Infinite Staircase Paradox

TonesInDeepFreeze

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? — Ludwig V

I made no judgement on that. Again:

Suppose there can be no completion of all the tasks. That does not entail that there is a finite upper bound to how many task can be completed. That is, suppose for some finite n you say that no more than n tasks can be completed. But n+1 tasks can be completed without contradicting that there can be no completion of all of them. So there is no n that is a finite upper bound to how many tasks can be completed.~~And your statement (at least as you wrote it) was that none of them could be completed, which is even more wrong.~~ [strikethrough in edit; the remark pertains to a different poster.]

To see that explained again, see Thomson's paper.

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? — Ludwig V

Indeed. Hence 2500 years of philosophers, mathematicians and scientists talking about it.

TonesInDeepFreeze

When we define the series, we have defined each and every step in the series — Ludwig V

I like to keep the word 'series' for sums per convergences, and the word 'sequences' for sequences.

It is not the case that when we define an infinite sequence we must individually define each entry in the sequence. Example:

Definition of sequence S:

The domain of S is the set of natural numbers. For every natural number n, S(n) = n+1.

That's a finite definition (all definitions are finite) of an infinite sequence.

Michael

I'm not firmly opining as to whether A implies B — TonesInDeepFreeze

Then let's rephrase P1 as a question.

If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then how many tasks are performed by 12:00?

nor as to whether B is possible. — TonesInDeepFreeze

This is where Thomson's lamp comes in. His argument is that if B is performed then a contradiction follows; the lamp can neither be on nor off at 12:00 but must be either on or off at 12:00. Therefore B is proven impossible.

I'm not sure whether the argument is modally valid — TonesInDeepFreeze

I'm pretty sure it is. But if you prefer:

P1. If (A) it is possible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) it is possible for infinitely many tasks to be performed by 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A

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. — TonesInDeepFreeze

See subjunctive possibility:

Logical possibility is usually considered the broadest sort of possibility; a proposition is said to be logically possible if there is no logical contradiction involved in its being true. "Dick Cheney is a bachelor" is logically possible, though in fact false; most philosophers have thought that statements like "If I flap my arms very hard, I will fly" are logically possible, although they are nomologically impossible. "Dick Cheney is a married bachelor," on the other hand, is logically impossible; anyone who is a bachelor is therefore not married, so this proposition is logically self-contradictory (though the sentence isn't, because it is logically possible for "bachelor" to mean "married man").

Metaphysical possibility is either equivalent to logical possibility or narrower than it (what a philosopher thinks the relationship between the two is depends, in part, on the philosopher's view of logic). Some philosophers have held that discovered identities such as Kripke's "Water is H₂O" are metaphysically necessary but not logically necessary (they would claim that there is no formal contradiction involved in "Water is not H₂O" even though it turns out to be metaphysically impossible).

Nomological possibility is possibility under the actual laws of nature. Most philosophers since David Hume have held that the laws of nature are metaphysically contingent—that there could have been different natural laws than the ones that actually obtain. If so, then it would not be logically or metaphysically impossible, for example, for you to travel to Alpha Centauri in one day; it would just have to be the case that you could travel faster than the speed of light. But of course there is an important sense in which this is not possible; given that the laws of nature are what they are, there is no way that you could do it. (Some philosophers, such as Sydney Shoemaker, have argued that the laws of nature are in fact necessary, not contingent; if so, then nomological possibility is equivalent to metaphysical possibility.)

I am claiming that supertasks are more than just nomologically impossible. They are either logically impossible or metaphysically impossible. I say the latter simply because it's the weaker claim.

Ludwig V

And your statement (at least as you wrote it) was that none of them could be completed, which is even more It is wrong. — TonesInDeepFreeze

In the quotation in that message, I made no statement. I just asked a question.

I made no judgement on that — TonesInDeepFreeze

Thank you for the answer to my question. It is very helpful.

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

I would have thought that "for any finite number of tasks, there may be a completion of all the tasks" does not imply that there may be a completion of all of infinitely many tasks and does not imply that there may not be a completion of all of infinitely many tasks.

