"Complete" does not apply to infinite series, by definition. — Ludwig V

An infinite series that has a sum (some might say the series is the sum) requires first having an infinite sequence (each entry in the sequence is a finite sum) that converges, and the sum is the limit. The sequence whose entries are 0, 1, 0, 1 ... does not converge. However, whatever you mean by 'complete', there are infinite series that have a sum.

An infinite sequence is a function whose domain is an infinite ordinal.

The infinite sequences in this context are:

(1) The function that maps n to to 1/(2^n)

(1) The function that maps n to either 0 or 1 (off or on) depending on whether n is odd or even

Everyone knows that tea is taken at at the tea time hour and that one is not to dawdle still drinking it, not even hypothetically, not even gedankenishly, past the tea time hour. — TonesInDeepFreeze

Yes. I discovered that after the tea-time hour, it turns into a grumpy tortoise.

too much public exercise of arithmetic would allow citizens to become too number savvy — TonesInDeepFreeze

Since then, however, it has been discovered that citizens will still get themselves into a hopeless muddle even if they practice all day. So the betting industry is safe.

We need only take it for granted that it does change at the rate stated in the puzzle. — TonesInDeepFreeze

The lamp puzzle doesn't require anything to occur in an infinitely small amount of time. — TonesInDeepFreeze

Yes. I was careless.

But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration. — TonesInDeepFreeze

But here's my problem. If I take one step, do I execute one task, or many? The argument of the paradox is that in order to take my step, I either must execute infinitely many tasks in a finite duration or fail to complete (or even begin) my step. I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations.

So it seems your analogy is between misuse of imaginary numbers and misuse of infinite numbers. — TonesInDeepFreeze

I think everyone agrees that there's misuse of something going on here. There's disagreement about what is being misused and how.

However, whatever you mean by 'complete', there are infinite series that have a sum. — TonesInDeepFreeze

I never meant to deny that.

Did Thomson make that argument? Was that part of his answer to the paradox? — TonesInDeepFreeze

No, he only argued that "talk of super-tasks is senseless."

I simply use this as a refutation by contradiction. If spacetime being infinitely divisible entails the metaphysical possibility of supertasks and if supertasks are metaphysically impossible then spacetime is not infinitely divisible.

Although on this point there is perhaps some ambiguity in the phrase "infinitely divisible," a point that Thomson addresses:

If something is infinitely divisible, and you are to say into how many parts it shall be divided, you have $\mathbb{N_0}$ alternatives from which to choose. This is not to say that $\mathbb{N_0}$ is one of them. And if something is infinitely divisible, then the operation of halving it or halving some part of it can be performed infinitely often. This is not to say that the operation can have been performed infinitely often.

So I suppose what I mean to say is that within some finite section of space and time there cannot be an infinite succession of halfway points.

In my mind this requires that there be some smallest unit of space and time, but I'm open to the possibility that it doesn't, and that I'm simply failing to fully grasp the notion of infinity. Perhaps this is related to the fact that $\mathbb{N_0}$ is infinite but that every $n\in\mathbb{N_0}$ is finite.

Although as I suspect that some quantum theory of gravity will be correct, and as such quantum theories require discrete spacetime (as far as I understand), I think that it's at least true that there is some smallest unit of space and time, even if it's not necessarily true.

Mathematics doesn't say there is no limit to the ways objects may be divided. — TonesInDeepFreeze

That really depends on what you mean by "object". If you mean "physical object" then mathematics doesn't say anything about them at all, and whether or not some physical object is infinitely divisible is a matter for empirical investigation.

As it stands one cannot have half a photon (or any point particle) and if spacetime is quantized then there is a limit to how far one can divide some section of it into two.

The pie is cut in half and placed in two different wrappings, as two different things, going to two different places. That these two different things were at one time united in a single source is irrelevant to the fact that after division they are two distinct things with two distinct centres of gravity.

The fact that we call the two things "halves" is just a feature of common vernacular. We know that they are not really each exactly a half, by any strict logical principles. It's just an approximation. Likewise, if we measure two things as 420 kg, we say that they are "the same weight", even though there is a discrepancy of a few grams here or there.

