The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined. — Ludwig V

That's precisely why supertasks are impossible.

The lamp is off at t₀. The lamp must be either on or off at t₁. But if the button has been pushed an infinite number of times between t₀ and t₁ then the lamp can neither be on nor off at t₁.

It's strange that some are taking the very thing that proves that supertasks are impossible as proof that the proof isn't a proof.

That's precisely why supertasks are impossible. — Michael

My point is that I think that the disagreement between you and @fishfry is about different ways to make the same point.

There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function. — Michael

Quite so. Wittgenstein made much of the endlessness of infinity and asked how it was possible. You may know what his answer is. If you don't, it is easy to look it up. (It would be far too long to try to outline it in this context and you likely know anyway.
My question is, how do we know that it is not possible to complete a recursive function (in the sense of writing or speaking each step that the function defines)? BTW, I don't think there is any particular problem about defining such a function, though you could argue that it is the result of misapplying an operation that is perfectly harmless in other contexts, like √2 or a self-referential pronoun like "I" or "this".

For example, proving that √2 is irrational doesn't rest on trying and failing to write it down, but on showing that the assumption that √2 is rational leads to a contradiction - reductio ad absurdum.
In the case of infinity the argument rests on mathematical induction. I understand that not everyone is happy with that argument but it seems OK to me. A recursive function is defined, but in such a way that its end is not defined.
So even if a sequence cannot be completed empirically, so to speak, there are ways of proving that it is endless with attempting the impossible refutation of writing down each step.

The code here is effectively the same as a recursive function.

My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V

I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible.

Conclusion: set theory is in violation of the law of identity. I've explained to you why this is the case. Do you agree with me? — Metaphysician Undercover

LOL.

No, I don't agree with you.

Surely, the contradiction is the result of the lack of any definition of the terminal state. If the terminal state could be a plate of spaghetti, why couldn't be a lamp that is neither on nor off?

I really cannot see what you two are arguing about. Why does the difference matter? — Ludwig V

Me either.

The plate of spaghetti is a great dramatic way of making the point that there is no definition. But the series is defined on the basis that its limit is 1. You can't derive 1/2 from a plate of spaghetti. — Ludwig V

Don't follow. The limit of 0, 1, 0, 1, ... can not be 1. Nor can it be 0. It's a sequence that has no limit.

My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V

I do not think @Michael and I are making the same point.

So why don't you conclude that the use in the context of the law of identity violates the use in the context of set theory? It seems to be an arbitrary choice. — Ludwig V

I'd agree except that the law of identity was first, set theory came along after. So set theory violated the law of identity, which was already established. If it was the other way around, then we'd want to look at the reasons why someone would be trying to enforce the law of identity, which seems like a useless, obvious, self-evident tautology, in violation of how set theorists were defining "same".

Well, if the law of identity is an obvious self-evident tautology, then it appears like there must be something wrong with set theory if it's in contradiction with what is obvious. We can see something like this in Aristotle's work to establish the law of identity. He claimed that the law of identity was necessary to battle against sophists who could logically demonstrate absurdities. If the base axioms of a logical system are contrary to what is self-evident tautology, then that system will be able to prove things which are contrary to what is obvious, one can prove the absurd.

The meaning of "same" depends on its context. — Ludwig V

Very true, but we need to pay attention to subtleties to avoid deception. Suppose you and I both drive the same make and model of car, and same year and colour as well. Notice, that "same" is being used in a qualified sense, referring to different properties "same make", "same model", "same year", "same colour". Now, suppose I say that I drive the same car as you. This would be incorrect. But why is it incorrect, because everything about it seems to be the same? The thing is that everything about it is not the same, only those named qualities are the same, and that's why it's incorrect to say that it is "the same" in that unqualified sense.

Now, when we say that "two sets are the same", this is incorrect for the same reason. Not everything about the two is necessarily the same, only the stipulated required qualities. So it is incorrect to say that the two sets are the same, in the unqualified sense, because some features like the ordering of the elements may be different.

I'd agree except that the law of identity was first, set theory came along after. — Metaphysician Undercover

Temporal priority is not logical priority.
Whether a proposition is obvious (self-evident) or (self-evidently) absurd are subjective.