It is not the case that when we define an infinite sequence we must individually define each entry in the sequence — TonesInDeepFreeze

I agree with what you say. But, nonetheless, we have defined each entry in the sequence. Or is there an entry that is not defined? I can't think of one.
My indirect response is:- I'm sorry. I should have explained.
I was thinking about "there is no finite upper bound to how many task may be completed in finite time." It occurred to me that that depended on how long each task takes. (And that depends on what the task is.)
Clearly, if each task takes 1 second, there is a limit to the number of tasks that can be completed in 2 minutes. But how long does it take to add 1 to a given number? Does it take less time or more to add 1,000,000 to a given number? How long does it take to switch Thompson's lamp on or off? How long does it take to divide a given number by 2 or 10?
And then,the proof that sqrt(2) is irrational is also a proof that no rational number is sqrt(2)? Do I have to show separately and individually that each rational number is not sqrt(2)? I think not, but I have proved, of each rational number, that it is not sqrt(2). Is this one task, or many? How long does it take, either to prove that sqrt(2) is irrational or to prove, of a specific rational number, that is not sqrt(2)?

It is not the case that when we define an infinite sequence we must individually define each entry in the sequence. — TonesInDeepFreeze

Certainly. But there are some points I am not clear about.

Example:
Definition of sequence S:
The domain of S is the set of natural numbers.
For every natural number n, S(n) = n+1.
That's a finite definition (all definitions are finite) of an infinite sequence. — TonesInDeepFreeze

Yes, but how long did it take? Have you not defined each individual member of the sequence and all the members of the sequence? Which members of the sequence are not defined? How many tasks have I completed?

I like to keep the word 'series' for sums per convergences, and the word 'sequences' for sequences. — TonesInDeepFreeze

Thank you. I struggle with that difference. I'm not sure that everyone is consistent. What term do you use for a member of the sequence. People seem to by using "stage" or "term". Then there's the difficulty that "0, 1, 0, 1, ...." has, in one way, two members, each of which occurs repeatedly, So what do we call the first "0" as distinct from the second "0"?

Hence 2500 years of philosophers, mathematicians and scientists talking about it. — TonesInDeepFreeze

Yes. They're still talking about Epimenides the Cretan (and variants), as well. I can't deny they are both fascinating and annoying.

Metaphysical possibility is either equivalent to logical possibility or narrower than it (what a philosopher thinks the relationship between the two is depends, in part, on the philosopher's view of logic).

quoted by @Michael
Which means that metaphysics does not have an authoritative definition that we can all use for communication purposes, apart from the recognition that it involves logic.
For that reason, I prefer to stick to logical and physical possibility. There at least appears to be some consensus about the use of those terms, even though philosophers will, from time to time, quarrel about their definitions as well.

Michael

@Metaphysician Undercover @TonesInDeepFreeze @fishfry

I've moved your discussion on set ordering and the meaning of equality to this discussion.

Metaphysician Undercover

↪Michael

Thank you sir.

TonesInDeepFreeze

This is where Thomson's lamp comes in. His argument is that if B is performed then a contradiction follows; the lamp can neither be on nor off at 12:00 but must be either on or off at 12:00. Therefore B is proven impossible. — Michael

But in that example, his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state.

And an earlier point:

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. — Michael

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it.

If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then how many tasks are performed by 12:00? — Michael

It would be difficult not to say that denumerably (thus infinitely) many are performed. But I am reserving full committment to that, as there may be a finer analysis depending on complications in the notions of 'task' and 'performed' and perhaps in the inscrutabiliy of the overall hypothetical context.

P1. If (A) it is possible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) it is possible for infinitely many tasks to be performed by 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A — Michael

That's okay (though, personally for me, modulo the residue of doubt that I mentioned about P1). But I wouldn't take P2 as a given without justification.