You argue by equivocation, confusing common vernacular with the logic of mathematics. If 1/2 in mathematics was allowed to be imprecise, as the baker cutting the pie in half is allowed to be imprecise, the problem of the op would not arise.

@TonesInDeepFreeze
This is the issue which Ludwig and I have been discussing. "One half" in practise does not have the same meaning as "1/2" in theory.

"One half" in practise does not have the same meaning as "1/2" in theory. — Metaphysician Undercover

I hate to be difficult, and I'm not really disagreeing, just amplifying. But I would like to add that if the pipe is cut in half lengthways, neither half is a (newly individuated) pipe. You have two gutters (or that is what I call them). And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours.

And that if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours. — Ludwig V

Why do you say that the two gutters are not distinct objects. A gutter is an object. Or do you not think so? Even if we call them halfpipes, the two are still separate objects. And if we can't even think of a name for them, we acknowledge that they are no longer united as one object, but are now two separate objects. Division of an object does not necessarily produce two new objects that are of the same type as the original, in fact it often does not. So there is no need to think that dividing a pipe ought to make two pipes. However, dividing an object in two always produces two new objects (as well as the waste material). The "waste" becomes an important feature often overlooked in systems theory and the conservation energy, as energy lost to inefficiency or entropy. But the fact of waste in any act of division nullifies the validity of the supertask.

Why do you say that the two gutters are not distinct objects. — Metaphysician Undercover

Perhaps I wasn't clear. I never intended to say that. I thought this was crystal clear:-

You have two gutters (or that is what I call them). — Ludwig V

However, dividing an object in two always produces two new objects (as well as the waste material). — Metaphysician Undercover

It depends what you mean by "divide" and by "object".

However, dividing an object in two always produces two new objects (as well as the waste material). — Metaphysician Undercover

This is not exactly wrong, but requires that you recognize that "division" and/or "object" may change their meaning in some contexts. That's why I said:-

... if I paint half the pipe blue and half red, the halves do not become objects in their own right, but remain halves of the same pipe, even though they are of different colours. — Ludwig V

I said that the two painted halves do not become objects in their own right, meaning separate, distinct objects. You may argue that this is not dividing the pipe, or that each half becomes a distinct object. I don't mind what you choose. This shouldn't be too difficult for you, since you said earlier:-

And, divisibility is dependent on the type of thing to be divided. Therefore, when it comes to division one standard does not fit all things, and the principles of division must be specifically designed for the different type of things to be divided. — Metaphysician Undercover

But the fact of waste in any act of division nullifies the validity of the supertask. — Metaphysician Undercover

Oh, there's no doubt that no-one could actually cut the pipe into halves, and then divide one of the halves into halves ad infinitum. But painting the pipe shows that it depends what you mean by "divide" and/or "object". You could say that painting the pipe is a theoretical, not a practical division; that would be a bit at odds with ordinary language, but we are not speaking ordinary language here.

But maybe we could notice that we are not actually cutting the pipe, but imagining cutting the pipe. So perhaps we could imagine cutting the pipe without any waste?

I said that the two painted halves do not become objects in their own right, meaning separate, distinct objects. You may argue that this is not dividing the pipe, or that each half becomes a distinct object. I don't mind what you choose. This shouldn't be too difficult for you, since you said earlier:- — Ludwig V

Oh, I misunderstood. You are painting the pipe without dividing the pipe. I guess we don't mean the same thing with "divide". I think of "divide" as "separate or be separated into parts; breakup; split". This is why I say that to divide something in two makes two distinct objects, because they are separated, each with its own centre of gravity.

But painting the pipe shows that it depends what you mean by "divide" and/or "object". — Ludwig V

I would say that painting a pipe two different colours is not a case of dividing the pipe. To use your terminology, you are distinguishing two halves without separating them. This does not qualify as "dividing". When I look at an object I can distinguish different parts of the object, and even draw lines on its surface, and all this is done without dividing the object.

I would say that painting a pipe two different colours is not a case of dividing the pipe. To use your terminology, you are distinguishing two halves without separating them. This does not qualify as "dividing". When I look at an object I can distinguish different parts of the object, and even draw lines on its surface, and all this is done without dividing the object. — Metaphysician Undercover

OK. So I guess measuring an object would count as "distinguishing different parts" of it even if the line that I draw does not correspond to any pre-existing difference or discontinuity in the object.

