Infinite Staircase Paradox

TonesInDeepFreeze

Got it.

Lionino

Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino

That is the advantage of platonism over nominalism on the matter of the application of mathematics. Each different nominalist program for mathematics will have a different solution for it — Azzouni's solution being deflationary (dissolving the issue).

I personally think the issue is misguided.

Ludwig V

I'll refer you to this: — TonesInDeepFreeze

I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.

I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. — TonesInDeepFreeze

Perhaps it would serve our purposes. I could probably get the point even if it isn't completely rigorous.

But let me explain why I need convincing.
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So for each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious.

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze

Well, it seems clear that at any specific time, it will be on or off depending on whether the button has been pushed an even number of times or an odd number of times since 11:00.

So at each of the times specified in the sequence, it will be on or off depending whether the number of times it has been pushed since 11:00 is odd or even.

rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.* — TonesInDeepFreeze

The contradiction is created here - specifically in the last two words, which make it impossible to know whether it has been pushed an even or odd number of times since 11:00.

Ludwig V

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze

Which one do you think should be rejected?

Ludwig V

For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals. — fishfry

If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe.

1 is a perfectly sensible probability. — fishfry

.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? (I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it.

Michael

Rather, infinitely divisibility along with the other premises entails a contradiction. — TonesInDeepFreeze

I think this is a misunderstanding of the problem.

Say we accept that Thomson's lamp entails a contradiction; the lamp can neither be on nor off at 12:00.

I take this as proof that having pushed a button an infinite number of times is metaphysically impossible.

You seem to take this as proof that having pushed a button an infinite number of times is metaphysically impossible only if the premises are true.

As an example, let's say that our button is broken; pushing it never turns the lamp on. In such a scenario we can unproblematically say that the lamp is off at 12:00. But this does not then entail that it is possible to have pushed the button an infinite number of times.

We can imagine a lamp with two buttons; one that turns it on and off and one that does nothing. Whenever it's possible to push one it's also possible to push the other, and so if it's possible to have pushed the broken button an infinite number of times then it's possible to have pushed the working button an infinite number of times. Given that the latter is false, the former is also false.

Having pushed a button an infinite number of times is an inherent contradiction, unrelated to what pushing the button does. Having the button turn a lamp on and off, and the lamp therefore being neither on nor off at the end, is only a way to demonstrate the contradiction; it isn't the reason for the contradiction.

Which is also why Benacerraf's response to the problem misses the mark.

The pseudocode I provided a month ago helps explain this:

var isLampOn = false

function pushButton()
{
  isLampOn = !isLampOn
}

var i = 120

while (true) {

  wait i *= 0.5
  
  pushButton()

}

echo isLampOn

isLampOn is only ever set to true or false (and never unset) but the echo isLampOn line can neither output true nor false. This demonstrates the incoherency in claiming that while (true) { ... } can complete.

Changing echo isLampOn to echo true does not retroactively make it possible for while (true) { ... } to complete.

Having pushButton() do nothing does not make it possible for while (true) { ... } to complete.

It is metaphysically impossible for while (true) { ... } to complete, regardless of what happens before, within, or after, i.e. neither of these can complete:

Code 1

var i = 120

while (true) { wait i *= 0.5 }

Code 2

while (true) { }

Ludwig V

Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino

Here's how I look at it. I think that everyone will agree that a formula is not about anything specific and, in itself is neither true nor false. x + y = z doesn't make any assertions, until you substitute values for the variables. So 2 +1 = 4 is false, but 2 + 3 = 5 is true. So there's a temptation to think it must be true of something. Hence realism. But 2 + 3 = 5 is itself like a formula in that once we specify what is being counted, it does make an assertion about the world - 2 apples + 2 apples = 4 apples. It is true of the world. Of course, 2 drops of water plus 2 drops of water doesn't make 4 drops of water, (until we learn to measure the volume of water). The domain of applicability and truth is limited.

fishfry

If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe. — Ludwig V

Ok forget that. But 0 and 1 are perfectly legitimate probabilities. After all if I roll a die, the probability is 1 that it will be either 1, 2, 3, 4, 5, or 6, Right? Nothing degenerate or unusual about that. And the probability is 0 that it will show 7.