Michael

But I wouldn't take P2 as a given without justification. — TonesInDeepFreeze

P2 is what Thomson's argument tries to prove. The lamp must be either on or off at 12:00, but if the button is pushed an infinite number of times between 11:00 and 12:00 then the lamp can neither be on nor off at 12:00. This is a contradiction. Therefore the button cannot be pushed an infinite number of times between 11:00 and 12:00.

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it. — TonesInDeepFreeze

This is the argument I am making:

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]

P3. If (C) it is possible for time to be continuous then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]

C3. Therefore, it is necessary for time to be discrete [from C2]

TonesInDeepFreeze

In the quotation in that message, I made no statement. I just asked a question. — Ludwig V

My mistake. Just now I put an edit note in the post.

"for any finite number of tasks, there may be a completion of all the tasks" does not imply that there may be a completion of all of infinitely many tasks and does not imply that there may not be a completion of all of infinitely many tasks. — Ludwig V

Right.

we have defined each entry in the sequence — Ludwig V

I am using 'define' in the exact sense of making a mathematical definition. A definition is a single formula. It follows from the definition of the sequence that each entry in the sequence is "determined" (for lack of a better word), but that is not to say that each has been individually defined.

there is no finite upper bound to how many task may be completed in finite time." It occurred to me that that depended on how long each task takes. — Ludwig V

I was speaking in the context of the completion times halving.

Does it take less time or more to add 1,000,000 to a given number? — Ludwig V

Less time or more time than what? And what is meant by "time to add a number"? Does it mean number of steps in some given adding algorithm?

How long does it take to switch Thompson's lamp on or off? — Ludwig V

The problem presented states the increasing rate of alteration. Maybe you're asking about how long it takes to jab the button. I don't find that to be relevant, since whatever times it takes, we need only assume that it happens within the durations given in the problem. For that matter, I don't think the particulars about buttons, jabbing, or especially about human acts such as fingers reaching to touch a device are relevant, as the problem could be entirely abstract, as what is essential only is that the lamp goes on and off at the increasing rate mentioned, or, for that matter, it's not essential even that it's a lamp or any other particular device (could be clown klaxon going off an on for all it matters) as long as there are alternating states, whatever they may be.

the proof that sqrt(2) is irrational is also a proof that no rational number is sqrt(2)? — Ludwig V

For all practical purposes, yes.

Do I have to show separately and individually that each rational number is not sqrt(2)? I think not, but I have proved, of each rational number, that it is not sqrt(2). — Ludwig V

You proved a universal generalization:

For all x, if x is rational then it is not the case that x^2 = 2.

Then, with universal instantiation, for any given rational number, for example, 1.4 we prove that it is not the case that (1.4)^2 = 2.

What term do you use for a member of the sequence. — Ludwig V

Which sequence? There are different sequences involved in the puzzles here.

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. *

* In discussion about the task problems, we are modifying a bit by not having 0 in the domains of the sequences. That is, we index starting with 1 rather than with 0.

A function is a certain kind of set of ordered pairs.

The members of a function are ordered pairs.

The domain of the function is the set of first coordinates of the ordered pairs. We call members of the domain "arguments for the function".

The range of the function is the set of second coordinates of the ordered pairs. We call members of the range "values of the function". I also call them "entries".

We say that an argument maps to a value.

The infinite sequences in this context are:

(1) The function that maps every natural number n>0 to to 1/(2^n)

This is the function {<1 1/2> <2 1/4> <3 1/8> ...}

(1) The function that maps every natural number n>0 to either 0 or 1 ("off or on") depending on whether n is odd or even

That is the function {<1 0> <2 1> <3 0> ...}

what do we call the first "0" as distinct from the second "0"? — Ludwig V

0 is 0. There are not different 0s. But with the function just mentioned, and 0 occurs as values for different arguments of the function. 0 is the value at the arguments 1, 3, 5 etc.

TonesInDeepFreeze

if the button is pushed an infinite number of times between 11:00 and 12:00 then the lamp can neither be on nor off at 12:00. — Michael

As I mentioned, that is a premise that you don't include in your own argument. As I mentioned:

"his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state."

