You may say that it is possible to divided indefinitely, but that does not mean that infinite divisibility is possible. Take pi for example. You can get a computer to produce the decimal extension for pi, "indefinitely", but you never succeed in reaching an infinite extension. Divisibility is the very same principle. Some mathematical principles allow one to divide indefinitely, but you never reach infinite division. That is because infinite division, therefore infinite divisibility, is logically impossible. — Metaphysician Undercover

Thomson makes a similar point:

People have, I think, confused saying (1) it is conceivable that each of an infinity of tasks be possible (practically possible) of performance, with saying (2) that is conceivable that all of an infinite number of tasks should have been performed. They have supposed that (1) entails (2). And my reason for thinking that people have supposed this is as follows. To suppose that (1) entails (2) is of course to suppose that anyone who denies thinking (2) is committed to denying (1). Now to deny (1) is to be committed to holding, what is quite absurd, (3) that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed. But no-one would hold (3) to be true unless he had confused logical possibility with physical possibility. And we do find that those who wish to assert (2) are constantly accusing their opponents of just this confusion. They seem to think that all they have to do to render (2) plausible is to clear away any confusions that prevent people from accepting (1).

What is wrong with that? — Ludwig V

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.

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.

Can you prove that it is impossible to complete a given recursive function?

Here's some pseudocode to demonstrate Thomson's lamp:


According to those who claim that supertasks are possible, after two minutes the echo isLampOn command will run. This is the first contradiction; the while (true) command ensures that this cannot happen.

But even assuming that echo isLampOn does run, it cannot output true and cannot output false, but also cannot output anything other than true or false. This is the second contradiction.

fishfry's solution is to inject some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so he is no longer addressing the problem as posed.

See also this.

Infinity minus one equals infinity

Would the above qualify as a paradox, or just be silly in "the" non abstract and possible realm but fit into the abstract and possible realm? Or the reverse? — kazan

Math (arithmetic etc) with infinity as a value or function argument is not valid.
An explanation why that's so is Hilbert's hotel explanation:
https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

@fishfry

You wanted something new so see the above.

I see no contradictions because division by 2 results in even number of total divisions.
Therefore if initial state of the lamp is off then it follows that the lamp will be off after 2 minutes.

Under assumption that those 2 minutes must pass the lamp will therefore be off with
And if those 2 minutes never pass due the nature of infinitely small wait time then that doesn't matter because we already know what logically will happen otherwise.

objections?

https://en.wikipedia.org/wiki/Thomson%27s_lamp

division by 2 ... results in even number of total divisions. — SpaceDweller

I don't know what you mean by this.

The wait(i *= 0.5) simply means that pushButton is called at successively halved intervals of time, i.e. it is called for the first time after 60 seconds, for the second time after a further 30 seconds, for the third time after a further 15 seconds, and so on.

Under assumption that those 2 minutes must pass the lamp will therefore be off with console.log(isLampOn) — SpaceDweller

Given what while (true) { ... } means, it is logically impossible for console.log(isLampOn) to ever run.

2 minutes must pass, and while (true) { ... } must still be running after those 2 minutes.

Here's an even simpler demonstration:


In this case there is no wait time between operations; the next occurs "immediately".

Does this script end in an instant, or does it continue forever? If it ends in an instant, does echo isLampOn output true or false?

But given the meaning of while (true), it must continue forever; echo isLampOn never runs.

If you think that this changes by introducing a wait time within the loop then you've obviously made a mistake.

I don't know what you mean by this. — Michael

Successive Division by even number results in even number of divisions, no matter how many times you divide some number by 2 total count of divisions will be even, not odd.
Therefore lamp state would not change after that count of divisions.

The wait(i *= 0.5) simply means that pushButton is called at successively halved intervals of time, i.e. it is called for the first time after 60 seconds, for the second time after a further 30 seconds, for the third time after a further 15 seconds, and so on. — Michael

Yes, but it will happen even count of times, that's my point.