Well, if the law of identity is an obvious self-evident tautology, then it appears like there must be something wrong with set theory if it's in contradiction with what is obvious. — Metaphysician Undercover

There are plenty of ways to formulate that law without using the word "same". In any case, "same" in that context just means "same object", so it isn't absolute. moreover, If you drive my car, you don't drive it at the same time.

The thing is that everything about it is not the same, only those named qualities are the same, and that's why it's incorrect to say that it is "the same" in that unqualified sense. — Metaphysician Undercover

There is no unqualified sense of "same".

Not everything about the two is necessarily the same, only the stipulated required qualities. — Metaphysician Undercover

We agree!

So it is incorrect to say that the two sets are the same, in the unqualified sense, — Metaphysician Undercover

There is no unqualified sense of "same".

He (sc. Aristotle) claimed that the law of identity was necessary to battle against sophists who could logically demonstrate absurdities. — Metaphysician Undercover

I'm sure that Aristotle would not object to my regarding that as not a logical argument.

Don't follow. The limit of 0, 1, 0, 1, ... can not be 1. Nor can it be 0. It's a sequence that has no limit. — fishfry

I'm sorry. I was talking about the convergent series. Didn't check

I do not think Michael and I are making the same point. — fishfry

Perhaps not. But if the last term in the series is not defined, contradictions are likely to follow from the attempt to identify it. Equally, if something gives rise to a contradiction, the definition will be faulty. So, if you are right, I need to ask why it matters.

The code here is effectively the same as a recursive function. — Michael

Not quite. The code specifies a process which must take time. The function does not.

I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible. — Michael

Thank you. I must have got confused.

I'm sorry. I was talking about the convergent series. Didn't check — Ludwig V

My confusion.

I do not think Michael and I are making the same point.
— fishfry
Perhaps not. But if the last term in the series is not defined, contradictions are likely to follow from the attempt to identify it. Equally, if something gives rise to a contradiction, the definition will be faulty. So, if you are right, I need to ask why it matters. — Ludwig V

There is no last term in any infinite sequence. There may (or may not) be a limit. Big difference.

There is no last term of the sequence 1/2, 1/4, 1/8, ...

It has a limit of 0. But 0 is not an element of the sequence.

Also the terminology I'm using is that a sequence is a comma-separated infinite list, 1/2, 1/4, 1/8, ...

A series is an infinite sum: 1/2 + 1/4 + 1/8 + ...

The elements of a sequence are not "premises," as @Michael seems to think. They're just numbers. Some sequences, like 1/2, 1/4, 1/8, ... have a limit, in this case 0. Other sequences, like 0, 1, 0, 1, ... can't possibly have a limit. That doesn't stop you from defining a "terminal state," which I've formalized as a value at $\omega$. But there's no need for there to be any logical relation between the sequence itself, and the arbitrarily-defined terminal state. That's why lamp on, lamp off, or lamp turns into a plate of spaghetti, are all equally valid terminal states to the sequence 0, 1, 0, 1 ...

Even if you insist that the terminal state must be either 0 or 1, there is no logical way to prefer one over the other.

There is no last term in any infinite sequence. There may (or may not) be a limit. Big difference. — fishfry

Yes. The exact status of 1 or 0 in these cases is more complicated than I realized.

Even if you insist that the terminal state must be either 0 or 1, there is no logical way to prefer one over the other. — fishfry

So can you help me to describe the role of 1 in defining the series 1/2, 1/4, ... when the limit state is 0? (Or indeed when it's the other way round?)

You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game.
— fishfry

No you can't. — Michael

What do you mean I can't? I already have, numerous times in this thread.

The point is that the sequence 0, 1, 0, 1, ... has no limit. We are free to define a terminal state that is not a limit, but is just a value assigned to a point beyond the sequence, traditionally notated as $\omega$. This value could be 0, it could be 1, or it could be a plate of spaghetti.

Now I will agree with you that we COULD make a rule that the terminal state must be 0 or 1. That rules out the spaghetti. But there is no logical preference between 0 and 1. Neither value could be the limit of the sequence, since that sequence does not have a limit.

