Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence?My point is that the stairs are countably infinite. Consequently, they COULD be counted, if we were traversing them. — Relativist
That's true. What puzzles me is why they are not dismissed out of hand. Someone earlier described them as fairy stories, and the writers seem to be able to wave a hand and create impossibilities, which would be magic, so that description makes sense. But it seems to me more like an illusion and the problem is then to understand how that illusion works.I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction. — Relativist
Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world?Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world — Relativist
Yes. How come anyone can't see that? Since the difference is the difference between simple addition and division followed by addition, I think it is then possible to see how people can be misled into thinking they are compatible - even that they must be compatible.The counter, with it's supertask has one way of counting out time, by dividing seconds into shorter and shorter increments, while the stopwatch is designed to measure an endless procession of seconds. The two are incompatible. — Metaphysician Undercover
there are no seconds unless measured out — Metaphysician Undercover
Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps.My point is that the stairs are countably infinite. — Relativist
It does, I'm quite aware. Just not in Zeno's argument.The article discusses the issue
I pretty much quoted exactly Black's remarks just above. Yes, the task is not complete by this finite definition despite every step having been taken, and that final step must be taken for your counter to have a defined value after a minute.Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...
Again, the stairs is utterly abstract. There's no kinematics to it. Not so with the tortoise. I can pass the tortoise, thus completing (by the 'all steps' definition) the supertask.The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete.
How does the abstract mathematics not account for the physical ability of me passing the tortoise?The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.
I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.PSA:
The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.
I'm trying to focus on the completion of all tasks and not on the measurement of a nonexistent value.That's what I see going on with the posters who focus only on the mathematical series.
But I think you have. Your attempted counter (or the color change thing in the recent post) treats it as a number, and suggests taking its modulus relative to base 10 or 3. What is the lowest digit of the number of the final step? If there is no such number, then the output of your scenario is undefined, which is very differnt from the digit counter displaying a value of 'undefined', or an undefined lamp state somehow violating the law of excluded middle by being in some state between on and off.I agree we can't treat infinity as a number, and haven't suggested you should.
Infinity means unbounded, which means there is a physical location and time interval.for any task n That's what makes it meaningful, and it only works if physicality is presumed not discreet.But for the supertask to be meaningful, you have to identify where infinity fits in the kinetic task description. I'm saying it entails a never-ending sequence of tasks. Identifying the limit doesn't make this disappear.
I can pass a tortoise without contradiction. That shows that at least one of three (two explicit, one implicit) premises are false. But it doesn't necessarily have to be the premise you just mentioned there, that supertasks are impossible.I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction.
I'm ignoring it because those contradictions arise from a 4th premise (that there is a final step), one which I don't accept.You seem to be avoiding the contradiction by ignoring the incompleteness of the infinitely many kinematic steps. The presence of the contradiction implies supertasks are logically impossible (not merely physically impossible).
Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics?What puzzles me is why they are not dismissed out of hand. — Ludwig V
Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'.A white box turns red when the Earth completes a half-orbit, turns blue when it completes another quarter-orbit, turns back to white when it completes another eighth-orbit, and so on.
What colour is the box when the Earth completes its orbit around the Sun? — Michael
Michael did very nicely with his first line in his reply.If someone would explain to me, in a way which makes sense, a better perspective, then I'd happily switch. — Metaphysician Undercover
Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'. — noAxioms
Correct.Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence? — Ludwig V
I think it's because they are interesting puzzles, and because they help teach certain concepts.That's true. What puzzles me is why they are not dismissed out of hand. — Ludwig V
Yes- that's a better way to describe it.Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world? — Ludwig V
The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable.More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?
I think they are interesting because they dangle the prospect of completing a task and persuade us to ignore the reality of the impossibility of the task.I think it's because they are interesting puzzles, and because they help teach certain concepts. — Relativist
Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics? — noAxioms
Maybe I've misunderstood what a supertask is. Are there not different kinds of cases?The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable. — Relativist
We can assign those numbers as we take each step. That's counting, and it's perfectly meaningful.Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps. — noAxioms
OK, but speed of light limitations put a physical limit on how fast the stairs can be descended, so that it eventually becomes physically impossible to descend a step in the prescribed period of time. The minimum size limitation also relates to a physical impossibility. But I'm making the stronger claim that it is logically impossible.Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask. — noAxioms
The entire exercise is abstract, but the scenario is written in terms of the kinematic (not abstract) process of descending stairs: each step is a motion, taking place in a finite amount of time.Relativist:"The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete."
Again, the stairs is utterly abstract. There's no kinematics to it. — noAxioms
Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1). The term (sequence) is not referring to the entire collection. The task is to reach the bottom of the stairs (as stated in the description in the first post of this thread). Perhaps you can already see that it's trivial: it's actually impossible to reach the bottom of the stairs, since there is no bottom to a staircase with infinitely many stairs.PSA:The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.