I can't imagine anyone denying that there is no immediate predecessor state, but some partisans who don't accept the argument deny that the state at 12:00 must be determined by an immediate predecessor state. So you must include the premise that the state at 12:00 must be determined by an immediate predecessor state.

TonesInDeepFreeze

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it.
— TonesInDeepFreeze

This is the argument I am making:

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens] — Michael

If you don't mean "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" then it should be considered scratched.

TonesInDeepFreeze

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]

P3. If (C) it is possible for time to be continuous then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]

C3. Therefore, it is necessary for time to be discrete [from C2] — Michael

The division of time mentioned in the thought experiment doesn't require continuousness of time; it only requires density time (via the density of the rationals).

Continuousness implies density, but density does not imply continuousness. So banning continuousness does not ban density. But you need to ban density of time.

You should have instead:

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]

P3. If (C) it is possible for time to be dense then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]
C3. Therefore, it is necessary for time to not be dense [from C2]

But:

(1) We may doubt P2.

(2) You have an unstated premise on which P2 is based, viz. that that state must be determined by an immediate predecessor.

(3) We may doubt P3.

Michael

As I mentioned, that is a premise that you don't include in your own argument. As I mentioned:

"his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state."

I can't imagine anyone denying that there is no immediate predecessor state, but some partisans who don't accept the argument deny that the state at 12:00 must be determined by an immediate predecessor state. So you must include the premise that the state at 12:00 must be determined by an immediate predecessor state — TonesInDeepFreeze

The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off.

Given the way lamps work, or at least the lamp in this example, if the lamp is off and the button is pushed then the lamp is turned on and if the lamp is on and the button is pushed then the lamp is turned off.

Nothing other than pushing the button can affect the lamp.

Scenario 1
The lamp is off at 10:00 and the button is only pushed at 11:00. Therefore the lamp is on at 12:00.

Scenario 2
The lamp is off at 10:00 and the button is only pushed at 11:00 and 11:30. Therefore the lamp is off at 12:00.

Scenario 3
The lamp is off at 10:00 and the button is only pushed at 11:00, 11:30, 11:45, and so on. Therefore the lamp is ? at 12:00.

No substitution for "?" is either derivable or consistent. Therefore the button-pushing procedure in scenario 3 is proven impossible in principle.

It's not a matter of continuousness but rather of density. — TonesInDeepFreeze

I don't know what this means.

If you don't mean "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" then it should be considered scratched. — TonesInDeepFreeze

I do mean that.

TonesInDeepFreeze

↪Michael

While you were posting, I revised my post to better explain about continuousness and density.

The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off. — Michael

That is not at issue. Rather, as I've said twice now, at issue is whether its state at 12:00 depends on there being an immediate predecessor state. Thomson assumes that it does.

Given the way lamps work, or at least the lamp in this example, the lamp is on if and only if the lamp was off and the button was pushed to turn it on, and (after having been turned on at least once) the lamp is off if and only if the lamp was on and the button was pushed to turn it off. — Michael

Not just that it was off and then turned on, but rather that it was off at time t1 and on at time t2. That is, that it's not just a matter of the lamp having been off previously but rather that there is an off state that is an immediate predecessor of the on state and that that extends to 12:00 too so that for the lamp to be on at 12:00 there must be an immediate predecessor state in which the lamp was off, mutatis mutandis for the lamp being off at 12:00. Thomson mentions this. It's a premise that needs to be stated.

If you don't mean "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" then it should be considered scratched.
— TonesInDeepFreeze

I do mean that. — Michael

(1) The first task is impossible to be performed. The second task is impossible to be performed. The third task is impossible to be performed ...

Quantified:

For all tasks, there is not a performance of any of them.

I think that is not what you mean.

(2) It is not possible for there to be a single performance of all the tasks.

Quantified:

There is not a performance that performs all the tasks.

I surmise that is what you mean.

I wouldn't write "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" because it can be understood in sense (1).

