I have more or less dropped out due to the repetitive assertions not making progress, but thank you for this post. — noAxioms

Thanks.

the set {1/2, 3/4, 7/8, ..., 1}
— fishfry
Interesting. Is it a countable set? I suppose it is, but only if you count the 1 first. The set without the 1 can be counted in order. The set with the 1 is still ordered, but cannot be counted in order unless you assign ω as its count, but that isn't a number, one to which one can apply operations that one might do to a number, such as factor it. That 'final step' does have a defined start and finish after all, both of which can be computed from knowing where it appears on the list. — noAxioms

Of course it's a countable set. It's a subset of the rationals, after all. You are right that it's not order-isomorphic to 1, 2, 3, ...

This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing. — noAxioms

Right. Exactly right. Point being that contemplating a set that includes an infinite sequence along with an extra point is nothing strange at all. And it serves as a nice conceptual model for supertask puzzles.

If Zeno includes 'ω' as a zero-duration final step, then there is a final step, but it doesn't resolve the lamp thing because ω being odd or even is not a defined thing. — noAxioms

There is no final step. There is a point at infinity. Not quite the same. Unless you allow the limiting process itself as a step. It's just semantics.

and we inquire about the final state at ω
Which works until you ask if ω is even or odd. — noAxioms

It's neither, and who's asking such a thing? Even and odd apply to the integers.

Anyway if this is repetitive feel free to not reply. I just go through my mentions everyday trying to reply best I can. And I do have a thesis, which is that the ordinal $\omega + 1$ is the proper setting for the mathematical analysis of supertask puzzles. So I'll repeat that every chance I get.

And as I keep explaining, the issue with supertasks has nothing to do with mathematics. Using mathematics to try to prove that supertasks are possible is a fallacy. — Michael

But I'm not doing that. I haven't been doing that. Are you deliberately misunderstanding me or am I being unclear?

"Using mathematics to try to prove that supertasks are possible is a fallacy"

Who did that? Are they in the room with us right now?

Yes. I got enough from it to realize a) that ω is one of a class of numbers and b) that it comes after the natural numbers (so doesn't pretend to be generated by "+1") — Ludwig V

Yes exactly. $\omega$ comes into existence via a limiting process. The idea is that the natural numbers are generated by successors, and the higher ordinals are generated by successors and limits. So we're adding a new rule of number formation, if you like. We go 1, 2, 3, ... by successors, and then to $\omega$ by taking a limit, then $\omega `=+ 1$, $\omega + 2$, etc., then eventually we get to $\omega + \omega$ by taking a limit, then we keep on going. I don't want to go too far afield, but the idea is that we can take successors and limits to get to all the higher ordinals.

This business about actions is what confuses people.
— fishfry
Certainly. That's what needs to be clarified, at least in my book. There's a temptation to think that actions must, so to speak, occur in the real world, or at least in time. But that's not true of mathematical and logical operations. Even more complicated, I realized that we continually use spatial and temporal terms as metaphors or at least in extended senses:- — Ludwig V

Right. A lamp that cycles in arbitrarily small amounts of time is not physical. A staircase that we occupy for arbitrarily small intervals of time is not physical. So trying to use physical reasoning is counterproductive and confusing. That's my objection to all these kinds of puzzles. People say there's a conflict between the math and the physics ... but as i see it, there's no physics either.

By the way, ω is the "point at infinity" after the natural numbers
— fishfry
What does "after" mean here? — Ludwig V

Follows in order. Given 1, 2, 3, 4, ..., we can adjoin $\omega$ "at the end." What do I mean by that? I mean that we extend the "<" symbol so that

1 < $\omega$, 2 < $\omega$, 3 < $\omega$, and so forth. So that conceptually, every natural number is strictly smaller than $\omega$. Does that make sense?

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work
— fishfry
Yes, but it seems to me that this is not literally true, because numbers aren't objects and a set isn't a basket. (I'm not looking for some sort of reductionist verificationism or empiricism here.) — Ludwig V

I can always form a set out of a collection of objects. Not following your objection.