I addressed this in my initial defence of Thomson here, and even more clearly below. — Michael

You can't have addressed it, since what I said is true.

You're claiming that "a plate of spaghetti" is a coherent answer to the question "is the lamp on or off after two minutes?" — Michael

It's consistent with the rules of the game. But if you add the rule that it must be 0 or 1, then either answer will do, and neither answer has any logical relationship to the sequence, since the sequence doesn't have a limit.

SEP is clear that Thompson was clear on this point, in full agreement with me.

So I think the confusion is yours. — Michael

You know, I lost track of whether I said this to you, or you said this to me. I'm sure we both feel that way. Can't we just agree to disagree? This poor deceased equine has been flogged well into the next life.

He discusses the sequence and its sum, but only to show its irrelevancy, hence the earlier quote. — Michael

All possible terminal states are equally irrelevant, because the sequence has no limit.

May I ask, you do you understand that? Do you understand that the sequence 0, 1, 0, 1, ... has no limit, and can not be made to have a limit?

From his paper:

What is the sum of the infinite divergent sequence +1, -1, +1, ...? Now mathematicians do say that this sequence has a sum; they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. — Michael

Completely irrelevant to anything. Nor do mathematicians say any such thing, except by using asymptotic density (which I believe SEP refers to as Cesaro summation).

You haven't proved your point. You haven't even made a point.

But there's no need for there to be any logical relation between the sequence itself, and the arbitrarily-defined terminal state. — fishfry

So, the terminal state not being defined does not prevent me defining one arbitrarily?
Isn't it the case that there is a requirement - that the terminal state not be defined by the function.

Have you even read Thomson's paper? This is the most relevant part:

There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. So if the lamp was originally off, and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on, according to Russell's recipe. After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

The answer to his question isn't "a plate of spaghetti", it isn't "0", it isn't "1", and it isn't "1/2".

The only coherent answers are "on" and "off" – except as he argues, it can't be either, and so therefore it is impossible, even in principle, to have succeeded in pressing the button an infinite number of times.

@Michael, This post may be of interest to you.

So, the terminal state not being defined does not prevent me defining one arbitrarily?
Isn't it the case that there is a requirement - that the terminal state not be defined by the function. — Ludwig V

This is the bit with the ordinals I was using earlier.

What is a sequence? A sequence is a function whose domain is the natural numbers 1, 2, 3, ...

For each natural number, there's a value, like 1/2, 1/4, 1/8, ...

We can conceptually adjoin a "point at infinity" to the natural numbers, traditionally called $\omega$ in this context, so that our extended natural numbers look like

1, 2, 3, 4, ... $\omega$.

This structure, the extended naturals, are called $\omega + 1$, because the natural numbers themselves are called $\omega$. That's just another name for the more usual notation $\mathbb N$, except that $\omega$ implies the set of natural numbers in their usual order.

So if a sequence like 1/2, 1/4, 1/8, ... is a function on the domain $\omega$, an "extended sequence" (not the official terminology) is a function on $\omega + 1$. Such a function might look like this:

1/2, 1/4, 1/8, 1/16, ..., 47

Here I just defined 47 as the terminal state.

Another possible completion is

1/2, 1/4, 1/8, 1/16, ..., plate of spaghetti

There is absolutely nothing wrong with that. I've just adjoined a plate of spaghetti to the rational numbers, and defined the terminal state of this particular sequence to be the plate of spaghetti.

Now in this case there is ONE completion that is "natural," namely:

1/2, 1/4, 1/8, 1/16, ..., 0

That completion is just as arbitrary as any other. But it has one supreme virtue: 0 happens to be the limit of the sequence. So that's why I call it natural.

The problem with the lamp is that the sequence 0, 1, 0, 1, ... has no natural completion. That's why we can define the terminal state of the lamp as on, or off, or a plate of spaghetti, or Cinderella's coach.

All completions are legal; and no completion is natural; because the lamp sequence does not have a limit.

You don't seem to understand what is being discussed at all. — Michael

Have a nice evening. I'll forego responding in kind.

