying to re-instate the line between, let us say, a mathematical reality and a physical reality - or between mathematical possibility and physical possibility. (I think we are agreed that what creates the difficulties here is the confusion of the two in the definition of the supertask.)

Remember the mathematical induction you brought up before? — Michael

I suggest that what creates the problem here is the idea that the mathematical induction is a process that takes time. Perhaps the temptation to do this derives from the analogy with a Hume's inductive process that creates so many issues about empirical laws or generalizations. But once you pose the challenge of actually executing the process to the bitter (and non-existent) end, or think that you can stipulate what happens at the end, you are enmeshed in contradictions. (Achilles' race with the tortoise has exactly the same issues, but in the medium of space rather than time.) This leads us to think that there is some sort of miracle involved in arriving at the fridge to get a beer. But the mathematical induction is one analysis of many that can be applied to either space or time, and does not in any way affect our walking about our kitchen or arriving on time at a party.

This leads us to think that there is some sort of miracle involved in arriving at the fridge to get a beer. — Ludwig V

There's no miracle. Motion isn't continuous; it's discrete.

There's no miracle. Motion isn't continuous; it's discrete. — Michael

Your argument doesn't prove that.

Your argument doesn't prove that. — Ludwig V

Others do, like Zeno's and Bernadete's.

There's no miracle. Motion isn't continuous; it's discrete. — Michael

Yes there is a miracle. The miracle is the start, the initial act, or force, which breaks the previous inertia, causing the new motion toward its end (in this case the beer in the (fridge).

In the case of the op, and also the lamp, the defined start is a rate of acceleration which will continue without an end. This is why it confuses it is acceleration without end. That is not an unusual way to define the effect of a force though, gravity is defined in this way (9.8 metres per second squared). The universal law of gravity describes an acceleration without end.

The commonplace nature of such a description, acceleration without end, makes it appear like there is no miracle involved with acceleration. But this is only because, in these commonplace activities, there is always contrary forces which negate the acceleration before the speed becomes infinite. There is commonly an end to acceleration. So this only serves to confirm our belief that there is no miracle involved with acceleration. However, the unintelligibility of examples like the op and the lamp, where no contrary forces are invoked, demonstrate to us, that acceleration truly is miraculous.

That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes. — Michael

Exactly. You don’t have infinity any more anywhere in the whole scenario if you want to talk about 2 seconds and what happens at 2 seconds or after.

What is infinite is the number of switching that would occur before two seconds. Once you are looking at second number 2, you have to ignore all of the premises and activity of the scenario. Midnight is not part of the conversation. It’s a limit that will never be reached so the state of the lamp at midnight is indeterminable because the function of the switching will never be operating at midnight.

Yes, of course you are right. It is no more help than saying that a table is an object. I was trying to re-instate the line between, let us say, a mathematical reality and a physical reality - or between mathematical possibility and physical possibility. (I think we are agreed that what creates the difficulties here is the confusion of the two in the definition of the supertask.) — Ludwig V

I think there are abstract things and concrete things. But physics these days pushes hard on the nature of physical things. Is there a philosopher in the house?

I understand that it has no end. That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes. — Michael

Who is pushing this button, by the way? And how? Magic again?

It's a premise of the thought experiment that there is a sequence of steps at successively halved intervals of time. You are now arguing that since such a premise is physically impossible, it's metaphysically impossible. You are wrong on the facts and wrong on the logic.

And its your premise, not mine.

0,1,0,...,10,1,0,...,0

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off. — Michael

It's a mathematical model of a lamp being turned on and off. What if instead of a lightbulb, we have a big LED that displays 0 or 1? You have an objection to that? Based on what?

I already showed you how to model the process using the inverse powers of 2. You ignored that, since it refutes your argument.

As a comparison, even though imaginary numbers have a use in mathematics it is more than just physically impossible for me to have -1‾‾√ — Michael

Tell that to an electrical engineer or quantum physicist, both of whom use imaginary and complex numbers as essential tools of their trades.

apples in my fridge; it is metaphysically impossible. — Michael

Wow. I guess you don't believe in negative numbers, fractions, or irrational numbers either. You are really flailing with these weak arguments.