{1/2, 3/4, ...} is a set, and {1} is a set, and I can surely take the union of the two sets, right?

{1/2, 3/4, ..., 1} is just a particular subset of the closed unit interval [0,1].

If you are not sure about what I'm saying we should stay on this point. I can definitely form a set out of any arbitrary collection of other sets. And each of 1, 2, 3, ... and $\omega$ can be defined as particular sets.

Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.
— fishfry
In that respect, yes. But I can't help thinking about the ways in which they are different. — Ludwig V

Of course {1, 2, 3, ..., $\omega$} is a different set that {1/2, 3/4, ..., 1}. But strictly in terms of their order, they are exactly the same. And with ordinals, all we care about is order.

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.
— fishfry
Yes. But it doesn't end in the sense that we can't count from any given natural number up to the end of the sequence. — Ludwig V

The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process.

I try not to mention this in public, but the fact is that I never took a calculus class, nor was I ever taught to think about limits or infinity in the ways that mathematicians sometimes do. I did a little formal loic in my first year undergraduate programme. Perhaps that's an advantage. — Ludwig V

You're far better off. People who take calculus and then engineering math end up confused about limits and the nature of the real numbers. Taking logic and not calculus is actually helpful, in that you haven't mis-learned bad ideas about limits.

Calculus is focussed on the computational and not the philosophical aspects of limits, and calculus students often end up a little confused about some of the technical details. I was actually referring to the other poster who you noted was talking past me and vice versa.

I have the impression that you don't think that they are mathematically possible either. (I admit I may be confused.) So does that mean you don't think that supertasks are possible? — Ludwig V

I've convinced myself both ways. On the one hand we can't physically count all the natural numbers, because there aren't enough atoms in the observable universe. We're finite creatures.

On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot.

I have no strong belief or opinion about supertasks. I have strong opinions about some of the bad logic and argumentation around supertasks.

I really don't see how there could be a staircase which is not physical. — Metaphysician Undercover

If I understood the OP, the walker spends arbitrarily small amounts of time on each step, 1/2 second, 1/4 second, etc. That violates the known laws of physics. So it's not a physical situation. It's a cognitive error to think we're contrasting math to physics. There is no physics in this problem.

That really makes not sense. However, just like in the case of the word "determine", we need to allow for two senses of "physical". You seem to be saying that to be physical requires that the thing referred to must obey the laws of physics. — Metaphysician Undercover

Well yeah. To be a fish a thing has to obey the known laws of fishes. Note that I include the word "known." Biologists could discover a new fish that extends our concept of what's a fish, just as physicists refine their laws from time to time. But a physical thing must obey the known laws of physics. This seems a very trivial point, i can't imagine what you mean by questioning it.

But the classic definition of "physical" is "of the body". — Metaphysician Undercover

Wasn't that a classic Star Trek episode? "Are you of the body?" And if you weren't, they zapped you with an electric stick.

And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law. — Metaphysician Undercover

Sorry, what? Given me an example of something that violates Newton's laws, unless it's an object large enough, small enough, or going fast enough to be subject to quantum or relativistic effects.

A freely willed action? Can you give me an example? You mean like throwing a ball? You kind of lost me here.

Therefore we have to allow for a sense of "physical" which refers to things which are known to violate the laws of physics, like human beings with freely willed actions. — Metaphysician Undercover

I'll be happy to consider any specific examples you have of human beings whose actions violate the laws of physics. If you mean actions caused by mentation, that's a bit of a puzzler, but I'm not sure how to violate the laws of physics. If I set out today to violate Newton's laws, I don't know how I could do that.

What is implied here is that the laws of physics are in some way deficient in their capacity for understanding what is "physical" in the sense of "of the body". — Metaphysician Undercover

Do you mean something like, "I think about raising my right hand and my right hand goes up, how does that happen?" If so, I agree that nobody understands the mechanism.

That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature. — Metaphysician Undercover

That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.