Michael, This post may be of interest to you. — fishfry

I'm afraid it's not, because it doesn't address the issue of supertasks.

For supertasks, we have this:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t₀
P4. The lamp is either on or off at t₁

Q. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

It makes no sense to answer this question with "a plate of spaghetti" or "$1\over2$".

It makes no sense to answer this question with "a plate of spaghetti" or "1/2". — Michael

I think you'll find that's because it makes no sense to answer the question.

In other words, it also makes no sense to answer the question with "on" or "off".

I understand why those answers seem more natural, but that's an illusion.

That completion is just as arbitrary as any other. But it has one supreme virtue: 0 happens to be the limit of the sequence. So that's why I call it natural. — fishfry

I understand that. What seems important to me is that the convergent series is the result of a calculation which involves 0 and 1, while "0,1, 0, 1, ..." doesn't involve any calculation at all. You could also have a series "a, b, a, b, ..." or "fish, chips, fish, chips, ..." The calculation involves numbers, but "0, 1, 0, 1, ..." only involves numerals.

I think you'll find that's because it makes no sense to answer the question.

In other words, it also makes no sense to answer the question with "on" or "off". — Ludwig V

The lamp is either on or off at t₁. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times.

The lamp is either on or off at t1. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times. — Michael

Exactly. The contradiction follows from the fact that no final state is defined.

I understand that. What seems important to me is that the convergent series is the result of a calculation which involves 0 and 1, while "0,1, 0, 1, ..." doesn't involve any calculation at all. You could also have a series "a, b, a, b, ..." or "fish, chips, fish, chips, ..." The calculation involves numbers, but "0, 1, 0, 1, ..." only involves numerals. — Ludwig V

Well, it's not a calculation. The limit, or the terminal value, is not necessarily the result of any logical process. I keep trying to explain this to @Michael. The limit of 0 does not "come from" the sequence 1/2, 1/4, 1/8, ... in the sense that you could put those numbers into a machine and 0 would pop out.

Rather, you are given the sequence; and given the limit; and you can apply a formal definition to see that 0 is indeed the limit of the sequence. It's conceptually sort of the other way 'round from thinking that the limit is the result of some logical process applied to the sequence. Although "find the limit of this sequence" is a common calculus problem, so there are heuristics and strategies to find limits of sequences that have closed-form representations. But not all sequences do, so my remark stands.

That's the think with the lamp. You can define the terminal state as on or off. Neither makes sense, and neither is forbidden. The sequence still stands on its own terms. On, off, on, off ... All that happens, as you count through 1, 2, 3, ... Then you arbitrarily assign a terminal state. On, off, or a plate of spaghetti.

@Michael just doesn't get this or just doesn't want to get this.

If Thompson were here I'd explain this to him. Thompson and @Micheal are making the same conceptual error. There is no logical relationship at all between a sequence, and some arbitrary terminal value. If the terminal value happens to be the formal limit of the sequence, that's a lovely special case that happens a lot in calculus class, where all the problems are designed to work out nicely for diligent students.

But it can't happen with the lamp. The sequence on, off, on, off ... does not have a limit

End of story as far as I'm concerned.

Temporal priority is not logical priority. — Ludwig V

I know, but if there is an existing law, and someone does something that is contrary to that law, then that person violates the law. That is a simple fact.

There is no unqualified sense of "same". — Ludwig V

I believe that what is attempted with the law of identity is to express an unqualified sense of "same". You seem to think it fails. Why?

It makes no sense to answer this question with "a plate of spaghetti" or "1/2 — Michael

The 1/2 was Thompson's idea. The plate of spaghetti is mine, inspired by Cinderella's coach, another omega sequence paradox that's not usually recognized as one.

The thing you and Thompson keep missing is that the lamp is not real. It violates the currently known laws of physics; and because it stipulates a circuit that can change state in arbitrarily small intervals of time as modeled by the mathematical real numbers, it's difficult to imagine, even speculatively, that the lamp could ever be physically realized even under far future physics.