.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. — Ludwig V

Are you talking about credence, perhaps?

So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? — Ludwig V

No. Only that the event is certain, in the finite case; or "almost certain" to happen, in the infinite case.

Can you give me an example of what you mean? What kind of statements are you applying probability to?

(I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it. — Ludwig V

Never mind probability. You just startled me by denying the legitimacy or sensibility of 0 and 1 as probabilities.

Metaphysician Undercover

I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential." — Ludwig V

I don't quite get what you mean here. Let's say there's something about reality which appears to be unintelligible. If we assign a name to that aspect, aren't we saying that there is actually something there which is unintelligible, and we've named it. This is to take a step further than simply that it appears as unintelligible.

I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. — Ludwig V

The point is that potential defies the laws of logic. That's why modal logic gets so complex, it's an attempt to bring that which defies the laws of logic into a logical structure.

The point I made, derived from Aristotle, is that whenever the lamp switches from on to off, or vise versa, there is necessarily a period of time during which it is changing (becoming). In other words, it is impossible that the switch from one to the other is instantaneous, and this is proven logically. In this 'mean time', the lamp is neither on nor off, and this defies the law of excluded middle. Dialethists would hold that it is both on and off, defying the law of noncontradiction.

Aristotle uses the concept of "potential" to explain his choice for defying the law of excluded middle rather than defying the law of noncontradiction. For him, the concept of "potential" is required to explain how something changes form having x property (being on), to not having x property (being off). "Potential" is a requirement of such a change, the thing cannot change without having the potential for change. However, this is a temporal concept, and the conclusion is that actualization requires a duration of time. So there is always a period of time between having x property (being on), and not having x property (being off).

What the lamp problem does not take into account, is that period of time between being on and off, during which it is changing. Assuming that the amount of time required to change from on to off, and vise versa, remains constant, then as the amount of time that the lamp is on and off for, gets smaller and smaller, the proportion of the time which it is neither, gets larger and larger. So at the beginning, when the time on and off are relatively long periods, the time of neither seems completely insignificant. But as the off/on actualization rapidly increases, the time of being on and off soon becomes insignificant in comparison to the time of being neither. The time of neither approaches all the time

A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc. — Ludwig V

You only say this, because the time of change in which the lamp is neither on nor off is so short and insignificant that it appears to be nil. Aristotle demonstrated logically that it cannot be nil. So when we say things like "lamps are a type of thing which must be on or off, and cannot be neither", this is a statement about how things appear to be, and this facilitates much of our talk about such things. But when we get down to the way that things actually are, the way that logic tells us they must be, we can see that this way of allowing appearances to guide our speaking is actually misleading.

I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time. — Ludwig V

I don't understand this. If a thing neither has nor has not the specified property, the excluded middle principle is violated (unless it's an inapplicable category). Potential itself neither is nor is not, and that's why we say it refers to what may or may not be. So "may or may not be" refers to the property we judge as in potential, and this says it neither is nor is not attributable to the thing.

TonesInDeepFreeze

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected.
— TonesInDeepFreeze
Which one do you think should be rejected? — Ludwig V

I don't proffer an opinion on that. But I can see that presumably the most likely candidate is "At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum." At least intuitively it is the ripest and lowest hanging fruit. Or put pejoratively, at least intuitively it is the sore thumb.

But logically we may reject any of them. None of them are logical truths (though, "At all times, the lamp is either Off or On and not both" would be logically true as a conclusion from defining 'On' as 'not Off'.

TonesInDeepFreeze

Say we accept that Thomson's lamp entails a contradiction; the lamp can neither be on nor off at 12:00.

I take this as proof that having pushed a button an infinite number of times is metaphysically impossible. — Michael

(1) If a set of premises G entails a contradiction, and for any member P of G we have that G\{P} does not entail a contradiction, then we are logically free to reject any member of G. None of the premises are logically true (except "Either On or Off and not both" as conclusion from a definition "'On' means 'not Off'"), so we can reject any of them. For example, we could reject "The lamp does not change from Off to On, or from On to Off, except by pushing the button."

