I'm not sure whether that doesn't amount to a contradiction or whether it is an entirely distinct issue. But it seems like that if that's the case, one doesn't get as far as a contradiction.(Some)... compound expressions suffer the fate I attribute to 'completed infinite sequence of tasks' and 'thinking robot'. What seems most notable about such compounds is the fact that one component (e.g., 'infinite sequence') draws the conditions connected with its applicability from an area so disparate from that associated with the other components that the criteria normally employed fail to apply. We have what appears to be a conceptual mismatch. — Benacerref on Supertasks
I hope you meant that actions taken outside the system are neither consistent nor inconsistent with the rules. Could we not express this by saying that the rules don't apply, or that it is not clear how to apply the rules, in the new context? — Ludwig V
I don't see it as a confusion of Michael. He is only rendering Thomson's setup. And I don't see Michael getting tripped up by the metaphorical use of a lamp and button. And I don't see Thomson as getting tripped up either. — TonesInDeepFreeze
That was a complete description. — Michael
There are no hidden assumptions. — Michael
P1 is implicit in Thomson's argument. Using the principle of charity you should infer it. As neither you nor Benacerraf have done so I have had to make it explicit. — Michael
The lamp is off at 10:00. The button is pushed 10100100 times between 10:00 and 10:01. Is the lamp on or off at 10:02? — Michael
Any reasonable person should infer that nothing else happens between 10:01 and 10:02. — Michael
Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — Michael
There is no Supreme Button Pusher arbitrarily willing the lamp to be on or turning it into a pumpkin. There is only us pushing the button once, twice, or an infinite number of times, where pushing it when the lamp is off turns the lamp on and pushing it when the lamp is on turns the lamp off. — Michael
This doesn't make sense. Each flip of the coin is an individual act, and it has a single outcome. Once the outcome is achieved, that outcome stands until there is another flip. The outcome "can't be both at different times", because a different outcome requires a different flip. However, there can be different outcomes from different flips. — Metaphysician Undercover
I think the problem is precisely that there is nothing to constrain the lamp and we want to find something. In theory, we could stipulate either - or Cinderella's coach. But we mostly think in the context of "If it were real, then..." Fiction doesn't work unless you are willing to do that. It's about whether you choose to play the game and how to apply the rules of the game. — Ludwig V
This seems to be more in tune with common sense, for what it's worth. The question is, why? I think it is because of the dressing up of the abstract structure. We assume the lamp has existed before the sequence and will continue to exist after it. — Ludwig V
So the fact that the sequence does not define it does not close the question and we want to move from the possible to the actual. But it is not clear how to do that - and we don't want to simply stipulate it. Perhaps that's because defining the limit of the convergent sequence as 1 - or 0, which have a role in defining the sequence in the first place, — Ludwig V
invites us to think in the context of the natural numbers (or actual lamps), whereas defining ω as the limit of the natural numbers does not. — Ludwig V
C1 is a premise.
