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 (heavenly body). If I say that Ringo Starr is Richard Starkey, I am saying that Ringo Starr is the same person as Richard Starkey.
If I say that Venus is Venus, I say nothing at all. But even then, it makes a big difference whether you are talking in a context in which Venus is a goddess or a planet.

The distance between your eyes is a whole. — Fire Ologist

I suppose you can. But then I can define as a whole anything I like. A spoonful of sugar. A rainbow. Six inches of two-by-four. The distance between my front door and the shop on the corner. What counts as a part is defined in relation to that. But each part is a whole in its own right. The leg of a chair. The branch of a tree. The handle of a door. Half of a penny. It's just a convenient trick of language.

Mind

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. — Fire Ologist

"Grabbed" from the physical world is a completely inappropriate metaphor. Nothing is grabbed. Something was defined. In any case, if the whole thing was "grabbed from the physical world", it follows that both halves of it were "grabbed". If they weren't, nothing was "grabbed".

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

The simple solution is to recognize the difference between an analysis and a dissection. A dissection physically separates an object into separate parts (and the parts then become wholes in their own right). An analysis has no physical impact on the object at all. One can analyse a distance into metres, centimetres, millimietrs or yards, feet and inches or any other units you like. You can analyse it into any fractions you like. All at the same time. The object doesn't change.

If I ask you what the minimum unit of space is, I can analyse that distance into fractions, however small it it is. Whether I can physically divide an object into those fractions is another question.

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? — Michael

I don't see how it's a problem for continuous motion that you can imagine something else taking place alongside of it.

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

Solution to what? I have yet to see a problem in need of a solution here.

The problem is that if motion is continuous and if the sensors are set up as stated then the lamp can neither be on nor off after the run is completed, which is a contradiction.

One or more of the premises is necessarily false. So either motion is not continuous or we cannot set up sensors at an infinite succession of halfway points. The latter would seem to suggest that there aren’t an infinite succession of halfway points and so would entail the former anyway.

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

That's not the case. "Object" is not implied. You are simply saying that whatever it is that the two names refer to, it is one and the same.

The issue with "=" in mathematics is that the meaning of, or what is referred to by, "2+2" is not the same as what is referred to by "4". Someone might stipulate by axiom, that these two do refer to the exact same thing, but that does not reflect the way that the symbols are commonly used in the application of mathematics. In this way the axiom would be false, in the sense of a false definition. That's why such an axiom is misleading. This is also the problem with formalism, what is stipulated by the formalist is not consistent with the way that logic is applied, therefore it is a false description of logic.

If I say that Ringo Starr is Richard Starkey, I am saying that Ringo Starr is the same person as Richard Starkey. — Ludwig V

Again, this is not true. When you say Ringo Starr is Richard Starkey, all you are saying is that these two names have the same referent. It is only upon analysis, if one seeks to determine whether it is true or not, or something like that, that one would determine that the two names both refer to a person.

The problem is that if motion is continuous and if the sensors are set up as stated — Michael

You are just restating - reimagining - Thompson's Lamp thought experiment, which has nothing to do with continuous motion as such (and repeating once more your baseless conclusion).

You are just restating - reimagining - Thompson's Lamp thought experiment, which has nothing to do with continuous motion as such — SophistiCat

I’m using Thomson’s lamp to show that continuous motion entails contradictions.

and repeating once more your baseless conclusion — SophistiCat

It’s not baseless. I’ve explained it quite clearly here and here and in many other comments.

I’m using Thomson’s lamp to show that continuous motion entails contradictions. — Michael

Well, as far as I can see, you haven't done either: you haven't demonstrated any contradictions in TL, nor linked it to continuous motion. But I am not going to wade into that trainwreck. I chimed in to comment on the TL. You can engage with that if you want, or leave it alone.

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.
— Fire Ologist
"Grabbed" from the physical world is a completely inappropriate metaphor. Nothing is grabbed. Something was defined. In any case, if the whole thing was "grabbed from the physical world", it follows that both halves of it were "grabbed". If they weren't, nothing was "grabbed". — Ludwig V

A single thing that can be grabbed is defined as you say as a unit. A single thing. Like one whole step.