(2) We still don't have a satisfactory definition here of 'metaphysically impossible'.

(3) For what it's worth, if I'm not mistaken, Thomson does not conclude the time is not infinitely divisible, but rather he weighs in against the notion of super-tasks.

As a coda to Thomson's argument, you rely on the premise "If time is infinitely divisible then the super-task is possible.' [not a quote of yours]

But we may reject that premise.

It seems to me that "the lamp super-task is executed" entails "time is infinitely divisible". But the converse - "time is infinitely divisible" entails "the lamp super-task is executed" - at least requires an argument.

And, it seems to me, that analysis is even more difficult because it involves modalities. First, we have to distinguish between "time is infinitely divisible" and "it is possible that time is infinitely divisible". Second, it's not really, "the lamp super-task is executed" but "it is possible that the lamp super-task is executed". The argument needs to checked whether the modal inferences are correct.

You seem to take this as proof that having pushed a button an infinite number of times is metaphysically impossible only if the premises are true. — Michael

No, I'm not claiming that.

let's say that our button is broken; pushing it never turns the lamp on. In such a scenario we can unproblematically say that the lamp is off at 12:00. But this does not then entail that it is possible to have pushed the button an infinite number of times. — Michael

Of course.

we can imagine a lamp with two buttons; one that turns it on and off and one that does nothing. Whenever it's possible to push one it's also possible to push the other, and so if it's possible to have pushed the broken button an infinite number of times then it's possible to have pushed the working button an infinite number of times. Given that the latter is false, the former is also false. — Michael

Whatever the validity of that, I don't see the point of it here.

Having pushed a button an infinite number of times is an inherent contradiction — Michael

It's not a contradiction in and of itself. Rather, it is inconsistent with the other premises (especially that the infinite number of executions occurs in finite time).

Having the button turn a lamp on and off, and the lamp therefore being neither on nor off at the end, is only a way to demonstrate the contradiction; it isn't the reason for the contradiction. — Michael

You would need to tell me the difference between a demonstration of a contradiction and the reason for a contradiction. For example, if someone asks "What is the reason that the unrestricted comprehension schema with the separation schema yields a contradiction?" then my best response would be to show a demonstration.

is also why Benacerraf's response to the problem misses the mark. — Michael

As far as I can tell, it's off-base because it doesn't address the premises of the lamp.

And related arguments against Thomson are that it is not problematic that the premises don't provide for concluding whether the lamp is Off or On at 12:00. But that misses the point that it is not that it is problematic that the lamp's state is undetermined, but rather that Thomson's argument shows that the lamp is neither Off nor On. Not that it is undetermined what the state is, but rather that is determined that the state is neither Off nor On.

/

I don't know enough about coding to have a comment on your pseudocode.

/

A while back you gave the argument in quite succinct form:

"P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1"

But you add to that argument two things:

(a) We must reject P5.

But each of the premises is required for the contradiction, so we can reject any one of them rather than P5. Granted, P5 does stand out as the candidate we would intuitively reject, but it is not logically required that it is the one we reject. For example, famously:

(U) ExAy yex

(S) ExAy(yex <-> ~yey)

yield a contradiction.

But to dispel the contradiction it is not logically required that we reject (U) to keep (S) when we could reject (S) to keep (U). We may have reasons for preferring that we reject (U) rather than (S), but that is not a demonstration that (U) is logically impossible or even that it is false.

(b) If time is infinitely divisible, then the super-task may be executed.

But that is not logically true either. It may be the case that time is infinitely divisible but still the super-task cannot be executed. Moreover, the modality "may be" slips in there, so the argument requires that it is made clear that the modal inference is permitted.

Michael

It's not a contradiction in and of itself. — TonesInDeepFreeze

An infinite sequence of operations is by definition an endless sequence of operations. An endless sequence of operations does not come to an end. That's what makes the premise of a supertask an inherent contradiction.

Having the operation be to push a button, and having this button turn a lamp on and off, is simply a way to make this inherent contradiction even clearer.

If you accept that this proves that this button cannot have been pushed an infinite number of times then what is the reasoning behind the claim that if some wizard steps in at 12:00 to magically turn the lamp on then this retroactively makes it possible to have pushed this button an infinite number of times? Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.