No pretend physics can allow for me to have an imaginary number of apples in my fridge and no pretend physics can allow for the above two mathematical sequences to model the state of a lamp over time. — Michael

You accept "on" and "off" but not one and zero? Again: replace the bulb with an LED that alternately flashes 1 and 0.

With Thomson's lamp, these are our premises:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at least once between t0 and t1
P6. The lamp is either on or off at t1

And these are our conclusions:

C1. If the lamp is on at t1 then the button was pushed to turn it on, prior to which it was off
C2. If the lamp is off at t1 then the button was pushed to turn it off, prior to which it was on
C3. The button was pushed n∈ℕ
�
∈
�
1
times between t0 and t1

These conclusions prove that a supertask is not performed. — Michael

I must have made a pretty good argument in my previous post, since in your response you went into wild irrelevancies and repeated arguments I've long lost interest in, and never engaged with a word I said.

You really reject the complex numbers, negative numbers, fractions, and irrationals? Is mathematical nihilism your last defense?

Tell that to an electrical engineer or quantum physicist, both of whom use imaginary and complex numbers as essential tools of their trades. — fishfry

I didn't say that imaginary numbers don't have a use. I said that I cannot have an imaginary number of apples in my fridge.

You really reject the complex numbers, negative numbers, fractions, and irrationals? — fishfry

No.

It's a premise of the thought experiment that there is a sequence of steps at successively halved intervals of time. — fishfry

Yes, and this premise is proven false. See refutation by contradiction and modus tollens.

I already showed you how to model the process using the inverse powers of 2. You ignored that, since it refutes your argument. — fishfry

It doesn't refute my argument because it doesn't address my argument.

My argument is that the lamp cannot be on unless it is turned on (whether by pushing a button or by magic), as argued above.

Therefore, off, on, off, ..., on makes no sense in context. The lamp being off must precede the lamp being on. Given that this is inconsistent with the premise that the button is pushed at successively halved intervals of time it is proven impossible in principle to push the button at successively halved intervals of time.

I said that I cannot have an imaginary number of apples in my fridge. — Michael

Eh. You have a collection of apples in your fridge whose count is referable to by the symbol of a natural number. You also have an apple in your fridge, which if rotated 90 degrees counterclockwise, could have its current orientation referred to as i multiplied by its previous one. It's not like either number is somehow more physical than the other one, people just intuit the naturals (and positive computable reals) to really "be there" whereas the other numbers aren't.

Whereas they live in rotations and waves and the columns and rows of accounting books. None of those concepts have, to my knowledge, a baked in proclivity in our wetware. Whereas natural numbers, counting, cardinality, size comparisons etc and their symbolic representations do.

At the very least you've got work to do in "dephysicalising" or "physicalising" the intuitions regarding number and processes you have.

It's not like either number is somehow more physical than the other one — fdrake

I’m not claiming otherwise. I’m only claiming that I cannot have an imaginary number of apples in my fridge.

That some number is sensible isn’t that it’s sensible to use that number in any and every context.

Similarly, the sequence 0, 1, 0, …, 1 might make sense in some contexts, but the sequence off, on, off, …, on doesn’t make sense in the context of a lamp being turned on and off by pushing a button.

I didn't say that imaginary numbers don't have a use. I said that I cannot have an imaginary number of apples in my fridge. — Michael

Well, you said that as if it makes a point. Which escapes my simple mind.

Yes, and this premise is proven false. See refutation by contradiction and modus tollens. — Michael

If after all this, your point is that Thompson's lamp is a silly thought experiment, at long last we are in full agreement. It doesn't tell you much about supertasks, though. Perhaps I should have tried to expand on that line of argument earlier. Thompson's lamp is not a good example of a supertask, because the sequence doesn't have a limit or any natural termination point.

Zeno getting up from the couch to get a snack in the kitchen, under the assumption of arbitrary divisibility of time and space, is a much better example. I have argued that that is a true supertask, one that is metaphysically possible.

But I was thinking about your button presses. I don't see why the sequence doesn't run at every time prior to midnight; and then at exactly midnight, the Supreme Button Pusher doesn't just flip a coin and turn the lamp on or off accordingly. Makes as much sense as the rest of the story.

It doesn't refute my argument because it doesn't address my argument. — Michael

It (my inverse powers of 2 model) shows that at any time prior to midnight, we know the state of the lamp. And that at midnight, it's undefined and entirely arbitrary.