I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all. — noAxioms
In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed. I mean that we know that Achilles will pass the tortoise and even calculate when with simple arithmetic (no infinities required). — Ludwig V
The lack of a defined number for the last task does not prevent completion (by the all-tasks definition), so I regard your statement as a non-sequitur.And so it is meaningless to claim that such a supertask can complete. — Michael
Several here have been defining completion effectively as measuring the value of the final task, and that instance I suppose differs from Zeno's that specifies no such requirement.Maybe I've misunderstood what a supertask is. Are there not different kinds of cases? — Ludwig V
Well, keystone suggested that Zeno denies this, and M-U suggests that time somehow stops due to the offense we've given it. Anyway, I agree with you, but it requires that implied premise that empirical evidence is valid.I mean that we know that Achilles will pass the tortoise
This suggests fallacious reasoning in the second presentation. Most of the fallacies I've seen posted seem to be based on the premise of there being a limiting step. It's why I like Bernadete's Paradox of the Gods (see post ~30) which explicitly leverages the lack of there being a limiting step, and drives that to a seemingly paradoxical result. That's a harder one to wave off.But then the same problem, presented in a different way, seems to suggest that it cannot.
Just not physically. Mathematically it can, but then the story mentions 'the bottom' which implies something final that 'no more stairs' does not. So it lacks rigor.The staircase ... gives us a task (going down the infinite stairs) that cannot be completed
I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory.But I'm making the stronger claim that it is logically impossible. — Relativist
Has always meant 'prostate specific antigen' to me. I get my PSA checked at least once a year.PSA
OK, 'act' is a step (go half the remaining way to the goal). 'task' is a goal (pass the tortoise).Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1)..
Yes, the PSA entails taking a final step. We agree infinity is not a number, so there is no final step.Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. — noAxioms
Show the PSA is false.I cannot buy into that PSA statement.
Why? The claim is indeed justified by the necessity of a final step for completion. Simply denying a final step is necessary doesn't make it so - you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.But I'm making the stronger claim that it is logically impossible.
— Relativist
I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory. — noAxioms
There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility. — Michael
Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time). — Michael
Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity. — Michael
If you want to say that supertasks are possible — Michael
but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it. — Michael
Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval. — Relativist
The limit of the series is "reached" only in the sense that we can reach a mathematical answer. — Relativist
The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. — Relativist
Deriving the limit just means we've identified where the sequential process leads. — Relativist
In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? — Relativist
The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else. — Relativist
The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal.Show the PSA is false. — Relativist
Simply asserting that such a step is necessary doesn't make it so, especially when it being the case directly violates the initial premise. That violation does very much demonstrate not only the lack of necessity of a final step, but the impossibility of it, given the premise.Simply denying a final step is necessary doesn't make it so
I don't know how the task being 'kinetic' changes the argument. You can phrase it as a n inertial object overtaking a slower one in frictionless space.you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.
Yes there are. A second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom". This occurs even if we don't measure it. — Michael
Yet again you can't seem to get beyond our use of labels to understand that our labels refer to things that exist and do things even when we're not around. — Michael
Your reading comprehension skills are also off. I never suggested converting you to some opinion other than the one which you hold. I simply suggests that you seem incapable of understanding alternatives, to the point where you don't understand people who presume one of these alternatives. — noAxioms
Or the PSA is correct, and the goal can't be met.The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal. — noAxioms
I'm not merely asserting it. You have to agree that a final step is necessary for completion when there are finitely many steps. Why would it matter if the number of steps is infinite?Relativist: "Simply denying a final step is necessary doesn't make it so."
Simply asserting that such a step is necessary doesn't make it so — noAxioms
Here's how: the infinity is manifested as a never-ending kinetic process.Relativist: "you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed."
I don't know how the task being 'kinetic' changes the argument. — noAxioms
I'm not sure it is even a puzzle if it is framed in terms of constant speeds by both. Let's say Achilles gives the tortoise a head start of 100 units of length, that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds. (This was suggested to me by a friend.) It seems OK to me, but perhaps I'm wrong to think that it will generalize.It depends on how the race is framed. It CAN be described as a supertask, wherein Achilles runs to a series of destinations, each established by where the tortoise is located when he begins each leg of the race. In that case, Achilles never actually reaches the turtle, he just gets increasingly closer. If you frame it in terms of constant speeds by both, then it's not a supertask - it's a different kind of puzzle. — Relativist
I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask.Or the PSA is correct, and the goal can't be met. — Relativist
Because a contradiction results from making that additional assertion. In the example given, it is a very direct contradiction.Why would it matter if the number of steps is infinite? — Relativist
If the process continues forever, by definition it isn't a supertask. It's a different process than the one being discussed.What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer. — Relativist
I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.Points on a number line exist concurrently (in effect). — Relativist
OK. I buy that. But this works mathematically as well, so 'kinetic' doesn't add anything. I can draw the worldlines of Achilles and the tortoise on some medium and all you get is two lines that cross at some point. The axes on the plot are x and t, so in this mathematical representation, the steps do not occur simultaneously, but are separate durations of time. What did 'kinetic' add to that?Steps in a kinetic process do not: they occur sequentially, separated by durations of time. — Relativist
OK, this has been about the stairway. There is no objective kinematics about that since it involves a space-like worldline, so the steps are not unambiguously ordered in time. The ordering of the steps becomes ambiguous due to relativity of simultaneity, and it becomes meaningless to use the word 'sequential' in this context.the Achilles/tortoise problem ... just clouds the issue with the stairway supertask. — Relativist
No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation.Or the PSA is correct, and the goal can't be met.