— TonesInDeepFreeze
It’s not, it’s a valid inference from the premises. — Michael
w can be defined such that it is the limit of the sequence of the natural numbers. — fishfry
But, just to be clear, we still need to prove that there exists a set such that every natural number is a member of that set, since that set is the domain of the aforementioned sequence. — TonesInDeepFreeze
wut? axiom of infinity. what's wrong with you tonight? — fishfry
Quite so. Except I thought that it had actually been done.ω can be defined such that it is the limit of the sequence of the natural numbers. — fishfry
Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ωth entry?Neither 0 nor 1 is the limit of the sequence of alternating 0's and 1's. — fishfry
Yes. My only point was that it is not a natural number, whereas 1 and 0 are. Hence, although both are limits of their respective sequences, as 1 or 0 also are, 1 and 0 are used in other ways in other contexts. This makes no difference to their role in this context and does not affect their role in other contexts, but does affect what we might call their meaning. ω is not used in any other context - so far as I know.w is a limit ordinal, and it is the ordinal limit of the sequence of all the natural numbers. — TonesInDeepFreeze
I agree that we can agree not to ask questions about the lamp outside the context of Thompson's story. But I'm not sure that an assumption really requires a justification. But, for the sake of argument, if I'm telling you a story about a real ball and the shenanigans the prince got up to, you would make that assumption. So if I'm pretending to myself that Cinderella's ball actually happened, I will make the same assumption. This is one reason why I prefer to stick to the abstract structure and shed the dressing up.What justifies such an assumption with regard to an entirely fictional lamp, coach, or pumpkin? — fishfry
Can I ask what your solution is? Just out of interest.My charity ran out long ago regarding this subject. The lamp is a solved problem. — fishfry
But actions which are outside of the rules are not contrary to the rules, so they are consistent with the rules. However, on thinking about it, I think my answer it that it depends on the rule. Sometimes the rule means that actions that are not permitted are forbidden and sometimes the rule means actions that are not forbidden are permitted. And sometimes neither.No, I mean they are inconsistent. To be consistent with the rules is to act according to the rules. Actions which are outside of the rules are not according to the rules, therefore they are inconsistent with the rules. — Metaphysician Undercover
Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. (If it could, it would not be an infinite sequence). The limit has to be something that is not an element of the sequence. It has to be, to put it this way, in a category different from the elements of the sequence. (I'm trying to think of a self-limiting activity, but my imagination fails me. Perhaps later.)Any reasonable person should infer that nothing else happens between 10:01 and 10:02. Even though this is a physically impossible imaginary lamp, and even though I haven't told you what happens at 10:02, it is poor reasoning to respond to the question by claiming that the lamp can turn into a plate of spaghetti. The correct answer is that because 10100100 is an even number, the lamp will be off at 10:02. — Michael
Quite so. But how does it help when we are thinking about an infinite sequence? As I understand it, the point is that the sequence cannot define it's own limit. — Ludwig V
Perfectly clear that you have stated nothing about 10:02. For all we know it turns into a pumpkin. — fishfry
Can't you see why I'm demanding that you write out, in one place, your entire description of the problem. That way you would be able to catch yourself making stuff up as you go. — fishfry
The premises don't not specify that the button is ever pushed. — TonesInDeepFreeze
The premises do not specify that there are only two states, unless, in this very hypothetical context we are clear that 'Off' is defined as 'not On', though it does seem reasonable that that is implicit. — TonesInDeepFreeze
Not quite. If the last stage of the supertask was on, it is not on spontaneously and without cause.1. The lamp can never spontaneously and without cause be on
2. If the supertask is performed, and if the lamp is on after the performance of the supertask, then the lamp being on after the performance of the supertask is spontaneous and without cause.
Therefore we must accept that the supertask cannot be performed. — Michael
If the last stage of the supertask was on... — Ludwig V
The problem is that whether the supertasks can be performed is not really the issue. — Ludwig V
I hope it makes better sense now.Not quite. If the last stage of the supertask was odd, it is not on spontaneously and without cause. — Ludwig V
So I don't see the point of your argument here — Ludwig V
Thank you. That is much clearer.The lamp cannot be on after the performance of the supertask and cannot be off after the performance of the supertask – precisely because there is no final button push and because the lamp cannot spontaneously and without cause be either on or off. — Michael
Nor is it. He talks about two instances of the game, and either outcome would be consistent - on its own. But they contradict each other and that's the problem. I don't rate that "refutation" any more than you do.Benacerraf claimed that the supertask being performed and then the lamp being on is not a contradiction. — Michael