My argument is that the lamp cannot be on unless it is turned on (whether by pushing a button or by magic), as argued above. — Michael

Ok. At midnight the Supreme Button Pusher flips a coin and turns the lamp on or off, accordingly. I don't see any problem.

Therefore, off, on, off, ..., on makes no sense in context. The lamp being off must precede the lamp being on. — Michael

Not at the termination point. Only prior to midnight. Not AT midnight. As I've indicated. It's fundamental to the nature of an infinite sequence with a termination point adjoined at the end.

Given that this is inconsistent with the premise that the button is pushed at successively halved intervals of time it is proven impossible in principle to push the button at successively halved intervals of time. — Michael

No no no no no. "the premise that the button is pushed at successively halved intervals of time" is only valid for the infinite sequence itself, and NOT for the termination point.

At midnight the Supreme Button Pusher flips a coin and turns the lamp on or off, accordingly. I don't see any problem. — fishfry

If the Supreme Button Pusher turns the lamp on at midnight then it must have been off before He turned it on, because that's what "turning on" means. If He turned it off at midnight then it must have been on before He turned it off, because that's what "turning off" means.

For the entire continuous time that the lamp exists – not just the time during which the button is pushed – it being off must precede it being on, regardless of magic or divine intervention. As such the sequence off, on, off, ..., on makes no sense, much like having an imaginary number of apples in the fridge makes no sense.

We must accept that the button can only be pushed $n \in \mathbb{N}_0$ times between t₀ and t₁.

But also note P1 in the argument above, implicit in Thomson's argument. You don’t get to just introduce God to deflect from the incoherent causal consequence of having pushed a button an infinite number of times.

Thompson's lamp is not a good example of a supertask, because the sequence doesn't have a limit or any natural termination point. — fishfry

A supertask is just performing an infinite succession of tasks in finite time. Pushing a button, which just so happens to turn a lamp on and off, is as good a supertask as any.

But let’s assume that pushing the button displays the time that the button is pushed – persisting until the button is pushed again – and is pushed (only) at successively halved intervals of time starting two minutes to midnight. What time is displayed at midnight?

The sequence approaches midnight but because the button is never pushed at midnight the display can never show midnight, but must show something because it’s never turned off. It always only displays the time that the button was last pushed, but in this scenario there is no last button push, entailing a contradiction.

As Thomson says, "the impossibility of a super-task does not depend at all on whether some ... sequence is convergent or divergent."

Others do, like Zeno's and Bernadete's. — Michael

I don't think that either of them suggests that space or time may be discrete. In any case, you seem to accept that that's a different topic.

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off. — Michael

That seems to me to be a good diagnosis of the issue with supertasks. All that is then needed to free people from the illusion that walking to the fridge can be mathematically analysed in many ways, none of which affect physical reality.

I think there are abstract things and concrete things. But physics these days pushes hard on the nature of physical things. Is there a philosopher in the house? — fishfry

The trouble is that many philosophers seem to be hypnotized by physics, and seem to forget that physicists develop their theories and conduct their experiments in ordinary human reality.

At the very least you've got work to do in "dephysicalising" or "physicalising" the intuitions regarding number and processes you have. — fdrake

Spot on. The difference between analysis (in the head) and dissection (on the bench) seems obvious, but turns out to be quite difficult to trace in certain situations.

Thomson's Lamp and similar supertasks can be placed in an alternate context simply by using time dilation. Assume the lamp goes on and off at increments of $\frac{1}{{{2}^{n}}}$ but the experiment is on a spaceship that travels at ${{v}_{n}}=c\sqrt{1-\frac{1}{{{4}^{n}}}}$. Then, on board the ship, as time quickly ticks down to zero, on Earth each such tick corresponds to 1, so that the Earthbound observer recognizes a tick each constant unit of time and the task goes on forever.

Just a passing thought on a thread that behaves in a similar fashion.

The trouble is that many philosophers seem to be hypnotized by physics, and seem to forget that physicists develop their theories and conduct their experiments in ordinary human reality. — Ludwig V

Well, physics got strange in the 20th century, leading to a lot of philosophy, good and bad. There's a philosopher of physics named Tim Maudlin whose videos were showing up on my YouTube feel a lot a couple of years ago, but not at all lately. He knows a lot of physics. Seems to be one of the few philosophers who does.