So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. Or as a plate of spaghetti, if you so desire. There's nothing about the lamp that's real. The lamp can turn into a plate of spaghetti exactly as Cinderella's coach turns into a pumpkin. By fiat in a fairy tale.

Please see my response to @Ludwig V above, where I expand on this theme.

https://thephilosophyforum.com/discussion/comment/907876

He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period. — keystone

The rest periods at step 1, instead of being 30 seconds long, is that really just 2 times as long of rest than at step 2? Or is it 4 times as long as step 3, or is it 6 times as long as step 4? In which case, this story has as much to do with increasing speed between steps as it does shortening rests on each step. So rest and motion are needed to place steps in flight beneath you. Is it the increasing speed that shortens the rest, or the shorter rests that increase the speed? How can any rest cause motion to increase?

And why not a bottomless pit if you are to reach an impossible bottom?

This whole image is that of a square circle. I don’t see what there is to resolve.

There can exist no infinite anything. Thingness, such as a step or a series of steps, is finitude. Infinity is not a thing to which you can add 1 or subtract from, certainly not when describing actual steps. Calling a thing or a series of things infinite, removes the thing or things from your sight, removes them from the chalkboard, and shows you the same infinity as imagined in the infinite series of fractions between step 1 and step 2. There is no such thing as a half step. Not is there a such thing as an infinite series of steps. There is only a whole distance later conceptually halved, as when you conceive of halving some existing whole step infinitely.

“The infinite” or “infinity” as a noun, is best used for dramatic effect. It’s not a thing, like a noun is best employed. “Infinitely” as an adverb, sets out some activity that, by definition, cannot conclude. Thereby banishing all finitude, which marks conclusion, such as a step, or a series of steps, or a noun.

“Halving” as a verb, like “stepping” as a verb, can be conceived of as continuing infinitely. But you never find the infinite. There need be no infinitely small fraction. Saying the stars and the atoms in the multiverse are infinite in number means you don’t know how to count them, so for dramatic effect, we invoke “the infinite”. But the infinite finds no home, no place in the physical world, in the form of the finite, save the mind that conceives of some activity that can continue infinitely.

We might as well start this by saying, “there were three steps to the basement, but before he took the first one Icarus had to get off the couch, but as he did so, he realized he had to first sit up, and then realized he had to move his legs to the floor, and increasing his effort between each new realization, he realized he had to move his first leg, by first turning his foot, after ending his knee… etc. infinitely, as he turned his eye and saw a corpse on the couch with him….

The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists… — keystone

No it doesn’t. There is no infinite “it” that could simultaneously do anything, such as exist.

So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. — fishfry

A supertask is not simply an infinite sequence.

With a supertask we are given some activity to perform and we assume that it is physically possible to perform this activity at successively halved intervals of time. We are then asked about the causal consequence of having done so.

We do not get to introduce additional (and nonsensical) premises such as "and then the lamp magically turns into a plate of spaghetti, prior to which the lamp was neither on nor off."

The lamp must be either on or off after two minutes. If the lamp is on after two minutes then it is on only because the button was pushed to turn it on, prior to which the lamp was off. If the lamp is off after two minutes then it is off only because the button was pushed to turn it off, prior to which the lamp was on. The supertask doesn’t allow for either of these scenarios and so is proven impossible in principle.

Rather, you are given the sequence; and given the limit; and you can apply a formal definition to see that 0 is indeed the limit of the sequence. It's conceptually sort of the other way 'round from thinking that the limit is the result of some logical process applied to the sequence. — fishfry

I take the point. I may not have stated it accurately enough, but the crucial thing, it seemed to me, is to realize that the limit is part of the definition from the start - not, as I think you're saying, something that is worked out from the sequence itself.

I believe that what is attempted with the law of identity is to express an unqualified sense of "same". You seem to think it fails. Why? — Metaphysician Undercover

If I say that Hesperus is Phosphorus, I am saying that they are the same object.