Given what while (true) { ... } means, it is logically impossible for console.log(isLampOn) to ever run. — Michael

Yes it's infinite loop, I misquoted that, regardless if the loop is finite or not count of divisions is even and so the result is known upfront.

This would not be the case if you divided by 3 or some other odd or real value.

You second example is completely different story though because there is no division involved.

Example with a finite number:

4 / 2 = 2
2 / 2 = 1

Count of divisions is even (2) because divided by even number (2) therefore lamp state does not change, it's off.
You continue this division to infinity and the result will be the same because you divide by 2.
ex. 1/2 = 0.5 / 2 = 0.25 etc... it doesn't matter how far you go.

I don’t understand your reasoning at all.

I'm not sure how to try better express myself.

But let's try division by 3 with same sample before:
4 / 3 = 1.33
count of divisions is odd so the lamp is ON,

You can continue dividing and it's uncertain that the result will be even.
I stopped with 1.33 because the result is lower than 3, same way I stopped in my previous example because 1 was lower than 2.

I have no problem acknowledging that 2+2=4. I have a problem with people who claim that "2+2" symbolizes the same thing that "4" does. — Metaphysician Undercover

2 + 2 and 4 symbolize the same set. You are the one strawmanning the claim that somebody says they're the same thing.

YOU are doing that. Not any mathematicians, unless they are speaking casually.

2 + 2 and 4 symbolize the same set. Not necessarily the same thing, whatever that might happen to mean in someone's lexicon. "Thing" is not a term of art in mathematics. Set is. I have explained this to you certainly more than ten or twenty times.

And so, I refused to accept your claim to have proven that "2+2" signifies the very same thing as "4" does. — Metaphysician Undercover

I have never made that claim. In your refusal to engage with a single thing I've ever said to you, you BELIEVE, in your heart of hearts, that I have claimed that 2 + 2 and 4 are the same "thing." I will credit you with sincerity for once.

But I have not said that, and if you will go back through my posting history as long as you want, you will never find that I ever said that.

I have said, repeatedly, that 2 + 2 and 4 represent the same set. And that, by the rules of the formal system known as set theory, is as true a statement as a statement ever could be true. It's as true as saying that bishops travel only on diagonals. It's not true by virtue of any property or aspect of the world. It's true by virtue of the rules of chess.

Likewise, the fact that 2 + 2 and 4 symbolize the same set, is derivable from the rules of set theory.

Simply put, if the right side of an equation does not signify something distinct from the left side, mathematics would be completely useless. — Metaphysician Undercover

Ah, the old tautology argument. "Since everything in math follows from axioms, the theorems are already inherent in the axioms, hence they add no new information, hence they are useless."

I counter that argument by asking whether the sculptor adds value in finding the statue within the stone.

Of course he does. The theorems follow logically from the axioms; but they do not necessarily follow obviously. It often takes great ingenuity and insight to derive theorems that are interesting, useful, and beautiful. That's mathematics.

You can say that I have a problem with formalism, — Metaphysician Undercover

I have observed with my own eyes your inability and unwillingness to engage with my Peano proof that 2 + 2 = 4, and the statement of the axiom of extensionality.

I don't "say" the sun rises in the east. I observe it to happen with great regularity. I don't "say" you have a problem with formalism; I observe it.

I will admit that you haven't failed to engage with formalism as many times as the sun has risen. But I'm willing to bet that's only because I haven't challenged you that many times.

because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. — Metaphysician Undercover

You are making a category error akin to complaining that bishops "make church laws, be a judge in church matters and to enforce observance of these laws," in response to my telling you that bishops move and capture along diagonals.

You are railing against a metaphysics that is not claimed by any practitioners of mathematics.

That is, to remove all content from a logical application, to have a logical system which is purely formal. If such a thing was possible we'd have a logical system which is absolutely useless, applicable to nothing whatsoever. — My vetaphysician Undercover

You're just making the unreasonable effectiveness argument. If math is a flagrant fiction, why's it so darn useful?