If the Supreme Button Pusher turns the lamp on at midnight then it must have been off before He turned it on, because that's what "turning on" means. If He turned it off at midnight then it must have been on before He turned it off, because that's what "turning off" means. — Michael

I understand your point. I really do. But you are expecting a limit point to have an immediate predecessor, when it does not.

Consider the sequence of times at which we "sample" the state of the lamp. I accept your point that the lamp exists continuously, so that (for example) if we want to know the state of the lamp at, say, 1/3 second before midnight, we would just note that at 1/2 second before midnight, it was off (I think we set it up that way earlier. On at 1 sec, off at 1/2, on at 1/4, all times before midnight.

So it's off at 1/2 sec before midnight, and it will stay off till we turn it on again at 1/4 sec before midnight.

Consider the infinite sequence 1, 1/2, 1/4, 1/8, ... Mathematically, it has the limit 0; and in the lamp puzzle, the limit 0 represents midnight; that is, zero seconds before midnight.

Now two things are true:

For any of the times in the sequence -- 1, 1/2, 1/4, etc. -- it has an immediate predecessor (except for 1 of course).

But 0 does not have an immediate predecessor.

It makes no sense to ask what is the state of the light "immediately before" time 0. Such a thing is not defined.

If you want to interpret this as meaning that the premises or conditions of Thompson's lamp makes it impossible; then I'm fine with that. I'm fine either way. I think you can just arbitrarily define to be the state at midnight as on, off, or turning into a plate of spaghetti.

But perhaps that's just another way to say that the problem is impossible. If so, I can live with that.

But tha does NOT mean that supertasks are impossible. It only means that THIS PARTICULAR supertask is impossible. There might be others that ARE at least metaphysically possible. I've argued this several times. A Zeno walk from my couch to my fridge is one such, under the (perhaps unrealistic, but at least metaphysically feasible) condition that time is modeled by the mathematical real numbers, infinitely divisible and Cauchy-complete.

Surely you must take my point. Even if the lamp is impossible, some other supertask might still be possible. Thompson's lamp is a rather poor example of a supertask, because its underlying sequence can not be made to converge to a limit.

For the entire continuous time that the lamp exists – not just the time during which the button is pushed – it being off must precede it being on, regardless of magic or divine intervention. As such the sequence off, on, off, ..., on makes no sense, much like having an imaginary number of apples in the fridge makes no sense. — Michael

I take your point, and for sake of discussion I'll agree that Thompson's lamp is impossible. I also think you can just complete it arbitrarily. But if you don't like that, then I'll agree it's impossible.

I do understand the point you think you are making. But you are just unhappy that the limit of a sequence has no immediate predecessor in the sequence, and that's just a mathematical fact that's true and that you don't like.

But also note P1 in the argument above, implicit in Thomson's argument. You don’t get to just introduce God to deflect from the incoherent causal consequence of having pushed a button an infinite number of times. — Michael

Ok. I'll agree for sake of discussion that Thompson's lamp is impossible.

A supertask is just performing an infinite succession of tasks in finite time. Pushing a button, which just so happens to turn a lamp on and off, is as good a supertask as any. — Michael

No, it's a terrible example of a supertask, because it can not possibly be made to have a limit.

Consider a Zeno walk from my couch to the kitchen. First I go 1/2 the distance. Then I go 1/2 the remaining distance, and so forth. So at each step I've covered 1/2, 3/4, 7/8, ... of the distance.

After one minute (say I go half the distance in half a minute, etc.) I am in the kitchen.

Now this is the same exact problem as with the lamp, namely that the limit of the kitchen has no immediate predecessor, just as the limit 1 of the sequence 1/2, 3/4, 7/8, ... has no immediate predecessor in the sequence.

But ending up in the kitchen has the virtue of at least being the limit of the sequence. So it's a natural solution. And, under the assumption that time is like the mathematical real numbers, it's a true supertask. I performed infinitely many steps in finite time.