I don't think you're really grasping what distinguishes a supertask from an abstract infinite sequence.

TonesInDeepFreeze

↪Michael

I didn't say they end.

Having the operation be to push a button, and having this button turn a lamp on and off, is simply a way to make this inherent contradiction even clearer. — Michael

Again, the contradiction comes from the conjunction of the premises. It is not a given that it is a contradiction in and of itself that infinitely tasks are executed in finite time. It's quite unintuitive that infinitely many tasks an be executed in finite time, but to show that doesn't entail that it is a contradiction in and of itself.

what is the reasoning behind the claim that if some wizard steps in at 12:00 to magically turn the lamp on — Michael

I never claimed any such thing. It's a straw man, even if unintentional.

What I said is that if we drop the premise that the lamp is only turned on by the button, then we don't get the contradiction. The point of that is that we are not logically obliged to reject only one certain premise. Note that I am not committing the fallacy of not addressing Thomson's premises. Rather, I am pointing out that rejecting one of the premises to avoid contradiction must then allow rejecting any other premise to avoid contradiction.

When I saw that you had put care into your numbered arguments, I surmised that you were interested in pursuing rigor. So in that regard, I'm examining all your reasoning.

Again (new numbering):

(1) Since none of the premises are logically true, but they yield a contradiction, we may reject any one of them to avoid contradiction. We are not logically bound to reject the one that happens to be least intuitive.

(2) Infinitely divisibility of time does not entail executability of denumerably many tasks in finite time, even though, executability of denumerably many tasks in finite time entails infinite divisibility of time.

(3) We don't have a satisfactory definition of 'metaphysical possibility' here.

(4) The argument is more complicated than appears with only a cursory look, since it involves the modality of 'possible'.

(5) If I'm not mistaken, Thomson does not conclude that time is not infinitely divisible. So, heuristically, we may wonder why that is if your conclusion actually follows as ineluctably as you claim.

(6) I wonder why you don't note my point about continuousness and density, which I mentioned to help sharpen your argument.

Michael

I didn't say they end. — TonesInDeepFreeze

A supertask is an infinite sequence of operations that ends in finite time.

Again, the contradiction comes from the conjunction of the premises. — TonesInDeepFreeze

One of the contradictions does; the state of the lamp at 12:00. This isn't the only contradiction. The other contradiction is the inherent contradiction of an endless sequence of operations coming to an end. The former is simply a tool to better demonstrate the latter.

Finding some way to resolve the former does not also resolve the latter.

Just in case you missed my edit to my previous post:

Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00.

(3) We don't have a satisfactory definition of 'metaphysical possibility' here. — TonesInDeepFreeze

See here.

All I mean by it is that supertasks are more than just physically impossible. No alternate physics can allow for them.

Michael

@TonesInDeepFreeze

As a different example, consider the grandfather paradox. I don't just take this as a proof that one cannot travel back in time and kill one's grandfather before one's father is born; I take this as a proof that one cannot travel back in time.

The premise of having one kill one's grandfather before one's father is born is just a tool to prove the impossibility, not the reason for the impossibility.

TonesInDeepFreeze

↪Michael

You answered pretty fast. That's your prerogative. But it make me wonder whether you're giving much thought to my remarks, as still it would be your prerogative not to. So I'll take the same prerogative.

The definition of a super-task is as you say. But your listed premises don't say anything about completion or ending.

One of the contradictions does; the state of the lamp at 12:00. — Michael

The contradiction is: The lamp is either On or Off T 12:00 and the lamp is neither On nor Off at 12:00.

But that contradiction comes from a set of premises, each of which is not logically true, and dropping any one of the premises blocks deriving the contradiction. It would help if you would at least tell me that you understand that.

The other contradiction is the inherent contradiction of an endless sequence of operations coming to an end. — Michael

(1) The premises don't say it comes to an end. It would help if you would at least tell me that you understand that.

(2) It is begging the question merely to declare it is a contradiction that denumerably many tasks can be executed in finite time. Indeed, the argument itself doesn't declare that it is a contradiction. Rather, the argument derives a contradiction from that premise along with other premises.