So now we have conceived of the unit. We’ve defined it as 1. As a whole.

Only now can we posit or “define” infinity. Only now can you keep the conception of the unitary whole and define half. Only once you have a single unit can you add to that unit more units infinitely. But at each step, if you refer to the physical thing, you have a finite number of units. And you can’t posit or define or conceive of half without reference to half of some other defined, conceived thing, and that thing must be a whole unit.

There is no infinity apart from the mind that conceives it. There are things apart from the mind that conceives of the unit.

There is no infinite thing to begin with. Only unitary wholes. And infinite staircase is an infinite finite unit - a square circle. There is no infinite thing, so pondering the paradoxes that arise from traversing an infinite distance, or descending and infinite staircase misapplies infinity to unitary whole, single, definable things.

Infinity applies to numbers. Numbers aren’t physical things, like stairs.

you haven't demonstrated any contradictions in TL — SophistiCat

Thomson does that himself in his paper. I am defending his paper and explaining why Benacerraf's response to it fails. See here where I first brought it up.

Perhaps you could explain which part of my (or Thomson's) reasoning you reject? You're a coder so perhaps you could even address the code here. Simply saying "it's wrong" is hardly a meaningful criticism. If that's all you have to say then I will simply reply with "it's right".

nor linked it to continuous motion. — SophistiCat

I did so in the post yesterday. Just as if we push a button an infinite number of times within two minutes the lamp can neither be on nor off after two minutes, if we run through an infinite succession of sensors when running a mile the lamp can neither be on nor off after finishing the run.

Given that the lamp must be either on or off, this is a contradiction, and so therefore it is proved that one cannot have run through an infinite succession of sensors.

Like I said, I don't want to get involved in that old trainwreck. I already said what I wanted to say about Thompson's Lamp. And since you don't want to engage with my posts, I think we are done.

Whether or not the mathematics mentioned here properly applies to the paradoxes mentioned, at least we should be clear about that mathematics. Adding to certain correct points made by the poster fishfry:

A sequence is a function whose domain is an ordinal. If the domain is finite, then the sequence is finite. If the domain is countable, then the sequence is countable. If the domain is w (read as 'omega' the set of natural numbers), then the sequence is denumerable. If the domain is uncountable then the sequence is uncountable. Sometimes we omit 0 from the domain so that there are sequences whose domain is the set of positive natural numbers.

Let t be the following sequence whose domain is the set of positive natural numbers:

t(n) = 0 if n is odd
t(n) = 1 if n is even

t is a mathematically defined sequence. It is not required to express its values per some other arithmetical formula.

t has no greatest member in its domain, there is no last value for t, and t does not converge.

Let x be any mathematical value. Let s be the following sequence whose domain is the set of positive natural numbers along with w itself (the domain of s is {n | n is a positive natural number or n = w}):

s(n) = t(n) if n is a natural number
s(n) = x if n = w

s is a mathematically defined sequence. It is not required to express its values per some other formula.

s has a greatest number in its domain, and the last value for s is x.

This is a way of saying that for any x, there is a sequence in which x is the last value.

A supertask is not simply an infinite sequence. — Michael

Understood.

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. — Michael

The lamp is not physically possible. Supertasks are not physically possible per currently known physics. Surely we agree on these two things. Yes? No?

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." — Michael

Well this is exactly the point I'm making. Why do we get to introduce a nonsensical premise such as a lamp and a switching circuit that can change states in arbitrarily small intervals of time, in contravention of the principles of quantum physics and electrical engineering?

My point is that once we've entered the realm of speculative fantasy, where do we stop? I say the lamp switches in arbitrarily small intervals of time, AND turns into a plate of spaghetti.

And you say ... what? That it CAN violate some laws of physics, but CAN'T violate others? Why?

The lamp must be either on or off after two minutes. — Michael

Suppose that I accept this additional stipulation to the problem. Then I say it's on at two minutes. Or I say it's off at two minutes. Both are consistent with the premises of the problem; and neither is to be preferred, since neither are the limit of the sequence. You can make it anything you like. I use the spaghetti example to illustrate the arbitrary nature of any stipulated terminal state. One answer is as good as any other.