You are not the first person to notice this, and you are not special for having noticed it. You are "special" in a different sense, for thinking you've had a profound insight, when it's such a commonplace. Bertrand Russell said, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." He was making the same point.

Attempts at formalism end up disguising content as form, — Metaphysician Undercover

I see the point you're making, but what of it? Do you object to chess because it redefines what bishops are? You're so against formalism that you reject it entirely? Why do you bother to read and write? Language is just an abstract formalism for notating thoughts, and we know words are just an approximation to thoughts. Why aren't you equally against language?

producing a smoke and mirrors system of sophistry, — Metaphysician Undercover

Just as language is, as some philosophers have argued.

which is riddled with errors, due to the inherent unintelligibility of the content, which then permeates through the entire system, undetected because its existence is denied. — Metaphysician Undercover

Just like language.

"Man and His Symbols," Jung. Humans use symbols. You are standing four-square against mathematical symbolism but not other forms of symbolism. Presumably you're not picketing art museums and book publishers.

Why is that?

You wanted something new so see the above. — Michael

Was that the pseudocode I was supposed to look at?

I entirely agree, with one exception. Code is a notation for a physical process executing on a finite computer. The mathematical lamp problem is the same code, but interpreted in abstract mathematical space where the process never ends. With that caveat I accept your pseudocode.

BUT!!!!!! You have not defined the terminal state. So why do you think there should be a sensible answer for what it is?

After all, there is no number that can serve as the limit of the sequence 0, 1, 0, 1, ...

I don't see how your expressing the problem in pseudocode adds anything. We all have agreed to it long ago, even before you wrote it down. That's the premise of the problem. But the question is about the terminal state, which is not defined.

What is wrong with that? — Ludwig V

I don't see how recursion qualifies as an infinite activity.

2 + 2 and 4 symbolize the same set. You are the one strawmanning the claim that somebody says they're the same thing. — fishfry

My point has always been that "same" in this context is not consistent with "same" in the context of the law of identity. So, to say " 2+ 2 and 4 symbolize the same set" is to use "same in a way which is in violation of the law of identity.

Whether we are talking about "same thing", "same set", "same number", or "same kick in the ass", is irrelevant. The point is that this specific use of "same" is very clearly in violation of the law of identity. If the law of identity indicates that only a thing can be said to be "the same", and you do not believe that a set is a thing, and you want to say that a set is the same, then I suggest that you do not agree with the way that "same' is used by the law of identity. Is this the case? Do you believe that mathematicians have a better definition of "same"?

If math is a flagrant fiction, why's it so darn useful? — fishfry

In case you have never noticed, fiction is extremely useful. I suggest you begin with a look at the obvious, deception. Deception demonstrates that fiction is very useful in convincing others, to help us get what we want from them. And, so is mathematics.

My point has always been that "same" in this context is not consistent with "same" in the context of the law of identity. So, to say " 2+ 2 and 4 symbolize the same set" is to use "same in a way which is in violation of the law of identity. — Metaphysician Undercover

In this context, same means that we may write the symbol "=" between the sets. The axiom of extensionality tells us when we may write A = B. If you prefer to be formal, we could simply never say, even casually, that A and B are "the same." Only that they satisfy the axiom of extensionality and therefore we may write A = B. And if you don't like the equal sign, we could use some other symbol.

Whether we are talking about "same thing", "same set", "same number", or "same kick in the ass", is irrelevant. The point is that this specific use of "same" is very clearly in violation of the law of identity. — Metaphysician Undercover

We need not use the word same if it bothers you. It's sufficient that the statement A = B is true (within set theory) exactly when A and B satisfy the axiom of extensionality. And "=" doesn't mean anything, it's just a symbol.

If the law of identity indicates that only a thing can be said to be "the same", and you do not believe that a set is a thing, and you want to say that a set is the same, then I suggest that you do not agree with the way that "same' is used by the law of identity. Is this the case? Do you believe that mathematicians have a better definition of "same"? — Metaphysician Undercover

Same is only being used casually. When pressed, we always revert to the definition of = given by the axiom of extensionality.