But let’s assume that pushing the button displays the time that the button is pushed – persisting until the button is pushed again – and is pushed (only) at successively halved intervals of time starting two minutes to midnight. What time is displayed at midnight? — Michael

Midnight. What time does YOUR clock say at midnight? Of course the clock says midnight at midnight.

You are just psychologically uncomfortable with the fact that the limit has no immediate predecessor in the sequence.

The sequence approaches midnight but because the button is never pushed at midnight the display can never show midnight, but must show something because it’s never turned off. It always only displays the time that the button was last pushed, but in this scenario there is no last button push, entailing a contradiction. — Michael

I'll concede your point that the lamp is impossible. That does not necessarily entail that supertasks in general are impossible. Only that this one is. But there is no time immediately before midnight, for the same reason there's no inverse integer power of 2 immediately before 1. It's just how infinitely divisible sets work.

As Thomson says, "the impossibility of a super-task does not depend at all on whether some ... sequence is convergent or divergent." — Michael

Oh, that is interesting. So I will take from this the happy conclusion that Thompson and I are thinking along exactly the same lines.

I don't know enough about supertasks to know if every supertask involves demanding that there be an immediate predecessor to the limit of a sequence.

But since there isn't an immediate predecessor to the limit of a sequence, I have to conclude that whoever is defining supertasks such that a mathematically impossible thing must become possible, to that extent, supertasks are impossible.

But they're just making a mountain out of a mathematical molehill. The ordinal $\omega$ does not have an immediate predecessor, and that's a fact.

So you are making this argument:

P1: Every supertask involves finding the immediate predecessor of $\omega$.

P2: $\omega$ does not have an immediate predecessor.

Therefore:

C1: There can be no supertasks.

I believe this is the heart of your argument. I'll make another run at the SEP article because it's interesting that Thompson said that convergence doesn't matter.

ps -- Ok, SEP says this:

Benacerraf (1962) pointed out a sense in which the answer is yes. The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. 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

So I have been expressing Benacerraf's point all along. Good to know. I may be wrong, but I'm at least in the ballpark of professional philosophical thought.

pps -- I love the example of the bouncing ball electrically connected to the lamp. That actually gives what I would call a natural solution in which the lamp is on, based on the fact that a bouncing ball eventually ends up on the ground.

ppps -- And:

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

But this is exactly what I've been saying all along.

So you are holding a position that many (some, several, whatever) philosophers disagree with, and you are arguing that I should break with them and come to your side. But the SEP article definitely shows that some philosophers are in complete agreement with the ideas I've been expressing all along.

the limit of a sequence has no immediate predecessor in the sequence, and that's just a mathematical fact — fishfry

I have always accepted this; it's the reason that the supertask is proven impossible.

A lamp being off must always precede it being on, and so the sequence off, on, off, ..., on cannot model a lamp.

Midnight. ... Of course the clock says midnight at midnight. — fishfry

It's not a clock. It's a screen that displays the time that the button was most recently pushed.

If the button is pushed at 23:00 then the screen will display 23:00 until the button is pushed again at 23:30, and then the screen will display 23:30 until the button is pushed again at 23:45.

The button is not pushed at midnight and so the screen will not display 00:00 at midnight, even though 00:00 is the limit of the sequence.

Thompson's lamp is a rather poor example of a supertask, because its underlying sequence can not be made to converge to a limit. — fishfry

A supertask is any infinite succession of tasks performed in finite time. Having that task be to push a button is as good a task as any, regardless of what pushing the button actually does.

I'll concede your point that the lamp is impossible. That does not necessarily entail that supertasks in general are impossible. — fishfry

The lamp has two buttons. One button turns the lamp on and off, the other does nothing. The same mechanism is responsible for pushing both buttons. If it is possible to have pushed one of these buttons an infinite number of times then it is possible to have pushed the other button an infinite number of times.

If Thomson's lamp proves that it is impossible to have pushed the button that turns the lamp on and off an infinite number of times then it proves that it is impossible to have pushed the button that does nothing an infinite number of times.

And we can replace pushing a button with literally any other task and the same conclusion follows. Therefore, Thomson's lamp proves that all supertasks are impossible.

I have always accepted this; it's the reason that the supertask is proven impossible.

A lamp being off must always precede that lamp being on, and so the sequence off, on, off, ..., on cannot model a lamp. — Michael

At exactly midnight, there is no time interval or delay prior to it that doesn't jump us back into the sequence. This is because midnight is a limit point of the sequence.