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. — Michael

Now you (and Thompson) are trying to reason logically about a hypothetical situation that is entirely fictional, namely a lamp that switches in arbitrarily small intervals of time. You accept that, and then try to flim-flam readers by appealing to readers' experience with actual lamps.

But the Thompson lamp is not an actual lamp! So you can not reason about it as if it were the lamp in your living room. There is no button, there is no bulb, there is no electric bill to be paid at the end of the month. There is only an entirely fictional situation that you are pretending to reason about as if it weren't entirely fictional.

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. — Michael

No. Not so. The terminal state of the lamp has nothing at all to do with what has come before. It's just like the mathematical sequence 0, 1, 0, 1, ..., to which I arbitrarily assign a terminal state (a state at the point at infinity) of 47. It's arbitrary. It's legal. It violates no laws of God or man.

The supertask doesn’t allow for either of these scenarios and so is proven impossible in principle. — Michael

You are as wrong as can be. I have explained this to you muliple times.

It's not a real lamp. It is not constrained to buttons and it's not plugged into a wall and connected to the electric company. It's a fiction.

Tell me this. Did Cinderella's coach turn into a pumpkin or not?

Do you not see that Thompson's lamp has the exact same ontological status as Cinderella's coach? It's a fairy tale. It's silly to try to reason as if it were a real thing, subject to the laws of this world.

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. — Ludwig V

Yes, and now I hope I did not overstate my case. I'm making an abstract philosophical point. The main thing is that the terminal state (the state at the point at infinity) bears no necessary relationship to the sequence that precedes it. But if we are given a sequence that does happen to have a limit, we can generally determine what the limit is. I don't recall how we got started on this, but the lamp sequence doesn't have a limit so the terminal state has no natural answer.

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

Pretty much sums up this thread.

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

We stop at the single issue being discussed: performing some action at arbitrarily small intervals of time. So taking the code here we assume that each line is run in an instant with the exception of the wait i *= 0.5 line which waits for the specified time in seconds. The logic of the code still behaves exactly as we would expect. Thomson is asking us what is output when echo isLampOn runs.

If your only solution is to insert the line isLampOn = 'a plate of spaghetti' after while (true) { ... } and before echo isLampOn then you are not answering the question as posed.

If you cannot make sense of the echo isLampOn line without inserting some arbitrary code before it then you must accept that it doesn't make sense for while (true) { ... } to complete. The arbitrary code you are trying to insert is a smokescreen to disguise this impossibility, exactly like your magic turning the lamp into a plate of spaghetti.

And you can’t posit or define or conceive of half without reference to half of some other defined, conceived thing, and that thing must be a whole unit. — Fire Ologist

Yes, but once you have defined your half, you can treat it as a unit and define a half of a half... and repeat indefinitely. What limits that process?

Infinity applies to numbers. Numbers aren’t physical things, like stairs. — Fire Ologist

That's certainly where are problems are. But you need to state this carefully. For example, there are no infinite natural numbers and while numbers are not physical things like stars, they do apply to physical things. The tricky point is that the idea of infinity is embedded in the number system, not some accidental additional property.

There is no infinity apart from the mind that conceives it. There are things apart from the mind that conceives of the unit. — Fire Ologist

Infinity is certainly not a concept and not a physical entity - I doubt that it should be called an entity at all. I would love to know what things apart from the mind "conceive of the unit".

That's not the case. "Object" is not implied. You are simply saying that whatever it is that the two names refer to, it is one and the same. — Metaphysician Undercover

Ostensive definition can only work if you know, or can work out from the context, what kind of thing (category) is being defined. When you gesture at a red car and say this is red, you will misunderstand if you take red to mean a car or a wheel or a heavy object.

Again, this is not true. When you say Ringo Starr is Richard Starkey, all you are saying is that these two names have the same referent. It is only upon analysis, if one seeks to determine whether it is true or not, or something like that, that one would determine that the two names both refer to a person. — Metaphysician Undercover

It's not enough to know that the two names have the same referent. You need to know, in Wittgenstein's phrase, where the referent "is stationed in the language".