If math is a flagrant fiction, why's it so darn useful?
— fishfry

In case you have never noticed, fiction is extremely useful. I suggest you begin with a look at the obvious, deception. Deception demonstrates that fiction is very useful in convincing others, to help us get what we want from them. And, so is mathematics. — Metaphysician Undercover

Then why are you disagreeing with me? If you say that mathematics is a useful fiction, that's exactly what I would say. And if you say it's a useful deception, that's fine.

May I ask, is chess similarly a useful deception? Language? You didn't respond to my point earlier that language is also a formal symbology that attempts to capture, however imperfectly. some aspect of abstract thought.

BUT!!!!!! You have not defined the terminal state. So why do you think there should be a sensible answer for what it is?

I don't see how your expressing the problem in pseudocode adds anything. We all have agreed to it long ago, even before you wrote it down. That's the premise of the problem. But the question is about the terminal state, which is not defined. — fishfry

The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times.

After all, there is no number that can serve as the limit of the sequence 0, 1, 0, 1, ... — fishfry

We're discussing the consequence of having pushed a button an infinite number of times, not the limit of some infinite sequence of numbers. These are two different things.

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

We need not use the word same if it bothers you. — fishfry

Great, I prefer the word "equal". It's better suited for that purpose. "Equal" generally allows that the two things which are said to be equal are not necessarily the same. "Same" is defined by the law of identity as indicating one thing only. That is the commonly expressed difference between "same" and "equal". "Equal" indicates a similarity of two things by both sharing an identifiable property, while "same" means that you are referring to one thing only.

Then why are you disagreeing with me? — fishfry

Generally I disagree with your wording, as indicated above. The axiom of extensionality indicates what is required for two sets to be equal, yet you state this as "the same". That I take as a mistaken use of words.

And if you say it's a useful deception, that's fine. — fishfry

I didn't say that though. I simply gave an example of how fiction is useful, one that was obvious. Many times fiction is used in ways not intended to deceive, like the use of counterfactuals in logic, for example. So, the issue is complex, because mathematics, like fiction in general has many uses.

May I ask, is chess similarly a useful deception? Language? You didn't respond to my point earlier that language is also a formal symbology that attempts to capture, however imperfectly. some aspect of abstract thought. — fishfry

These are broad generalizations which I can't relate to because I do not accept them as valid generalizations, so I do not reply. For example, you say that language "attempts to...". But language doesn't attempt anything, individual people attempt to do things with the use of language. And, there is such an extremely broad range of things which people attempt to do with language, that it doesn't make sense to make the generalization that what people attempt to do with language is to capture "some aspect of abstract thought".

I imagine that a supertaskist might agree with the formalist or intuitionist that supertasks aren't mathematically or logically defensible, whilst nevertherless insisting that reality contains supertasks, by arguing that mathematical logic is the map rather than the territory, and by conceding that they are only using mathematics as a descriptive tool for expressing their beliefs, rather than as a prescription for justifying their a priori beliefs in super-tasks.

For instance, a supertaskist might appeal to the fact that one cannot say how many moments of time has passed during a minute, or how many physical operations took place in one's computer to sum 1 + 1, and they might appeal to this inability to measure, divide and count experience or events as grounds for being open minded to the idea that space-time is a literal continuum.

But in that case, how does the supertaskist propose identifying what isn't a super-task? If super-tasks are to have empirical meaning and inferential value, the supertaskist must delineate task from super-task, but how can they delineate them on a non-ad hoc basis?

We need not use the word same if it bothers you.
— fishfry

Great, I prefer the word "equal". It's better suited for that purpose. — Metaphysician Undercover

I could say I was using the word same in a casual sense.

Or, I could say that when i comes to sets, two sets are the same if they have the same elements. Same set. Equal sets or same set. It's the same.

"Equal" generally allows that the two things which are said to be equal are not necessarily the same. — Metaphysician Undercover

In set theory, two sets that are equal are the same set. Two numbers that are equal are the same number. You're playing word games.

