The scenario describes a fictional, physical process. The lesson is that the defined supertask (the fictional, physical process) is logically impossible, but this isn't apparrent when considering only the mathematical mapping.There is no physical process. — fishfry
The lesson is that the defined supertask (the fictional, physical process) is logically impossible, — Relativist
Are you suggesting that supertasks cannot be completed?the process of counting steps is not completable — Relativist
Agreed, but most importantly: (4) apply those intuitions to (the original) experiments.The process is:
1) Have fuzzy intuitions;
2) Study some math;
3) Develop far better intuitions. — fishfry
I like where you're going with this. To navigate between the staircase and omega (and back), one must leap over infinite steps. This concept becomes more palatable if we consider that the steps become progressively smaller towards the bottom. However, let me try to rephrase your perspective: Icarus requires a finite number of strides to reach the bottom and a finite number to return to the top, thus avoiding any supertask. When Icarus adds 1/2, then 1/4, then 1/8, he gets bored and chooses to make a final leap. On his final leap, instead of adding an infinite series of smaller terms, he simply adds another 1/8 and reaches omega, where his calculator displays exactly 1. In this case, the infinity in the paradox describes the steps which he potentially could have traversed (and seen), not what he actually did (and saw). Since he never actually observed all steps, he is in no position to confirm that there were actually infinite steps...but there could have been...potentially. Paradox solved?It's only a finite number of steps back, even from infinity. — fishfry
I'm asserting that an infinite process is necessarily never completed - by definition.the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone
The lesson is that the defined supertask (the fictional, physical process) is logically impossible,
— Relativist
The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible? — fishfry
Good. Then we're on the same page!I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist
The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed. — Relativist
No it doesn't. — Michael
I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state. — noAxioms
The cuts themselves are the points (think Dedekind cuts).Not sure of the difference. If I cut a string, I don't get points, I get shorter strings. — noAxioms
One can observe a superposition directly? Please share a link.You can under some interpretations. — noAxioms
What I aim to demonstrate is that there is a scenario where local motion is possible and continuous without involving supertasks. This occurs in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"
If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation. — noAxioms
If the universe is discrete, then Zeno's paradoxes cannot occur as he described them. What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.Necessary only if the first premise is to be accepted. — noAxioms
I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term. — Relativist
However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step? — Relativist
By definition, a limit is not reached, it is approached. — Relativist
Take the scenario here:
After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.
We can sum the geometric series to determine that the limit is 60 seconds. The claim some make is that this then proves that this infinite sequence of events can be completed in 60 seconds.
However, then we ask: what colour is the square when this infinite sequence of events is completed? — Michael
As per the setup, the square can only be red, white, or blue, and so the answer must be red, white, or blue. — Michael
However, as per the setup it will never stay on any particular colour; it will always turn red some time after white, turn blue some time after red, and turn white some time after blue, and so the answer cannot be red, white, or blue. This is a contradiction. — Michael
The conclusion, then, is that an infinite sequence of events cannot be completed, — Michael
and the fact that we can sum the geometric series is a red herring. — Michael
To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible. — Michael
we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: — Michael
"Not defined" does not mean that you are free to choose the result. — SolarWind
Which solution has n = n+1? — SolarWind
Certainly not 42. — SolarWind
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.As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal ω+1
+
1
. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers. — fishfry
The limit of the series is "reached" only in the sense that we can reach a mathematical answer. The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. Deriving the limit just means we've identified where the sequential process leads. In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? 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.By definition, a limit is not reached, it is approached.
— Relativist
That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.
In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.
It's not reached by a single step. Rather, it's reached by the limiting process itself. — fishfry
This is equivalent to asserting that 'infinity' is the largest integer. Does nobody else see that making such an assertion is going to lead to contradiction? It doesn't mean that there cannot be an unbounded thing.if a physical process ends, there has to be a final step. — Relativist
This depends on one's definition of completing a process. The SEP article on supertasks has this to say about it:I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist
And a different page than me.Good. Then we're on the same page! — keystone
Zeno's argument is that X is possible, and another that X is not possible.(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible. — keystone
OK, so now we have point cuts separating shorter strings, each with nonzero extension.The cuts themselves are the points (think Dedekind cuts).
Any interpretation that denies wave function collapse has everything in superposition at all times. One simply finds ones self in superposition with the observed state. So I observe both the dead and the live cat, presuming that "I" dong the observing is the same person as the person a moment ago with the closed box.One can observe a superposition directly? Please share a link.
Moton is change of postion over time. The block universe very much has that for any moving object. The worldline of that object is a different spatial locations at different times. All of Zeno's arguments still apply, and are still contradictory.in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).
The first premise would be demonstrably false. The second premise (that supertasks are impossible) would be moot, but arguably true then.If the universe is discrete, then Zeno's paradoxes cannot occur as he described them
You seem to do this by reducing the universe to a point (your 'photo'), which is not something that is continuous. A point in time at least, which is the same as denial of time at all.What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.