An interesting indeterminate comment. But I think that the impossibility of the final cycle before the limit does put paid to it. It's all about what "complete" means in the context of infinity. Benacerraf, it I've read him right, allows that Achilles can be said to complete infinitely many tasks in a finite time, but argues (rightly) that Thomson's lamp is a different task and suggests to me that he is inclined not to allow that conclusion in that case.The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency. — SEP on Supertasks
Quite so. Except I thought that it had actually been done. — Ludwig V
Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ωth entry? — Ludwig V
As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, — Michael
Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer. — fishfry
I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though.Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea. — fishfry
Oh, yes, I get it. I think.There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., ω such that ω is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ... — fishfry
I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42.No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. — fishfry
So we say that all limited infinite sequences converge on their limits. Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit.Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. In this case 47 is still the value of the "extended sequence" function at ω. I call it the terminal state. — fishfry
I have completist tendencies. I try to resist them, but often fail.I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke." — fishfry
The issue is about how to perform a thought experiment - how much of reality you can import into the story. — Ludwig V
I'm glad you agree. And you are right to go on to consider choices we could make.That's exactly right. — Metaphysician Undercover
That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. Actually, if the task is suspended before it is concluded for any reason, no conclusion can be drawn either way. So I would think that we have to say that the passing of time is a presupposition of the problem. So I wouldn't use this case as an argument against the infinite divisibility of time (or space, in the case of Achilles). (Actually, following our earlier argument, I'm inclined to see that as a mathematical or conceptual proposition, rather than a fact about the real ("physical") world.)However, if we attempt to prove that the amount of time must pass, we run into problems, like those exposed by Hume, namely a lack of necessity in the continuity of time. — Metaphysician Undercover
That's interesting. Do you mean a proof that the amount of time must pass in reality, or a proof that the amount of time must pass in the story? If the former, then we do have a problem. But if the latter, I would argue that the amount of time must pass in order for the conclusion to be drawn. — Ludwig V
OK. I'm with you that far. Comment:-Now, we add a bit of "reality". Achilles will pass the tortoise, the allotted amount of time will pass. So we see that what we take for "reality", is inconsistent with, or contradicts what the thought experiment asks us to consider. — Metaphysician Undercover
Yes. What you are doing is applying the actual context (reality) of the story, but instead of drawing on "common sense", drawing on philosophy. That seems to be not unfair, given that Zeno drew a rather radical philosophical conclusion in direct contradiction with "common sense". (He doesn't even have the grace to compromise by dismissing change as an illusion.) Thomson is different because all he wants to conclude is that supertasks are impossible. That's one thing I've never grasped - If supertasks were possible, what philosophical conclusions would follow?However, "because there always has been" does not provide proof that there will continue to be into the future. — Metaphysician Undercover
Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue.Then it is actually going so slow in comparison to the other time frame, that a very large number of switching can occur in a very short time, and so on as it approaches an infinite amount. — Metaphysician Undercover
If we have made a continuous uninterrupted journey from A to B we can be said to have covered all the stretches described in the first premise; that is, our motion can be analyzed as covering in turn AA', A'A", etc. (his italics) — Benacerraf on Supertasks p. 766
That's where the thought experiment isn't a piece of fiction like a fantasy. Aesop's Fables are also not just a piece of fiction; we are meant to draw conclusions about how to live our lives from them. So "It's just a silly story" is not playing the game. This story wants us to draw a conclusion about how reality is. — Ludwig V
So, instead of rejecting the idea that time is infinitely divisible, you are turning to Hume and arguing that anything can happen. Maybe you are on stronger ground here. I think some people would feel that you are importing more reality than the rules allow. But I can't be dogmatic about that because I don't really know what the rules are - and I'm certainly not going to argue with Hume - perhaps I'm just shirking a long complicated argument, because I don't think he's right, even though he has a point. — Ludwig V
Yes. I don't know how this would play with actual Relativity Theory. But in any case, I don't think that resolves the problem. Why? Because it doesn't actually get Achilles to the finishing line. In the case of Thomson's lamp, it doesn't get to the crunch point when the time runs out. In other words, it postpones, but doesn't resolve, the issue. — Ludwig V
That's why I insist that the convergent sequence is not about space or time, but about the analysis of space and time. — Ludwig V
I don't understand your question. — Michael
Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off. — Michael
We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible. — Michael
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