"Same" is defined by the law of identity as indicating one thing only. — Metaphysician Undercover

Same set. That's what it means.

That is the commonly expressed difference between "same" and "equal". — Metaphysician Undercover

Commonly meaning you've said it a couple hundred times, to little effect. You've had the extensional/intensional distinction pointed out to you many times to no avail.

"Equal" indicates a similarity of two things by both sharing an identifiable property, while "same" means that you are referring to one thing only. — Metaphysician Undercover

Well if we're talking about sets, equal and same are the same. And I'm talking about sets.

Generally I disagree with your wording, as indicated above. — Metaphysician Undercover

Sets that are equal are the same set.

The axiom of extensionality indicates what is required for two sets to be equal, yet you state this as "the same". That I take as a mistaken use of words. — Metaphysician Undercover

They're the same set. If you can give me an example of two sets that are equal yet not the same set, well ... you can't.

I didn't say that though. I simply gave an example of how fiction is useful, one that was obvious. — Metaphysician Undercover

So math is a useful fiction.

Many times fiction is used in ways not intended to deceive, like the use of counterfactuals in logic, for example. So, the issue is complex, because mathematics, like fiction in general has many uses. — Metaphysician Undercover

You are agreeing with me.

These are broad generalizations which I can't relate to because I do not accept them as valid
generalizations, so I do not reply. — Metaphysician Undercover

You object to symbolic notation for math. Do you object to symbolic notation for music? For thought and speech? For chess?

For example, you say that language "attempts to...". But language doesn't attempt anything, individual people attempt to do things with the use of language. And, there is such an extremely broad range of things which people attempt to do with language, that it doesn't make sense to make the generalization that what people attempt to do with language is to capture "some aspect of abstract thought". — Metaphysician Undercover

You are avoiding the issues I've raised.

The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times. — Michael

You can't push a button an infinite number of times. This is not a physical experiment. It's purely an abstract mathematical exercise.

You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. It's not a real light bulb, it's not driven by a real circuit. You're confusing yourself on this point.

We're discussing the consequence of having pushed a button an infinite number of times, not the limit of some infinite sequence of numbers. These are two different things. — Michael

You can't push a button an infinite number of times. It's not a real button, it's an abstract thought experiment about an entirely fictional entity.

And since when can't we use math to model some real or fictional situation? Did you complain when Maxwell wrote down his equations of electromagnetism? Hey it's electricity and magnetism buddy, it's not math. Surely you are not trying to make this absurd claim, are you?

The sequence 0, 1, 0, 1, ... which clearly has no limit, is a perfect mathematical model of this problem. You can define the terminal state -- that is, the state at $\omega$, as 0, or 1, or a plate of spaghetti. It makes no difference to the original sequence.

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

That's funny, because according to SEP, he used the asymptotic density of the sequence 0, 1, 0, 1, ... to argue that the final state must be 1/2. He made that exact argument using that exact sequence. So you are mistaken, because Thompson has used the exact same reasoning I did. Please read this section:

https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp

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

You're confusing yourself on this point.

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

So I think the confusion is yours.

That's funny, because according to SEP, he used the asymptotic density of the sequence 0, 1, 0, 1, ... to argue that the final state must be 1/2. He made that exact argument using that exact sequence. So you are mistaken, because Thompson has used the exact same reasoning I did. — fishfry

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

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.

Thompson's views sound intuitionistic, in that he didn't apparently consider the "completion" of button-pressing to have a meaningful answer in relation to his thought-experiment. Indeed, if one formally treats Thompson's Lamp as being an unfinishable choice-sequence, which is an object equivalent to what computer-science calls a "stream", then Brouwer's weak axiom of continuity explicitly forbids the construction of any theorem that postulates a property of Thompson's stream that isn't decidable in a finite number of steps. Indeed, Brouwer can be understood as inventing intuitionism to explicitly forbid the informal interpretation of total functions as representing super-tasks, which are concepts that are incompatible with temporal intuition.