Again, the example I gave:

AxEy yex

ExAy(yex <-> ~yey)

entails a contradiction, but it doesn't entail that either of the above is itself a contradiction.

Even most minimally:

P
~P

entails a contradiction, but that doesn't entail that either P or ~P is a contradiction.

It would help if you would at least tell me that you understand this.

Just incase you missed my edit to my previous post:

Let's even assume for the sake of argument that this wizard will only appear with a probability of 0.5, and that this is determined only at exactly 12:00, i.e after the performance of the supertask. It must already be possible for the supertask to be performed for him to even appear, and so his appearance cannot retroactively make the supertask possible, even if half the time it resolves the secondary contradiction regarding the state of the lamp at 12:00. — Michael

There's no just in case that you missed my own post. You missed that I said I don't argue in any such way that is knocked down as a straw man as you have.

/

The link doesn't go to a defininition. It merely says that metaphysical possibility may be logically possibility and that there's another notion that the article describes ostensively. So is it just the same as logical possibility, and if not what is a proper definition that is not merely ostensive?

Finding some way to resolve the former does not retroactively resolve the latter.

Finding some way to resolve the former does not retroactively resolve the latter. — Michael

So, yes, clearly taking your prerogative to answer so quickly did result in your not even taking note of what I said, let alone taking a moment to understand it.

I am not "finding some way to resolve the former [to] retroactively resolve the latter." You can read my post again to see my explanation.

Michael

The definition of a super-task is as you say. But your listed premises don't say anything about completion or ending. — TonesInDeepFreeze

The infinite button pushes ends after two hours. That's the premise of Thomson's lamp (albeit minutes in his specific case). In his own words, "after I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?".

If, as per the premise, I only push the button at 11:00, 11:30, 11:45, and so on ad infinitum, then I am no longer pushing the button at any time after 12:00. My infinite button pushes has allegedly ended.

The very thing we're discussing is the possibility of supertasks, i.e. can an infinite sequence of operations end in finite time?

The contradiction is: The lamp is either On or Off T 12:00 and the lamp is neither On nor Off at 12:00.

But that contradiction comes from a set of premises, each of which is not logically true, and dropping any one of the premises blocks deriving the contradiction. It would help if you would at least tell me that you understand that. — TonesInDeepFreeze

That's one of the contradictions. If one drops or adds or changes any premises, e.g. by stipulating that the lamp spontaneously and without cause turns into a pumpkin at 12:00, then you have resolved the contradiction regarding the state of the lamp at 12:00, but doing so does not then allow for the possibility of supertasks; it does not resolve the contradiction in claiming that an infinite sequence of button pushes has come to an end.

It is begging the question merely to declare it is a contradiction that denumerably many tasks can be executed in finite time. — TonesInDeepFreeze

It's simply true by definition. An endless sequence of operations cannot end. An infinite sequence of operations is an endless sequence of operations. An infinite sequence of operations cannot end.

It merely says that metaphysical possibility may be logically possibility and that there's another notion that the article describes ostensively. So is it just the same as logical possibility, and if not what is a proper definition that is not merely ostensive? — TonesInDeepFreeze

I'm not the authority on the matter. I am simply arguing that supertasks are more than just nomologically impossible. I use the phrase "metaphysical impossibility" rather than "logical impossibility" simply because it's the weaker claim. Call it hedging my bets if you will.

TonesInDeepFreeze

P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. The lamp is either on or off at t1

But actually, if we look at the mechanics of the inferences toward the contradiction, we see that we need that the lamp is never both on and off. So I would write.

P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t0
P5. The button is pushed at successively halved intervals of time between t0 and t1
P6. At all times the lamp is either on or off and not both.

Then we derive a contradiction:

C1. The lamp is either on or off, and the lamp is neither on nor off.

Now, what premise do we delete to avoid the contradiction? Since none of them are logically true, and each is needed in the derivation of the contradiction, we may delete any one of them.

Michael

↪TonesInDeepFreeze

P5 is an inherent contradiction, just as travelling back in time is an inherent contradiction.