But I've already discovered that Benacerraf and others have viewed this problem exactly as I do. So my position is valid, or you should take your objections up with them. The problem does not defined the lamp state at midnight and it can be anything you like.

If the button is pushed at 23:00 then the screen will display 23:00 until the button is pushed again at 23:30, and then the screen will display 23:30 until the button is pushed again at 23:45. — Michael

Yes I agree with your point that if we look at the lamp between two consecutive inverse powers of 2, the state is determined by the earlier point.

But you are trying to apply that same reasoning to the limit point, and you can't do that. Every interval around the limit point necessarily contains all but finitely many members of the sequence. There is no tiny little interval before midnight where the sequence has ended, leaving the lamp in a particular state. That's just a mistaken intuition on your part.

A supertask is any infinite succession of tasks performed in finite time. Having that task be to push a button is as good a task as any, regardless of what pushing the button actually does. — Michael

Your logic is bad. I shouldn't eat one of those poisonous Japanese pufferfish, but that doesn't mean I should turn down a nice tunafish salad sandwich.

But now that I think of it, I no longer even accept that the lamp is "impossible" via your logic. You just haven't got a good intuition for the limit of a sequence. Any step backward, no matter how tiny, jumps over all but finitely many members of the sequence.

So if the lamp is on at midnight, there IS a prior moment at which it was off, and vice versa. Just not an immediately preceding point, because there is no such thing.

The lamp has two buttons. One button turns the lamp on and off, the other does nothing. The same mechanism is responsible for pushing both buttons. If it is possible to have pushed one of these buttons an infinite number of times then it is possible to have pushed the other button an infinite number of times. — Michael

Yes, there are as many even inverse powers of 2 as odd ones.

If Thomson's lamp proves that it is impossible to have pushed the button that turns the lamp on and off an infinite number of times then it proves that it is impossible to have pushed the button that does nothing an infinite number of times. — Michael

It proves no such thing. And having discovered that Benacerraf totally agrees with me, I don't have much reason to continue to assert the same points I've made over and over. Clearly, reasonable people agree with my point of view. Benacerraf is still alive, perhaps you can take this up with him.

And we can replace pushing a button with literally any other task and the same conclusion follows. Therefore, Thomson's lamp proves that all supertasks are impossible. — Michael

If repeatedly asserting a falsehood were proof, we'd all be geniuses.

As a side note, you’ve proven that reasoning with you about how infinity works is a supertask.

Therefore, Thomson's lamp proves that all supertasks are impossible. — Michael

Now this conversation can resume towards its ever ever-approaching, but never accomplished, conclusion.

But I've already discovered that Benacerraf and others have viewed this problem exactly as I do. So my position is valid, or you should take your objections up with them. The problem does not defined the lamp state at midnight and it can be anything you like. — fishfry

I addressed his paper a month ago. See also here for a more formal argument.

As for the screen, it can only ever display the time that the button was most recently pushed. If the button is only ever pushed at 23:00 then at midnight the screen displays 23:00. Notice that the rule about how the screen works still applies at midnight, long after the button has been pushed. This rule also still applies at midnight if the button is only ever pushed at 23:00, 23:30, 23:45, and so on.

And because the button is not pushed at midnight the screen cannot display 00:00 at midnight, even though 00:00 is the limit of the sequence.

There is no tiny little interval before midnight where the sequence has ended, leaving the lamp in a particular state. — fishfry

This is why the supertask is impossible. A lamp can only be on if something causes it to change from off to on. This is as true at midnight as it is at any time before and after midnight. It’s a fundamental fact about lamps. Therefore a lamp cannot be modelled by the sequence off, on, off, …, on.

As a side note, you’ve proven that reasoning with you about how infinity works is a supertask. — Fire Ologist

I understand how infinity works. The problem is that fishfry doesn't appear to understand how a lamp works.

Not only is no answer to the second question deducible from the premises (unlike the answer to the first question), no answer consistent with the premises is possible. — Michael

That’s what I was saying about the lamp a couple weeks ago and you were disagreeing with me and telling me how time works and how lamps work.