It is not possible for the first dancer to do a flip today, for the second dancer to do a flip tomorrow, and so on.

I would take that to mean that none of the dancers can do a flip on their appointed day.

Michael

Not just that it was off and then turned on, but rather that it was off at time t1 and on at time t2. That is, that it's not just a matter of the lamp having been off previously but rather that there is an off state that is an immediate predecessor of the on state and that that extends to 12:00 too so that for the lamp to be on at 12:00 there must be an immediate predecessor state in which the lamp was off, mutatis mutandis for the lamp being off at 12:00. Thomson mentions this. It's a premise that needs to be stated. — TonesInDeepFreeze

I also revised my post after posting it.

The three implicit premises are:

1) The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off.

2) If the lamp is off and the button is pushed then the lamp is turned on and if the lamp is on and the button is pushed then the lamp is turned off.

3) Nothing other than pushing the button can affect the lamp.

It therefore follows that if the lamp is on then it is on only because it was off and the button was pushed to turn it on. It makes no sense for a lamp to be on without having been turned on, and in this specific case it can only be turned on by pushing a button when it is off.

Therefore if the lamp is on at midnight then it is on only because it was off and the button was pushed to turn it on. This is impossible if the infinite sequence 11:00, 11:30, 11:45, ... models the times that the button is pushed.

(1) The first task is impossible to be performed. The second task is impossible to be performed. The third task is impossible to be performed ...

Quantified:

For all tasks, there is not a performance of any of them.

I think that is not what you mean.

(2) It is not possible for there to be a single performance of all the tasks.

Quantified:

There is not a performance that performs all the tasks.

I surmise that is what you mean.

I wouldn't write "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" because it can be understood in sense (1).

It is not possible for the first dancer to do a flip today, for the second dancer to do a flip tomorrow, and so on.

I would take that to mean that none of the dancers can do a flip on their appointed day. — TonesInDeepFreeze

I am saying that it is impossible in principle for the infinite sequence 11:00, 11:30, 11:45, ... to model the times that the button is pushed.

TonesInDeepFreeze

↪Michael

That seems okay at face value. But since you've put the argument in a list, I'd make explicit all the premises.

I'd like to read Benacerraf's paper that disputes that there can't be a state at 12:00, and the papers mentioned in Thomson's paper for required context.

Lionino

You are interested in exploiting that to define metaphysics. — Ludwig V

I just found this when reading of p-zombies on SEP:

Still, many physicalists hold that what guarantees the impossibility of zombies is ‘metaphysical’ necessity. Typically they maintain that states of phenomenal consciousness are identical with physical states, and that these identities are necessary a posteriori as argued by Kripke (see e.g. McLaughlin 2005, and for criticism, Stoljar 2000). But the vocabulary of possibility and necessity is slippery. For example there is disagreement over whether logical and metaphysical possibility are different (section 3.1 below); when Kripke (1972/80) writes of ‘logical’ and ‘metaphysical’ possibility he seems to use those words interchangeably (Yablo 1999: 457n.), and some use ‘logical’ where others prefer ‘conceptual’ (Chalmers 1999: 477); compare Latham 2000, 72f.).

They hold there is an identity that is metaphysically necessary, and it is metaphysically necessary because it is a a posteriori necessity.

fishfry

I'd like to read Benacerraf's paper that disputes that there can't be a state at 12:00 — TonesInDeepFreeze

I wandered over and happened to notice this remark. I can say something about it. I was pointed at Benacerraf's paper a few weeks ago, and since I had developed the same solution, I felt vindicated and at the very least sane. I know Benacerraf from his great paper, What Numbers Could Not Be, and has a lot of credibility with me.

So I feel qualified to explain his point.

Here is Benacerraf's argument in my math-y conception. It's the same argument.

The lamp problem is best modeled as a function defined on the ordinal $\omega + 1$ , with output in the set $\{0, 1\}$ . (I don't recall if Benacerraf explicitly uses this mathematical approach but his argument is the same ).