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

I think that there some rules that apply in fiction (imaginary stories), because the story needs to have plausibility. But I don't know how to work out what they are. Coleridge, I think it was, said that there needs to be a "suspension of disbelief" for any fiction to work. The reader/audience needs to co-operate and not ask awkward questions. But there are limits. There needs to be some realism for the story to be recognizable at all.

s has a greatest number in its domain, and the last value for s is x. — TonesInDeepFreeze

I don't quite see why x is the last value, nor why you think that defining the set in that way gets round the point that w is not derived from the criterion from which all the other numbers in that set are derived.

Yes, but once you have defined your half, you can treat it as a unit and define a half of a half... and repeat indefinitely — Ludwig V

Exactly! You have to take the thing you call a “half” as a single whole unit before you can take some measure again. A half is just a measure conceived of after there is a unit. Only unitary whole things can be touched or stepped on, like a step.

Only unitary whole things can be touched or stepped on, like a step. — Fire Ologist

I'm missing something here. When I step on a step, do I step on the whole step, or just a part of it. When I sit on a chair do I sit on the whole of it. It depends how you interpret the words, that's all.

I don't quite see why x is the last value, nor why you think that defining the set in that way gets round the point that w is not derived from the criterion from which all the other numbers in that set are derived. — Ludwig V

I merely defined two different functions with two different domains.

x is the last value in the sense that it is the value of the last member of the domain of s where the domain of s is ordered in the usual manner for an ordinal.

The function s is the union of the function t with {<w x>}. There is no mathematical requirement of "deriving by a same criterion". The function s does not have the same presumed domain as in the thought experiment, but when it is asked, "What would be the case if there were a greatest member of the domain (a last point in time)?" then we see that the thought experiment itself does not require that the value of the function at that greatest member of the domain be an x other than 0 or 1 since the thought experiment itself makes no mention of a greatest member of the domain.

If, as suggested, the thought experiment is a kind of fiction, then we see the fiction we're told doesn't mention anything about a last moment in the sequence of moments, so the fiction does not disallow us from extending to another fiction in which there is a last moment and such that the value of the action at that last moment is whatever x we want it to be. That is, at all the moments mentioned in the fiction, the lamp is on or it's off. That doesn't preclude another fiction in which there is a last moment in which any number of things can be the case: (1) The lamp is on, (2) The lamp is off, (3) The lamp explodes into bits and is off, or (4) The lamp expands to the size of the sun and is on, or (5) the lamp transforms into a pepperoni pizza.

[RETROACTIVE EDIT: The above paragraph is incorrect. Whether explicitly or implicitly it is understood that Thomson's lamp requires that the lamp is on/off at time t only if it at some before time before t it was off/on, then the button was pushed on/off and not off/on again before time t. Therefore, another fiction in which that is not the case is not a fiction that addresses Thomson's lamp. This edit also carries forward for any other similar comments I might have made.]

/

Infinity is certainly not a concept — Ludwig V

At least in mathematics, we have the adjective 'is infinite' so that there is not a set called 'infinity' but rather many sets having the property of being infinite. (There are also such things as points of infinity in different number systems and things like that; but this is different from the notion that there is one particular object in mathematics called 'infinity'.)

Meanwhile, in other fields of study, such as philosophy, writers do speak of a concept of infinity. I am not opining here whether, despite philosophical discussions of infinity, infinity is or is not a legitimate concept.


/

Regarding another poster:


Ringo Starr is not a name. Richard Starkey is not a name.

'Ringo Starr' is a name. 'Richard Starkey' is a name.


The denotation of 'Ringo Starr' = the denotation of 'Richard Starkey'

Ringo Starr = Richard Starkey

Ringo Star is Richard Starkey


The sense of 'Ringo Star' not= the sense of 'Richard Starkey'

'Ringo Starr' not= 'Richard Starkey'

'Ringo Starr' is not 'Richard Starkey'

I’m just saying the notion of an infinite staircase is impossible to conceive as steps and groups of steps are unitary wholes, and infinity never unifies or finishes multiplying. You can’t apply infinity to finite things. There is no infinite number of steps between the 1 yard line and the 2 yards line. There is a single yard. You can mathematically take the single yard and mathematically divide it in half, and take one of the halves and divide it…infinitely. But that has nothing to do at all whatsoever with taking a one yard physical step on a football field.