By contrast, Benacerraf changed the goal posts by giving Thompson's Lamp a formal treatment from the perspective of classical analysis, in which he interpreted Thompson's Lamp as being an incomplete description of an extended abstract function with a point at infinity, whose value can be chosen as being continuous with some property of the original sequence. However, Bencerraf's formal interpretation doesn't have a temporal interpretation in the sense demanded by the intuitionists. And his methodology runs into problems in situations where the function to be completed has conflicting notions of continuity, as in The Littlewood-Ross Paradox.

In set theory, two sets that are equal are the same set. — fishfry

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?

I don't think it has anything to do with formalism or intuitionism or anything like that.

Our starting premises are:

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

Thomson asks the following question:

Q1. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

Compare with:

Q2. If the button is pushed an odd number of times between t₀ and t₁ then is the lamp on or off at t₁?
Q3. If the button is pushed an even number of times between t₀ and t₁ then is the lamp on or off at t₁?

Benacerraf claims that we can simply stipulate that the lamp is on at t₁.

There are two problems with this.

The first is that we cannot simply stipulate the answer. I cannot stipulate that the answer to Q2 is "off" and I cannot stipulate that the answer to Q3 is "on". The answers must be deducible from the premises.

The second problem is that the following is deducible from the first three premises:

C1. If the button is pushed an infinite number of times between t₀ and t₁ then the lamp is neither on nor off at t₁

This is because if the button is pushed at least once then for the lamp to be on at t₁ the button must have been pushed to turn and leave it on for t₁, and for the lamp to be off at t₁ the button must have been pushed to turn and leave it off for t₁, neither of which are possible if the button is pushed an infinite number of times between t₀ and t₁.

C1 contradicts P4.

Given that P4 is necessarily true, it follows that the antecedent of C1 is necessarily false. So it is metaphysically impossible to have pushed the button an infinite number of times.

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use e and i, because why not?

The logic of the lamp just has nothing to do with numbers at all. — Michael

Yes, and that sounds identical to the philosophy of intuitionism :) It doesn't matter what type of object is associated with the lamp's output.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises, but which requires changing the original problem by adding an point at infinity to accommodate such a completion.

Are you arguing that Thompson's sequence is finishable hypothetically, but without possessing a definite end value? — sime

No, I'm saying that it isn't completable, even hypothetically. The fact that there is no definite end value is just one way to demonstrate that it isn't completable; the lamp must be either on or off after two minutes, but if the button has been pushed an infinite number of times before then then it cannot be either.

In which case your argument would be closer to constructive mathematics based on intuitionistic logic, rather than to intuitionism. — sime

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use $e$ and $i$, because why not?

The logic of the lamp just has nothing to do with some sequence of additions and subtractions. The code here properly demonstrates the logic.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises — sime

I think it does. We need to examine the process in reverse, and remember that the lamp is on iff the lamp was off and the button was pushed to turn it on. We're discussing a supertask after all, not simply the infinite sequence {off, on, off, ...}.

If the lamp is on at t₁ then either:

a) the button was pushed to turn the lamp on before t₁ and then it was left on until t₁, or
b) the button was pushed to turn the lamp off before t₁ and then it was left off until t₁ when the button is pushed to turn it on

Neither (a) nor (b) are possible given the defined supertask – the lamp is never left either on or off – therefore the lamp cannot be on at t₁. And then the same reasoning shows that it cannot be off at t₁ either. Yet it must be either on or off. This is a contradiction.

This fact has nothing to do with one's interpretation of mathematics (and nothing to do with the limit of some proposed infinite sequence of numbers).

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

It seems pretty clear that "same" does not have the same (!) meaning in the context of set theory that it has in the context of the law of identity. How could it? The definition that applies in the context of the law of identity is inapplicable to the context of set theory, and vice versa. 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.
The meaning of "same" depends on its context. There are many other concepts of which that is true.

The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times. — Michael

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.

You have not defined the terminal state. So why do you think there should be a sensible answer for what it is? — fishfry

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?

You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. It's not a real light bulb, it's not driven by a real circuit. — fishfry

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.