$\omega + 1$ can be visualized as the sequence

0, 1, 2, 3, ...; $\omega$

where 0, 1, 2, 3, ... ranges through all the natural numbers; and there is a "point at infinity" after them all, traditionally called $\omega$ in this context. It can of course be formalized as an ordinal but that's not even important in this context.

If it makes anyone happy in the context of this problem, we can model it as

1/2, 3/4, 7/8, ...; 1

where the values are times getting ever closer to 1, with 1 being the mathematical limit, as well as the time on a clock. Or 1/2, 1/4, 1/8, ...; 0 where the times are seconds before midnight, and 0 is exactly midnight.

I analogized that last idea to Cinderella's coach, which is a beautiful coach at 1/2, 1/4, 1/8, ... seconds before midnight; but becomes a pumpkin at midnight. It's not usually thought of as such, but Cinderella's coach is a supertask puzzle. If Cinderella checks her watch at each time 1/2, 1/4, etc., it's always a coach. How does it become a pumpkin?

It's the same mathematical insight that applies to most of these supertask puzzles. Sequences don't ever "reach" their limit, but every sequence can be assigned an arbitrary terminal state. Sometimes the terminal state is the limit of the sequence, and sometimes it's not. You can always assign a terminal state because it's just a mathematical function. It's a function defined not on $\omega$ , as infinite sequences usually are; but on $\omega + 1$ , to include a terminal state.

We can use this structure to model the lamp. For each natural number 0, 1, 2, 3, ... the output alternates on/off, which we can denote as 1, 0, 1, 0, ...

So the lamp problem defines the function for each of the inputs 0, 1, 2, 3, ... but does not specify the value at $\omega$ . Therefore we are free to define it any way we like.

Both outcomes are entirely consistent with the premises of the problem.

Additionally, unlike in some supertask puzzles, in this case there is no natural or preferred solution. That's because in neither case can the terminal value be the limit of the sequence.

Contrast this, for example, with the sequence 1/2, 3/4, 7/8, ...; 1. That notation says that there's an infinite sequence; and after the entire sequence, we have a value, 1.

Now just as before, the assignment of the value of the terminal state is arbitrary. But in this particular case, the value 1 is natural, in the sense that 1 is the mathematical limit of the sequence. This idea of adjoining the limit of a convergent sequence to the end of the sequence is a convenient and natural formalism.

With the lamp, there is no possible way to assign a terminating value that makes any particular sense. Instead, absolutely any answer will do. On, Off, or as I facetiously said earlier, a plate of spaghetti; to emphasize the arbitrariness of the choice.

I've visualized the lamp problem that way for a long time. And when I found out that Benacerraf gave the same argument (minus the spaghetti), I was happy.

That's Benecerraf's argument in my words. Here's his paper.

Tasks, Super-Tasks, and the Modern Eleatics

Ludwig V

There is no truth of the matter, because it is a matter of deciding how to apply the rules to a situation which they were not designed to cater for. — Ludwig V

Which sequence? There are different sequences involved in the puzzles here. — TonesInDeepFreeze

I was thinking of a term that would apply to sequences in general.

0 is the value at the arguments 1, 3, 5 etc — TonesInDeepFreeze

I think I can work with that. But I see that you used "entry" elsewhere. That's simpler.

I was speaking in the context of the completion times halving. — TonesInDeepFreeze

I didn't realize that is the context. Then most of what I said is irrelevant.

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. — TonesInDeepFreeze

I agree with all of that. There is a twist, of course. The sum is not the total addition of all the entries, but the limit of the total addition of all the entries. The total addition of all the entries up to a specific point will converge on/with the sum.

For that matter, I don't think the particulars about buttons, jabbing, or especially about human acts such as fingers reaching to touch a device are relevant, as the problem could be entirely abstract, as what is essential only is that the lamp goes on and off at the increasing rate mentioned, or, for that matter, it's not essential even that it's a lamp or any other particular device (could be clown klaxon going off an on for all it matters) as long as there are alternating states, whatever they may be. — TonesInDeepFreeze

I agree with all of that. That's why I ended up formulating the problem in terms of 1 and 0 alternating. Unfortunately, that doesn't resolve everything. But I need time to work out how to articulate this reasonably clearly. Sorry. I will get back to you.