Sounds Bergsonian. Actual movement is indivisible but the mathematical modelling of movement is infinitely divisible.

We stop at the single issue being discussed: performing some action at arbitrarily small intervals of time. So taking the code here we assume that each line is run in an instant with the exception of the wait i *= 0.5 line which waits for the specified time in seconds. The logic of the code still behaves exactly as we would expect. Thomson is asking us what is output when echo isLampOn runs. — Michael

Can you answer this question that I asked earlier? What is the difference in the ontological status between Thompson's lamp and Cinderella's coach?

If your only solution is to insert the line isLampOn = 'a plate of spaghetti' after while (true) { ... } and before echo isLampOn then you are not answering the question as posed. — Michael

But it's not my only solution. I've said (several times) that "Lamp is on" and "Lamp is off" are also valid solutions. Just about anything in the universe, physical, abstract, or fictional, is a valid solution.

After all the times I've explained this to you, I don't mind if you say, "You're wrong." But how on earth can you say that MY ONLY SOLUTION is X? I have already said that ANYTHING AT ALL can be a solution; and that if you insist that on/off are the only legal solutions, then either one is valid.

I don't mind having a difference of opinion, but you seem to have not read a single thing I've said.

If you cannot make sense of the echo isLampOn line without inserting some arbitrary code before it then you must accept that it doesn't make sense for while (true) { ... } to complete. The arbitrary code you are trying to insert is a smokescreen to disguise this impossibility, exactly like your magic turning the lamp into a plate of spaghetti. — Michael

You accuse ME of magic! YOU are the one with the lamp that switches state in arbitrarily small intervals of time. I'm playing by YOUR rules, which clearly allow magic.

Of course I have already made that point to you many times as well.

I think that there some rules that apply in fiction (imaginary stories), because the story needs to have plausibility. But I don't know how to work out what they are. Coleridge, I think it was, said that there needs to be a "suspension of disbelief" for any fiction to work. The reader/audience needs to co-operate and not ask awkward questions. But there are limits. There needs to be some realism for the story to be recognizable at all. — Ludwig V

You've solved my problem. I need the willing suspension of disbelief to converse with @Michael. Indeed, that's the question I asked him. What's the difference between Thompson's lamp and Cinderella's coach? Why am I supposed to treat the lamp as if it has a button, or is operated by a computer program (that likewise can wait for arbitrarily small time intervals), or is subject to some rules of rationality but not others?

I would gladly put these questions to Thompson, but Wiki says he died in 1984. I dare say that if there's an afterlife, he is now eating his plate of spaghetti.

mathematical modelling of movement is infinitely divisible — Heracloitus

Modeling is not physical, so the models built with infinity will never pose a problem when descending stairs. There is no paradox because the paradox seeks to mix actual stairs with modeling.

But it's not my only solution. I've said (several times) that "Lamp is on" and "Lamp is off" are also valid solutions. — fishfry

The lamp is on only if the button was pushed to turn it on, prior to which the lamp was off. Even if you want to introduce magic it is on only if magic turned it on, prior to which the lamp was off.

So if you want to say that the lamp is on after two minutes then you must accept that at some final time prior to two minutes the lamp was off. This is a simple logic fact.

Except the supertask doesn't allow for this. So, as Thomson argues, the lamp cannot be on after two minutes. And for the same reasoning cannot be off after two minutes. This is a contradiction and so the supertask is proven impossible in principle.

The lamp is on only if the button was pushed to turn it on, prior to which the lamp was off. — Michael

According to what principle of physics?

If the lamp can violate the known laws of physics, what are the limits of its magic?

Or, if you want to introduce magic, it is on only if magic turned it on, prior to which the lamp was off. — Michael

But I didn't. YOU (and Thompson, who attained his terminal state in 1984 and is now enjoying his spaghetti) introduced magic. I'm just playing by your rules.