Previously I wrote:

"Thomson says [...] there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time."

I may have erred there. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations. — Ludwig V

The puzzle supposes infinitely many segments. Of course, if you deny the supposition of the puzzle, then it may be easy to dispense the puzzle. But one may wish not to take the easy way out but instead grapple with the puzzle under the suppositions it makes. Of course, one can hold that there are empirical or even physics theoretical bases to hold that there is a finite upper limit to how many times distance and duration may be divided, but one might still wish to drive a harder bargain, which is that the question is not that of physical but rather of logical possibility.

he only argued that "talk of super-tasks is senseless." — Michael

Where in the paper does Thompson say that?

Mathematics doesn't say there is no limit to the ways objects may be divided.
— TonesInDeepFreeze

That really depends on what you mean by "object". If you mean "physical object" then mathematics doesn't say anything about them at all, and whether or not some physical object is infinitely divisible is a matter for empirical investigation. — Michael

I had in mind not 'object' but that 'ways of dividing' is vague. Or maybe the poster just refers to the fact that there are many different divisors. And, yes, among the reals there are uncountably many divisors ("ways to divide").

This is less directed to you than it is directed to the poster who has ridiculous, ignorant ideas about mathematics. These are points that are too terribly obvious to anyone of even barely adequate intelligence and education, but the poster drags discussion down to the level that these things need to be made explicit:

Regarding numbers, when we say 'divide x by y to get z', that is an instruction for a procedure. That procedure merely upholds that x/y is the unique number z such that y*z = x. That does not imply, for example, that from the number x we to create two other numbers - one x/2 and another x/2. Rather, x/2 is itself a number. There are not two x/2, each one a separate object made by dividing x.

However, when mathematics is applied, then a material object may have x number of units, such as x number of square inches, or x number of pounds, or x number of grains of salt. Then, when we say, for example, "divide the object in half", we mean that we will have two different objects, each with x/2 square inches, or x/2 pounds, or x/2 grains of salt, as the case may be.

Those two different senses of "divide" are not in contradiction when a reasonable person considers the two different contexts.

The poster claimed that I equivocate about this. On the contrary, I am clear of quite clear of the distinction and none of my comments employ any equivocation regarding it.

As it stands one cannot have half a photon (or any point particle) and if spacetime is quantized then there is a limit to how far one can divide some section of it into two. — Michael

The paradoxes discussed don't require splitting such things as photons. And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.

And then the poster cites that halving material object is merely an approximation, as if that has any bearing here.

Oh please! Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. A person who drags in the fact that material objects are only measured approximately as if that refutes anything in a discussion such as this one is a person who has the mentality of a juvenile.

Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times. The poster brought in a big fat stupid red herring with his pointless point that measurements of material objects are not exact.

OK. So I guess measuring an object would count as "distinguishing different parts" of it even if the line that I draw does not correspond to any pre-existing difference or discontinuity in the object. — Ludwig V

I don't understand this at all.

The poster claimed that I equivocate about this. On the contrary, I am clear of quite clear of the distinction and none of my comments employ any equivocation regarding it. — TonesInDeepFreeze

The problem is, that when we divide a material object such as a pie, in half, it is never a perfect division with two perfectly equal parts. There is always some degree of approximation. Dividing in half in mathematics is perfect, no approximation. Therefore "half" has a different meaning in the theories of mathematics, from the meaning it has in practical usage. So your example of dividing a pie, which necessarily involves an approximation, to demonstrate "half" in the context of theoretical mathematics, which is necessarily a division of perfect precision, is simple equivocation.

The paradoxes discussed don't require splitting material objects. — TonesInDeepFreeze

This is exactly why your example of dividing a pie in half, is a case of arguing through equivocation.

I didn't argue that dividing a pie in half proves anything about the thought experiments.

And the poster again states his big fat stupid red herring about approximation, thus deserving a restatement by me:

"Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times." The poster argues like a child.