They hold there is an identity that is metaphysically necessary, and it is metaphysically necessary because it is a a posteriori necessity. — Lionino

Well, I don't want to quarrel about a name. Call it metaphysics or call it a posteriori necessity, my account of the problem derives from Wittgenstein, Ryle and Toulmin. That's what matters.

There is no truth of the matter, because it is a matter of deciding how to apply the rules to a situation which they were not designed to cater for. — Ludwig V

The same comment applies to much of the discussion in this thread as well. Ryle seems to have thought it applies to all philosophical problems, but I wouldn't go that far.

We are being asked about the causal consequence of having carried out a supertask. — Michael

My problem is that I don't understand what carrying out a supertask is. So I can't even think about the consequences of carrying one out. @TonesInDeepFreeze, This is the best that I can do to articulate what bothers me.

Michael

With the lamp, there is no possible way to assign a terminating value that makes any particular sense. Instead, absolutely any answer will do. On, Off, or as I facetiously said earlier, a plate of spaghetti; to emphasize the arbitrariness of the choice. — fishfry

The first sentence is true and is the proof that "supertasks are senseless" (as Thomson says).

The second sentence is false. As mentioned several times, the implicit premises are that the lamp continues to exist (as a lamp) at midnight and that nothing other than pushing the button can turn the lamp on or off.

We are being asked about the causal consequence of having performed a supertask. Your introduction of magic or God or some other entity or event at the end is a red herring.

This is abundantly clear with the pseudocode I offered a month ago:

var isLampOn = false

function pushButton()
{
  isLampOn = !isLampOn
}

var i = 120

while (true) {

  wait i *= 0.5
  
  pushButton()

}

echo isLampOn

The logic of this does not allow for echo isLampOn to output true or false or 'a plate of spaghetti'.

Your "solution" is to inject some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so you are no longer addressing the problem as posed.

fishfry

The first sentence is true and is the proof that "supertasks are senseless" (as Thomson says). — Michael

@TonesInDeepFreeze asked for Benacerraf's argument, of which I provided my own version without any change in meaning. I also linked Benacerraf's paper per Tones's request. That was my motivation for posting.

As mentioned several times, the implicit premises are that the lamp continues to exist (as a lamp) at 12:00 and that nothing other than pushing the button can turn the lamp on or off. — Michael

This is a red herring of no relevance. I've responded to the button pushing argument as many times as you've mentioned it. If "nothing other than pushing the button turns the lamp on or off," then at midnight, the button pusher pushes the button and turns the lamp on or off, per your premise. The terminal state is On or Off. In either case "the button was pushed" since you insist on it. No matter the final state, the button was pushed, and your premise is satisfied. Neither On nor Off is to be preferred since the sequence 1, 0, 1, 0, ... has no limit. Thompson himself makes this observation in his paper.

I did enjoy our earlier conversation, but no additional typing on my part could be of any use to you. If it could be, it would have already been.

I regard Thompson's lamp as a solved problem. I'll leave it at that, and refer you to my existing posts on the subject.

Michael

If "nothing other than pushing the button turns the lamp on or off," then at midnight, the button pusher pushes the button and turns the lamp on or off, per your premise. — fishfry

He doesn't push the button at midnight. He only pushes it at 23:00, 23:30, 23:45, and so on. This is an explicit premise of the problem.

Also, pushing the button will only turn the lamp on if it was off and so to say that the button was pushed at midnight to turn it on is to say that the lamp was left off after having performed the supertask, which is false.

fishfry

He doesn't push the button at midnight. He only pushes it at 11:00, 11:30, 11:45, ...