Sorry, what? You don't believe that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? You don't believe in calculus? You are arguing a finitist or ultrafinitist position? What do you mean?
Of course if you mean real world events, I quite agree. But your three-state lamp is not a real world event, it violates several laws of classical and quantum physics, just as Thompson's two-state lamp does. — fishfry
So if you wish to define a final state, you can make it anything you like. I choose pumpkin. — fishfry
The contradiction is very obvious. I'm surprised you persist in denial. The supertask will necessarily carry on forever, as the sum of the time increments approaches 60 seconds, without 60 seconds ever passing. Clearly this contradicts "60 seconds will pass". — Metaphysician Undercover
An ordinary stopwatch is started.
After 30 seconds a white box turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.
When the stopwatch reaches 60 seconds, what colour is the box? — Michael
Wrong. The statement (the completion of a consecutive series of physical steps entails a final step) is necessarily true. When we consider this statement in conjunction with a statement about the series being "complete" (in terms of convergence) we introduce a contradiction. This is the point! These statements cannot both be true, but both are entailed by the scenario.if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer. — noAxioms
The SEP article says:But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires. — noAxioms
I agree with this, but this simply ignores the implication of the physical process of step-counting. For the scenario to be coherent, BOTH view of completeness have to be true. But they aren't - so the scenario is actually incoherent.I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed. — noAxioms
Yes- and that's because the role of infinity in the task. The task entails a sequence of events, so the infinity can only mean an infinite chain of events - one after another without end.the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone
No they mustn’t. — Michael
The physical process of descending stairs is not a supertask. I couldn't think of a way to make it a supertask, even by making each step smaller. A supertask has no final (or first, respectively) step, so by counterexample, the assertion "there has to be a final step." is incorrect.if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer.
— noAxioms
Wrong. The statement applies universally to the physical process of descending stairs. — Relativist
I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent.A contradiction is introduced when this statement ("a completed step counting entails a final step)
I notice the SEP article correctly doesn't claim that the last step is taken.The SEP article says:
"... From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity"
Agree. But the only attempted step counting processes are examples like the lamp or Michael's digit counter, and those examples are not physical. The Achilles example can be physical, but it isn't counting anything.As I noted above, a physical, step-counting process that completes must entail a final step.
There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step?Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction.
Kind of like I ignore the green ball in the bag, yes.I agree with this, but this simply ignores the implication of the physical process of step-counting.
I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number.For the scenario to be coherent, BOTH view of completeness have to be true.
:up:No they mustn’t. — Michael
If you truly believe that an increment of time exists without being measured, tell me how I can find a naturally existing, already individuated increment of time. — Metaphysician Undercover
Once again, M-U cannot comprehend a view outside his own idealistic assumptions. — noAxioms
but it is a fact that 60 seconds of time can pass without anyone looking at a clock or a stopwatch. — Michael
https://en.wikipedia.org/wiki/Unit_of_time#:~:text=The%20base%20unit%20of%20time,oscillations%20of%20the%20caesium%20atom.the second, defined as about 9 billion oscillations of the caesium atom. — Wikipedia
Billions of years passed before humanity evolved, and this isn't some retroactive fact that only obtained when humanity started studying the past. — Michael
I don't know whether you're arguing for some kind of antirealism or if you're failing to understand a use-mention distinction. — Michael
Regardless, the arguments I am making here are directed towards the realist who believes that supertasks are possible. — Michael
My point is that the stairs are countably infinite. Consequently, they COULD be counted, if we were traversing them.I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent. — noAxioms
Yes, the sequence of defined temporal points (1/2, 1/4, 1/8...) is a series, but the mathematics that identifies the limit does not take into account the kinematics of the task. Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world - regardless of whether or not you wished to consider it.The series (say the time needed to complete all tasks) converges. The count does not.
It fits this definition:The physical process of descending stairs is not a supertask. — noAxioms
The goal of removing all the marbles will therefore never be met if there are at least 2 green marbles, and it will rarely met even if there is only 1. How does this relate to a supertask that allegedly completes?Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete. — noAxioms
The article discusses the issue:I notice the SEP article correctly doesn't claim that the last step is taken. — noAxioms
Yes, it's a cop-out because it ignores the kinematic process. Stating this in terms of the PSA gives you something specific to address, if you want to not cop out.Relativist: "Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction."
There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step? — noAxioms
I agree we can't treat infinity as a number, and haven't suggested you should. 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.Relativist: "For the scenario to be coherent, BOTH view of completeness have to be true."
I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number. — noAxioms
If we take the term “1 year” as an example, the Earth orbiting the Sun does not depend on us measuring it. It just orbits it, independently of us. — Michael
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
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