Just two of many now:

(1) The thought experiments don't depend on exactitude of division. Only a person who hasn't thought about the matter would overlook that one divisor might be 2 and the next divisor 2.1 and the next divisor 10, and we'd still have diminishing lengths.

(2) After several iterations of the challenge in threads, the poster still won't say what he supposes is the inherent ordering of, for example, the members of the Beatles, as lack of facing that challenge illustrates that the poster cannot sustain his attack on the notion of the extensionality of sets, either in everyday life or in mathematics, as the principle of extensionality is basic to the mathematics he presumes, without any education in the subject, to refute.

By the way, the poster, in his earlier remark about mathematics, has it exactly backwards, as usual with him, just as he has it backwards with me. Mathematics doesn't itself make claims about whether material objects can be divided any number of finite times without bound on the finite number ("infinitely divisible") but rather that there is no finite upper bound on reiterating division of numbers. That is a distinction that I hold myself; it's not a distinction that, contrary to the poster's confusion, I equivocate about.

Where in the paper does Thompson say that? — TonesInDeepFreeze

And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case. — TonesInDeepFreeze

It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp.

So if these arguments prove that contradictions follow if we assume that there is no smallest unit of space and time then as a refutation by contradiction it is proven that there is some smallest unit of space or time.

Of course, if you deny the supposition of the puzzle, then it may be easy to dispense the puzzle. But one may wish not to take the easy way out but instead grapple with the puzzle under the suppositions it makes. — TonesInDeepFreeze

Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice.

There are not two x/2, each one a separate object made by dividing x. — TonesInDeepFreeze

That's right. The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts, taking the case of cutting something into pieces as the model. It obviously doesn't apply to numbers, or to space or time.

It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp. — Michael

No, it's confusing theory with practice, abstract with concrete and not understanding that infinity means endless (but not necessarily limited)

Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice. — Ludwig V

Combined with another premise, it yields a paradox.

Thomson outlines the paradox:

Premise 1. To compete the main task requires completing infinitely many subtasks.

Premise 2. Infinitely many subtasks cannot be completed.

Conclusion. The main task cannot be completed.

But the main task can be completed. So one of the premises must be false.

One may say that Premise 1 is false. But that is a kind of "jumping the gun" when we would want to examine whether Premise 2 is false and Premise 1 is true. To say that Premise 1 is false on the grounds that it yields a falsehood skips that it's not Premise 1 alone that yields a falsehood, but rather it's the conjunction of Premise 1 and Premise 2 that yields a falsehood.

So Thomson does not right away say that it is Premise 1 that is false but rather he examines arguments for and against both premises.

Also Thomson argues:

"It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, 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."

That is, if there is a finite upper bound to how many times division can be executed, then there is some finite number k such that division can be executed k number of times, but division cannot be executed k+1 number of times. But why? Again, it's a leap to say that the answer is that otherwise there would be the false conclusion of the paradox, as the false conclusion might stem from the other premise of the paradox. Also, keeping in mind that Thomson is not arguing about the fact of the matter as to divisions of whatever, but rather about conceivability.

And I would consider that it's reasonable that the problem need not be contained to what happens to be true or false per some theory of physics, but rather that the problem of tasks may be abstract so that we may wish to resolve the paradox while granting the logical possibility or logical impossibility of the premises.

And Thomson's first point is one of logic, whatever the truth of falsity of the premises:

It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts — Ludwig V

Who makes that assumption?

not understanding that infinity means endless (but not necessarily limited) — Ludwig V

There are different philosophical notions of infinity. But I don't know what specific approach to the paradoxes you think depend on a certain notion of infinity that, on its own terms, carries a misunderstanding. At least in context of mathematics that is mentioned in the puzzles, we have definitions:

S is finite if and only if there is a 1-1 correspondence between S and some natural number.

S is infinite if and only if S is not finite.

S has an upper bound in S if and only if there is a member of S that is greater than or equal to all members of S.

S has an upper bound if and only if there is an x such that x is greater than or equal to every member of S.

S has a lower bound in S if and only if there is a member of S that is less than or equal to every member of S.

S has a lower bound if and only if there is an x such that x is less than or equal to every member of S.

g converges to L if and only if for any d, there is an n such that for all k greater than or equal to n, |g(k) - L| is less than d.