Also, pushing the button will only turn the lamp on if it was off and will only turn the lamp off if it was on. — Michael

I appreciate your enthusiasm, but surely you can see that nothing new has been said between us for a long time. We could just reread each other's old posts. I hope you'll forgive me for declining to engage. If you had better insight into infinite sequences and their limits, you wouldn't be troubled by what happens at midnight. I say that because the code you posted exactly reflects your misunderstanding of the nature of infinite sequences. Think of Cinderella. Why does the coach turn into a pumpkin? It does so because that's the premise of the story. The lamp is in some state at midnight because you say it is. And it doesn't matter which.

Michael

↪fishfry

I understand how infinite sequences and limits work, as did Thomson. That is why I understand that an infinite sequence of button pushes before midnight is inconsistent with the lamp being either on or off at midnight.

The problem is that you seem to fail to acknowledge how lamps work.

fishfry

The problem is that you seem to fail to acknowledge how lamps work. — Michael

Lamps that switch state in arbitrarily small intervals of time? I missed that day in lamp school.

The lamp is as much a fairy tale as Cinderella's coach, which is why I use that example. It's a magic lamp. You are the one invoking magic, and then acting like it's a real lamp to try to argue a pointless point. That's another source of your confusion.

"How lamps work?" Lamps don't work that way! Maybe YOU missed that day in lamp school.

Michael

Lamps that switch state in arbitrarily small intervals of time? — fishfry

This is the assumption we allow for to examine the possibility of supertasks.

But it is still the case that the lamp cannot arbitrarily be on (whether at midnight or any other time). It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ω.

fishfry

This is the assumption we allow for to examine the possibility of supertasks.

But it is still the case that it cannot arbitrarily be on. It can only be turned on by pushing a button when it is off. You continually ignore this fact when you talk about the mathematical value ω. — Michael

Jeez I didn't even finish editing before you snapped back. Have a nice evening. You are trolling me now. Doing a good job of it.

Michael

But since you've put the argument in a list, I'd make explicit all the premises. — TonesInDeepFreeze

Sure.

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 22:00
P5. The lamp is either on or off at midnight

C1. If the button is only ever pushed at 23:00 then the lamp is on at midnight
C2. If the button is only ever pushed at 23:00 and 23:30 then the lamp is off at midnight
C3. If the button is only ever pushed at 23:00, 23:30, 23:45, and so on ad infinitum, then the lamp is ? at midnight

It is essential to note that we are being asked about the causal consequence of having performed the supertask, not simply asked for the value of ω in the infinite sequence off, on, off, ...; ω.

No substitution for "?" in the consequent of C3 is either derivable from or consistent with the premises and the antecedent of C3. Therefore we must accept that it is impossible in principle to have performed the supertask.

The following pseudocode provides a demonstration of C3 and is the correct way to interpret the logic of Thomson's lamp:

var isLampOn = false

function pushButton()
{
  isLampOn = !isLampOn
}

var i = 120

while (true) {

  wait i *= 0.5 // seconds
  
  pushButton()

}

echo isLampOn

We are being asked what value is output by the line echo isLampOn (which allegedly will run after two minutes). The problem is that it cannot be true and cannot be false but cannot be anything other than true or false.

Benacerraf's solution is akin to injecting 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.

Lionino

↪Ludwig V

For clarification, my quoting of your original comment is to add context to my post, not to agree or disagree with you.

Metaphysician Undercover

The division of time mentioned in the thought experiment doesn't require continuousness of time; it only requires density time (via the density of the rationals). — TonesInDeepFreeze

Infinite density of time. What could that possibly mean? "Density" implies two distinct substances, as a ratio (weight per volume for example). Are you suggesting that time consists of two distinct substances, in a relation which is infinitely dense? That seems absurd.

I know your sophistry TIDF. When cornered in an argument, instead of retreating, you create an absurd concept, by combining words in an incoherent, irrational way, hoping that the incoherency will go unnoticed. This would allow you to slither away through the crack between the infinitely dense particles of time. This is better represented as the irrational space between the rational numbers, and the slippery sophist slides out of that possible world, through the irrational exit.

Infinite Staircase Paradox

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