So if you want to say that the lamp was on after two minutes then you must accept that at some final time prior to two minutes the lamp was (left) off. — Michael

Not at all. I've given many counterexamples. The sequence 0, 1, 0, 1, ... has no limit. If I define its terminal state (that is, its value at the point at infinity) as 47, that's perfectly legal. And as sensible as any other terminal state.

The supertask doesn't allow for this. — Michael

Of course it does, and I have repeatedly explained how.

A supertask is not simply an infinite sequence of numbers.

In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect.

It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause – even if it's magic.

If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off.

Given this, if the lamp is on at t₁ then either:

a) it was turned and left on prior to t₁, or
b) it was turned and left off prior to t₁ and then turned on at t₁

But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false. Therefore the lamp is not on at t₁. Similar reasoning shows that the lamp is not off at t₁ either.

Meanwhile, in other fields of study, such as philosophy, writers do speak of a concept of infinity. I am not opining here whether, despite philosophical discussions of infinity, infinity is or is not a legitimate concept. — TonesInDeepFreeze

I'm afraid there was a typo in my last post. I posted "Infinity is certainly not a concept", which is rubbish. I meant to post "Infinity is certainly a concept". Apologies.

If, as suggested, the thought experiment is a kind of fiction, then we see the fiction we're told doesn't mention anything a last moment in the sequence of moments, so the fiction does not disallow us from extending to another fiction in which there is a last moment and such that the value of the action at that last moment is whatever x we want it to be. That is, at all the moments mentioned in the fiction, the lamp is on or it's off. That doesn't preclude another fiction in which there is a last moment in which any number of things can be the case: (1) The lamp is on, (2) The lamp is off, (3) The lamp explodes into bits and is off, or (4) The lamp expands to the size of the sun and is on, or (5) the light transforms into a pepperoni pizza. — TonesInDeepFreeze

Quite so.

In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect. — Michael

I'm not at all clear how the ordinary logic of cause and effect would apply in the context of hypothetical physical laws. But we are clearly not dealing with the ordinary physical world, and that leaves us free to imagine anything at all.

Modeling is not physical, so the models built with infinity will never pose a problem when descending stairs. There is no paradox because the paradox seeks to mix actual stairs with modeling. — Fire Ologist

The first sentence is fine. I don't get the second sentence. You seem to be saying that the paradox is real. But mixing up actual stairs with models of stairs just produces a confusion, so the paradox is just an illusion - in my opinion.

You've solved my problem. I need the willing suspension of disbelief to converse with Michael. Indeed, that's the question I asked him. — fishfry

Yes, I realized that and was hoping to produce a formulation that would allow a more constructive discussion.

Sounds Bergsonian. Actual movement is indivisible but the mathematical modelling of movement is infinitely divisible. — Heracloitus

I don't know about Bergson. I think it is clearer to distinguish between "physical division (separation)" and "mathematical division (analysis)".

I don't recall how we got started on this, but the lamp sequence doesn't have a limit so the terminal state has no natural answer. — fishfry

There are two cases in play at the moment - "0, 1, 0, 1, ..." and "1/2, 1/4, 1/8, 1/16.." Comments switch between them without always being clear. You are, however, quite right that the first sequence doesn't have a limit and the second one has what we could call a natural limit.

I'm not at all clear how the ordinary logic of cause and effect would apply in the context of hypothetical physical laws. But we are clearly not dealing with the ordinary physical world, and that leaves us free to imagine anything at all. — Ludwig V

If it's on at t₁ then either it was left on before t₁ or it was left off before t₁ and then turned on at t₁.

This is a straightforward logical point that does not depend on what the physical laws are.

To make it very simple, Thomson's lamp proves that these premises are logically inconsistent:

P1. The lamp is turned on and off only 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 successively halved intervals of time between t₀ and t₁
P6. The lamp is either on or off at t₁

This is a straightforward logical point that does not depend on what the physical laws are. — Michael

So Thompson's lamp is not merely physically or metaphysically impossible, but logically impossible.

P6. The lamp is either on or off at t1 — Michael

But which is not defined.