If g converges to L, the L is the limit of g.

So, regarding the infinite sequences in Thomson's lamp problem:

The range of the sequence 1, 2, 3 ... has no upper bound in the range, but it has an upper bound as far as ordinals go (the least upper bound is wu{w}).

The sequence 1, 2, 3 ... does not converge.

The range of the sequence 1, 1/2, 1/4 ... has no lower bound in the range, but it has a lower bound (the greatest lower bound is 0).

The sequence 1, 1/2, 1/4 ... converges to 0.

The range of the sequence 1, 0, 1 ... has an upper bound and a lower bound in the range.

The sequence 1, 0, 1 ... does not converge.

Those are just the ordinary mathematical definitions, which (in some sense) we don't evaluate as true or false but rather they're just stipulative definitions, and as such, they don't carry any "misunderstandings".

Thank you for the quote. His statement of his view that the notion of super-tasks is nonsense is wedged in a chain of reasoning, and somewhat hedged by saying 'if'. It's odd the way he slipped it in rather than stating it more centrally, since it would seem to be the most important conclusion in the paper. But for practical purposes, yes, we would take him as holding that notion of super-tasks is nonsense. Though it is not clear to me how he reached that conclusion (but I admit not really following certain parts of the paper).

But two points he does stress are the faulty inference from "no finite upper bound to the number of subtasks that can be completed" to "there is a completion of an infinite number of subtasks" and (2) that there are good grounds for doubting the premises of the paradox and that if there is misunderstanding in in those grounds than they are found in the "mathematical solutions" too (if I'm not mistaken, he has in mind that those faults are an incorrect understanding of the notion of an infinite sum).

And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.
— TonesInDeepFreeze

It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp.

So if these arguments prove that contradictions follow if we assume that there is no smallest unit of space and time then as a refutation by contradiction it is proven that there is some smallest unit of space or time. — Michael

My reaction to that is the same as I wrote above to Ludvig V.

But that has no bearing on the principle of division. — TonesInDeepFreeze

Yes it surely does have bearing on "the principle of division". If, the principle of division indicates that when we divide a 6 kg item into two equal parts, we will have two pieces of 3 kg, yet when we repeatedly carry out the procedure, the closest we can get is two parts of 2.99999 kg, then the principle of division is proven to be false. It would be a theory which has been falsified by empirical evidence.

The principle of division does not indicate that a material object can be perfectly divided in two. Whether a pie can be perfectly divided so that the masses of the resulting pieces are exactly equal is not a mathematical question.

I predict that the crank will just say his incorrect argument over again.

And notice no response from the crank to the point that "there is no finite bound to the number of calls to division" does not require that each division be by the same divisor, as the paradoxes discussed don't require that the divisors all be the same.

And still interested in what is supposed to be the inherent ordering of a set such as the set of bandmates in the Beatles. If that question can't even be addressed by the crank then his claim that sets have inherent order (thus that the mathematical notion, and even the everyday notion, of sets is wrong) is not sustainable.

It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks. — TonesInDeepFreeze

So from this we make the following argument:

P1. If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then infinitely many tasks have been completed by 12:00
P2. It is impossible to have completed infinitely many tasks
C1. Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on.

So what prevents the performance of the first task at 11:00, the second at 11:30, the third at 11:45, and so on? One answer is that it is metaphysically necessary that time is discrete with some smallest unit of time. At some point the time between task n and task n+1 is equal to the time between task n+1 and task n+2.

Is there another answer that allows for continuous time?

(1) We may question P2.

(2) C1 doesn't follow from P1 and P2. And it contradicts the point you quoted. Again, even if there is no completion of all of infinitely many subtasks, it is not entailed that there is a finite upper bound to how many may be completed, so, a fortiori, it is not entailed that each of the subtasks is not completed.

(3) Thompson argues, "It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, 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."

(4) If I'm not mistaken, Thompson recognizes physical possibility and logical possibility, which are at least fairly well understood, but he doesn't mention metaphysical possibility. That's not to say that the notion of metaphysical possibility should be ruled out, but only that it requires explication.