— Relativist
I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask. — noAxioms
Fair enough, I misstated it. The process does not continue forever, however there is no end to the process.If the process continues forever, by definition it isn't a supertask. — noAxioms
My point was that the kinematic stair-stepping process has a temporal element that is not reflected in a number line.Points on a number line exist concurrently (in effect).
— Relativist
I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept. — noAxioms
No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation. — Relativist
Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.
This shows that no specific halfway step reaches the goal, which is the same as saying that the goal cannot be reached in a finite number of steps.
It seems that every post seems to attempt finite logic on an unbounded situation. If you accept that motion is possible, there is a flaw in at least one of the premises.
— Relativist
Yea, I do, don't I? I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. Attack them if you will. The do require some axioms that are not obvious, so there's a good place to start. Nevertheless, I can do more than just handwave, by several unrelated methods.You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal.
There is a temporal end to it, a final moment if not a final step.The process does not continue forever, however there is no end to the process.
There is a bijection yes. It does not imply that both or neither completes.But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.
The 'process' can go beyond the end of the line despite it ending before the goal. This is sort of a different issue since you're putting an uncountable set of points between 0 and 1. Why not just 1/2, 1/4, ...The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process.
Disagree. The kinematic process isn't restricted to only points on the number line.The goal is therefore unreachable by the kinematic process.
This is an example of a supertask:
I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.
According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.
Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.
It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote. — Michael
You're continuing to argue against a position I don't hold. Why are you doing this? — fishfry
I would, however, disagree with you that being inconsistent with known physics is the same as logical impossibility. — fishfry
Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks. — Michael
I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility. — Michael
No physical law can allow for an infinite sequence of events to be completed. — Michael
The very concept of an infinite sequence of events being completed leads to a contradiction. — Michael
To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense. — Michael
If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity. — Michael
Do you deny infinite mathematical sets? — fishfry
Mathematically that's not true. The set {1, 2, 3, 4, ...} contains all the natural numbers, but there's no last number. — fishfry
And besides, eternal inflation posits a temporally endless universe. It's speculative, but it's part of cosmology. Serious scientists work on the idea. So at least some scientists are willing to entertain the possibility of a physically instantiated infinity. — fishfry
You keep repeating that, but you have no evidence or argument. — fishfry
1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)
You don't find this contradictory? — noAxioms
Sure. You have to agree the PSA is true for finite tasks. Is there something different about infinite tasks? It doesn't seem so: consider the process: stepping increasingly closer to temporal point in time 1, but the process never actually reaches it. So the goal is unreachable by the process.Demonstration that immediate contradictions arise from denying either of the premises or presuming your conclusion 3 is also more than just handwaving. — noAxioms
No need. I understand that the math shows that the series reaches a point of convergence at time 1. However: the kinematic process never actually reaches time 1. That's why the series doesn't adequately account for the kinematic process -and why I've stressed we need to examine the process, not just do the math on the mathematical series.I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. — noAxioms
On the contrary, there's a logical impediment to reaching the goal through the process: the process does not reach time 1.no impediment to the reaching of the goal has been identified, — noAxioms
I'm actually basing my claims on real analysis, which analyzes the characteristics of real numbers - including the associated infinities.You do seem to heavily rely on definitions that come only from finite logic — noAxioms
That makes no sense. The process does not have a final moment. because there are infinitely many moments prior to time 1. There is no end to the series of kinematic steps, in spite of the fact that the mathematical series converges.There is a temporal end to it, a final moment if not a final step. — noAxioms
Why not?Relativist: "But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes."
There is a bijection yes. It does not imply that both or neither completes. — noAxioms
No it can't - that is logically impossible. The process entails taking steps with increasing shorter durations: 1/2 second, 1/4, 1/8,.... The process can only approach 1, it can never reach it.Relativist: "The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process."
The 'process' can go beyond the end of the line despite it ending before the goal. — noAxioms
No! Each new step is half the duration of the last step, and this halving process has no end.. The kinematic process isn't restricted to only points on the number line.
No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them. — Michael
That's not relevant to the claim I'm making. — Michael
I'm saying that if I have finished counting the members of some set then some member must be the final member I counted. — Michael
However: the kinematic process never actually reaches time 1 — Relativist
↪jgill That's true, but that just makes it physically impossible. I think it's stronger: logically impossible. — Relativist
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