The lamp being neither on nor off at t₁ and killing one's own grandfather before one's father is born are secondary contradictions to prove the inherent contradictions.

The possibility of P5 does not depend on whether or not P1-P4 are true, e.g. having the button be broken does not make it possible to push the button an infinite number of times within two hours.

TonesInDeepFreeze

↪Michael

You're stuck thinking I'm making a certain kind of argument, but I am not. You're not thinking about what I've specifically written, as probably you take me to be making a version of other arguments around. If you're not going to take my arguments as given, then there's no rational inquiry to be had.

But I'll address these two:

"after I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?". — Michael

But completion is not in your premises.

I am no longer pushing the button at any time after 12:00. My infinite button pushes has allegedly ended. — Michael

That is begging the question. It is begging the question to rule that there can't be denumerably many tasks executed in finite time. You haven't proved: If there are denumerably many tasks executed in finite time then there is an end to their executions. So you have to either prove it or add it as a premise.

I use the phrase "metaphysical impossibility" rather than "logical impossibility" simply because it's the weaker claim. — Michael

It's not much of a claim if it is not defined.

Michael

But completion is not in your premises. — TonesInDeepFreeze

This is my argument.

Notice the antecedent of C6: "If the button is only ever pushed at 11:00, 11:30, 11:45, and so on ad infinitum...".

If I am only ever pushing the button at these times then I am not pushing the button at 12:00 or at any time after 12:00. Therefore, my (infinite/endless) button-pushing has ended by 12:00.

That's what supertasks are. They are an inherent contradiction, irrespective of what pushing the button actually does. It is as metaphysically impossible to have performed a supertask on a broken button as it is metaphysically impossible to have performed a supertask on a working button. Having the button turn the lamp on and off, like killing my grandfather before my father is born, is just a means to better demonstrate this impossibility and not the reason for it.

TonesInDeepFreeze

P5 is an inherent contradiction — Michael

You haven't paid attention to my answer to that. Now, you're arguing by mere assertion and repeated mere assertion.

Moreover, if P5 is deemed in and of itself contradictory, then we don't need an argument with other premises to derive a contradiction. If P5 is, from the outset declared a contradiction, then we don't have to bother with the bloody lamp at all.

You're not thinking about my points as it seems you just lump them in with other people's arguments. You are skipping key points. You argue by question begging and mere assertion. And now you link me to a post from eleven days ago that I had already addressed in detail.

As to C6, you put the halving ad infinitum as an antecedent in a conclusion, which is fine. But it's equivalent to just making it a premise. And I mentioned that a while ago, and mentioned why it is more stark to make it a premise, but you ignored that too. Your terse argument that I've recently mentioned and your argument eleven days ago are essentially the same: It doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same.

Anyway, we're going in circles as you skip my arguments while reasserting your own. When I saw that initially you were taking care to make numbered arguments, I got interested and thought you might be open to more scrutiny. But I can see you're not, as you only keep repeating assertions and not actually thinking about the replies to them. It seems your primary interest is to persist that you are rigtht and not to truly think through objections. So, bye for now.

Michael

It doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same. — TonesInDeepFreeze

Yes, it makes no difference if it's an antecedent in a conclusion or as a premise. Either way, the supertask is the completion/end of an infinite/endless sequence within finite time (e.g. I have stopped pushing the button by 12:00) and is an inherent contradiction, irrespective of what the task is. Having the task be to push a button that turns a lamp on and off is just a means to demonstrate the impossibility of a supertask and not the reason for its impossibility, and neither having the button be broken nor having the lamp spontantously turn into a pumpkin allows for the supertask to be possible.

TonesInDeepFreeze

it makes no difference if it's an antecedent in a conclusion or as a premise. — Michael

No, it doesn't. It's called 'the deduction theorem'. For example:

P, Q, R |- S

is equivalent with

P, Q |- R -> S

Michael

↪TonesInDeepFreeze

Huh? I'm reiterating/agreeing with your claim that "it doesn't matter whether you put the halving ad infinitum as an antecedent in a conclusion or as a premise - it's logically the same"?

TonesInDeepFreeze

↪Michael

I was rushing. My mistake.

Infinite Staircase Paradox

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