(1) We may question P2. — TonesInDeepFreeze

P2 is what Thomson tries to prove by introducing his lamp. Having performed infinitely many tasks entails a contradiction (the lamp must be either on or off but cannot be either).

(2) C1 doesn't follow from P1 and P2. — TonesInDeepFreeze

It seems to be a straightforward modus tollens.

If A then B
B is impossible
Therefore, A is impossible

As far as I can tell, you're committing the very conflation that Thompson warns about and as you quoted my paraphrase of it.

Think of the difference between "For all, there exists" and "There exists for all". Even if there does not exist a completion of all the subtasks, it does not follow that there there does not exist a completion of each of them. Just like that there is no completion of all of infinitely many additions but there is a completion of any finite number of them.

Where is the conflation in my argument? I'll set it out more clearly:

P1. If (A) the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) infinitely many tasks have been completed by 12:00
P2. B is impossible
C1. Therefore, A is impossible

The argument is certainly valid; it's modus tollens. So are you saying that A doesn't entail B or are you saying that B is possible?

The conflation is just as I stated it.

But your new argument is quite different from the previous argument, and, as far as I can tell, it does not suffer the conflation.

But your new argument is not modus tollens.

Modus tollens is:

A -> B
~B
Therefore ~A

Your new argument is:

A -> B
~pB
Therefore ~pA

The antecedent and consequent of the conditional are A and B; they are not ~pA and ~pB.

I'm not sure whether the argument is modally valid (I'm very rusty in modal logic).

/

I'm not firmly opining as to whether A implies B nor as to whether B is possible.

First, though, what sense of 'possible' is meant? Thomson discusses physical possibility and logical possibility. If I'm not mistaken, he doesn't mention metaphysical possibility. Of course, discussion doen't have to be limited to Thomson's context, but 'metaphysical possibility' requires even more explication.

Anyway, I tend to favor that A is correct. But I tend to think Thomson may well be on the right track when he argues that it is not logically impossible to execute infinitely many steps in finite time.

But that does not imply that there may be a completion of all of infinitely many tasks. — TonesInDeepFreeze

S has a lower bound in S if and only if there is a member of S that is less than or equal to every member of S.
S has a lower bound if and only if there is an x such that x is less than or equal to every member of S. — TonesInDeepFreeze

The range of the sequence 1, 1/2, 1/4 ... has no lower bound in the range, but it has a lower bound (the greatest lower bound is 0). — TonesInDeepFreeze

Thank you for the clarification. I must admit, I was a bit puzzled by "bound". I'm used to "limit". This clarifies something that was puzzling me - how one could describe the relationship between the 0 and 1 to the steps of the series. This seems to work very well.

Again, even if there is no completion of all of infinitely many subtasks, it is not entailed that there is a finite upper bound to how many may be completed, so, a fortiori, it is not entailed that each of the subtasks is not completed. — TonesInDeepFreeze

Are you suggesting that it might be the case that all of infinitely many tasks can be completed? What would the last task be?
On the other hand, perhaps we should accept that when Achilles catches the tortoise or finishes the race, he has completed all of infinitely many tasks. That might need some explaining, though, wouldn't it?

If I'm not mistaken, Thompson recognizes physical possibility and logical possibility, which are at least fairly well understood, but he doesn't mention metaphysical possibility. That's not to say that the notion of metaphysical possibility should be ruled out, but only that it requires explication. — TonesInDeepFreeze

There is the possibility that he doesn't recognize metaphysical possibility. Not everyone does.

P1. If (A) the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) infinitely many tasks have been completed by 12:00
P2. B is impossible
C1. Therefore, A is impossible — Michael

Here's a thought. When we define the series, we have defined each and every step in the series (or, if you prefer, each and every member of the relevant set.) Defining a step is a task. Suppose we do that at 10:55. So we completed infinitely many tasks, not only before 12:00, but before 11:00. So it is not impossible to complete infinitely many tasks before 12:00.
What's more, adding together the first step and the second step is a task. I can calculate the limit (sum) of the infinite series, well before 12:00.
Obviously, I haven't defined the last step, or the penultimate step, but that wasn't the challenge.
Or have I misunderstood the mathematics?