“The infinite” or “infinity” as a noun, is best used for dramatic effect. It’s not a thing, like a noun is best employed. “Infinitely” as an adverb, sets out some activity that, by definition, cannot conclude. Thereby banishing all finitude, which marks conclusion, such as a step, or a series of steps, or a noun. — Fire Ologist

You are right, Language is a great trap here. I would like to use "endless" or "endlessly" and even "endlessness" instead. That would make it more difficult to talk about conclusions. But we are lumbered with a world which uses "infinity". Natural language allows this, but has no guard rails to prevent us from talking nonsense.

But the infinite finds no home, no place in the physical world, — Fire Ologist

The difficulty here is that it is possible to defined an infinite series in a finite frame, which leads people to think of apply the abstract idea to the physical world. Sometimes that works, as in physics, so we can't just say that such ideas have no place in the physical world.

But you never find the infinite. There need be no infinitely small fraction. — Fire Ologist

Yes, we do. We don't find them by failing to count them, but through various arguments. The proofs that π or sqrt(2) or that there is no largest natural number are all well established. So is the possibility of a convergent series.

There is no such thing as a half step. — Fire Ologist

True, if you are thinking of a staircase. But nobody would contest that. But if you think of the distance between my eyes, you can certainly divide that by 1/2 or 1/4 or...

But if you think of the distance between my eyes, you can certainly divide that by 1/2 or 1/4 — Ludwig V

The distance between your eyes is a whole. You need to grab that finite whole thing first from the physical world to then posit the concept of half of that whole. The half wasn’t grabbed from the physical world. Otherwise it would have been a smaller whole distance to start again. The half-distance comes after a whole is firmly in hand. Let’s say it’s two inches between eyes. You can’t identify half that distance by referring to your face. You call some smaller distance half, numerically, in reference to the numerical value the refers to the whole distance. The whole distance refers to your eyes.

Imagine someone says in order to walk from the goal line to goal line in football, you first have to walk one-hundredth of a distance, but before that you have to walk one-third of the distance to that first one-hundredth mark, etc. etc. infinitely.

Calling these smaller distances fractions is semantics with reference to mathematical concepts. No fraction ever exists. What exists would be one whole distance from goal to goal, one whole yard, one whole foot… we can rename the measures fractions by referring them to some greater whole, but then we need to have the greater whole first before we can measure a fraction. We must walk the entire football field first before we can conceive of a whole yard being 1/100th.

Infinity is like that. It can’t refer to a physical, identifiable thing or be contained in an object. It can only refer to numbers, which are concepts. There are no infinite series of steps.

The interesting thing about Thompson's Lamp thought experiment is that it produces an indeterminate outcome where one might not expect it. If you take an ordinary lamp with an on/off switch, you could think of it as a simple deterministic system, as long as switching is determinate. You can always tell whether the lamp is on or off at any given time t if you know the state of the lamp at some earlier time, plus how many times the lamp was switched between then and t. Another way to tell could be by inquiring whether the lamp was on or off just before t and whether the switch was activated at t. And yet, in this thought experiment with a determinate sequence of switchings, determinism breaks down at a certain point. There is no mystery as to how that happens, but one is still left with a feeling that something funny is going on.

However, it is difficult to see how one could make much metaphysical hay out of Thompson's Lamp. Unlike Zeno's thought experiments, which deal with examples of ordinary motion, Thompson's Lamp is blatantly unrealistic. It is not like anything in this world. Its "motion" is not even piecewise-continuous. So what if it's indeterministic? Who said that it must be? Neither intuitions nor physics are of much help here.

The most one can do here is argue that while, of course, Thompson's Lamp is not physically possible, it is possible in a "metaphysical" sense. And at the same time, its indeterminate state is not possible in that same sense. But that argument would be hard to put across (and I haven't seen anyone seriously try). Just what kind of goofy sense of possibility slices the problem in just such a way? And why would we be obligated to take it seriously?

Unlike Zeno's thought experiments, which deal with examples of ordinary motion — SophistiCat

Continuous motion suffers from the same problem. We can imagine sensors at each successive half way point that when passed turn a lamp on or off. Is the lamp on or off when we finish our run?

The simple solution is to say that motion isn’t continuous. Discrete motion at some scale is a metaphysical necessity.

Quantum gravity and quantum spacetime are viable theories so it’s not a hard pill to swallow.