If we are using a limit within which there is an infinitely decreasing time interval, or infinitely decreasing distance interval, then, regarding a question about some state of time or some state of position at the limit, “no answer to the question is deducible from the premises.”

That’s it. No one can give a crap about the lamp or time or lights on or or off at 2 minutes, at 7 minutes, because it is nowhere in the universe of the premises. You don’t even need to fix the limit at precisely 2 minutes. It can be slightly before two minutes and the premises will still not admit of the existence of that limit.

Any discussion of the bottom of an infinite staircase, of the state of the lamp at two minutes, or whether Achilles can beat the tortoise, cannot be deduced from the premises.

It’s like we are approaching agreement, but you are quibbling about half of what I say, and then we approach agreement again and you quibble about half again.

Or you are very simply not explaining yourself.

You clearly understand the facts - time in minutes and seconds, halving fractional functions, the difference between on and off. You do understand infinity.

But you seem to disagree with how these facts must clearly relate to one another in order to make some point about “completing super tasks” and “contradiction”.

If you are saying the lamp must be on or off at two minutes because that’s how time works, and that’s how lamps work, then you are not recognizing that “no answer to the question is deducible from the premises.”

You are assuming something that you are not stating, not stating something clearly, not defining supertasks, or wrong about one or more of these things if you think the lamp has to be on or off at 2 minutes because that’s how lamps work.

And because lamps are either on or off at all times, but you can’t deduce the state from the premises, we don’t have a contradiction. The premises of switching a lamp on and off at ever decreasing intervals of time simply do not speak to the state the lamp will be at when two minutes or more elapse. It just means that from your infinite fractional function “no answer to the question is deducible from the premises.”

And because lamps are either on or off at all times, but you can’t deduce the state from the premises, we don’t have a contradiction. — Fire Ologist

Not only is no state deducible from the premises, no state is consistent with the premises.

The lamp is turned on if and only if the button is pushed when the lamp is off.
The lamp is turned off if and only if the button is pushed when the lamp is on.

Neither of these are possible at t₁ if a supertask has been performed between t₀ and t₁. But the lamp must be either on or off at t₁. This is the contradiction.

Not only is no state deducible from the premises, no state is consistent with the premises. — Michael

But no state being consistent with the premises doesn’t create a contradiction. At two minutes, the lamp becomes a duck and flies away. That doesn’t contradict the premises, because the premises never touch the lamp at two minutes. Two minute lamp is utterly a new whole scenario, needing new premises to start being discussed at all.

That doesn’t contradict the premises, because the premises never touch the lamp at two minutes. — Fire Ologist

Yes they do. These are our background premises:

P1. Between 00:00 and 23:59, nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 00:00
P5. The lamp is either on or off at 12:00

Only then do we consider the manner in which the button is pushed.

If the button is only ever pushed at 11:00 then the lamp is on at 12:00.

If the button is only ever pushed at 11:00, 11:30, 11:45, and so on, then the lamp can neither be on nor off at 12:00. This is a contradiction.

Note especially P1 which is where fishfry and Benacerraf go wrong.

P1. Between 22:00 and 01:00, nothing happens to the lamp except what is caused to happen to it by pushing the button — Michael

I think you mean between 22:00 and 23:00, assuming the two minute mark is 0:00.

Why do you need to redefine the premises? We are getting nowhere over and over again.

It’ time zero and two minutes later that are the limits. Two minutes later is a theoretical, because it is never actually reach by halving the prior interval starting at zero time, one minute time, one and half minutes time, one and three quarters, etc.

Precisely two minute lamp is outside the scenario. Period. Whatever state or non-state you assign or can’t assign to it, is not a function of the half timed lamp switching scenario. You are ducking the issue.

I think you mean between 22:00 and 23:00, assuming the two minute mark is 0:00. — Fire Ologist

No I don't. I mean exactly what I said.

Why do you need to redefine the premises? We are getting nowhere over and over again.

It’ time zero and two minutes later that are the limits. Two minutes later is a theoretical, because it is never actually reach by halving the prior interval starting at zero time, one minute time, one and half minutes time, one and three quarters, etc.

Precisely two minute lamp is outside the scenario. Period. Whatever state or non-state you assign or can’t assign to it, is not a function of the half timed lamp switching scenario. You are ducking the issue. — Fire Ologist