The laws of physics are a human creation, intended to represent the laws of nature, that is the goal, as what is attempted. — Metaphysician Undercover

Agreed.

And, so far as the representation is true and accurate, physical things will be observed to obey the laws of physics, but wherever the laws are false or inaccurate, things will be observed as violating the laws of physics. — Metaphysician Undercover

Still waiting for specific examples. I believe the muons were misbehaving a while back and it made the news. Of course there are things we don't understand, like dark matter, dark energy, a quantum theory of gravity.

Evidently there are a lot of violations occurring, with anomalies such as dark energy, dark matter, etc., so that we must conclude that the attempt, or goal at representation has not been successful. — Metaphysician Undercover

Ok. Scary that you and I are thinking along the same lines. What is your point here with respect to the subject of the thread?

Sure, it's a conceptual thought experiment, but the interpretation must follow the description. A staircase is a staircase, which is a described physical thing, — Metaphysician Undercover

The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.

just like in Michaels example of the counter, such a counter is a physical object, — Metaphysician Undercover

Which counter? The lamp? The lamp is not physical. No physical circuit can switch in arbitrarily small intervals of time.

and in the case of quantum experiments, a photon detector is a physical object. And of course we apply math to such things, but there are limits to what we can do with math when we apply it, depending on the axioms used. The staircase, as a conceptual thought experiment is designed to expose these limits. — Metaphysician Undercover

It's designed to confuse people who mis-learned a little calculus and don't know what's allowed or disallowed by the laws of physics.

OK sure, but that's a limit created by the axioms of the mathematics. So it serves as a limit to the applicability of the mathematics. The least upper bound is just what I described as "the lowest total amount of time which the process can never surpass". Notice that the supposed sequence which would constitute the set with the bound, has already summed the total. This is not part of the described staircase, which only divides time into smaller increments. It is this further process, turning around, and summing it, which is used to produce the limit. The limit is in the summation, not the division. — Metaphysician Undercover

Ok I guess. No walker can traverse a staircase as described by the premises of the problem. So if I said the staircase was not physical, I should have said the walker is not physical. Better?

It is very clear therefore, that the bound is part of the measurement system, a feature of the mathematical axioms employed, the completeness axiom, not a feature of the process described by the staircase descent. The described staircase has no such bound, because the total time passed during the process of descending the stairs is not a feature of that description. This allows that the process continues infinitely, consuming a larger and larger quantity of tiny bits of time, without any limit, regardless of how one may sum up the total amount of time. Therefore completeness axioms are not truly consistent with the described staircase. — Metaphysician Undercover

I don't see why not. The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1, and then to ask what is the final state. Which, as I have pointed out repeatedly, is not defined, but could be defined to be anything you like.

However, since our empirical observations never produce a scenario like the staircase, that inconsistency appears to be irrelevant to the application of the mathematics, with those limitations inherent within the axioms. The limitations are there though, and they are inconsistent with what the staircase example demonstrates as logically possible, continuation without limitation. Therefore we can conclude that this type of axiom, completeness axioms, are illogical, incoherent. — Metaphysician Undercover

I'm sorry that you fine the completeness axiom of the real numbers incoherent. On the contrary, the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.

The real problem is that as much as we can say that the staircase scenario will never occur in our empirical observations, we cannot conclude from this that the incoherency is completely irrelevant. — Metaphysician Undercover

The premises violate the known laws of physics, specifically the claim that we can know the walker's duration on each step even though that duration is below the Plancktime.

We have not at this point addressed other scenarios where the completeness axioms might mislead us. Therefore the incoherency may be causing problems already, in other places of application. — Metaphysician Undercover

Modern math is incoherent. Is it possible that you simply haven't learned to appreciate its coherence?

Who did that? Are they in the room with us right now? — fishfry

See here:

As Salmon (1998) has pointed out, much of the mystery of Zeno’s walk is dissolved given the modern definition of a limit. This provides a precise sense in which the following sum converges:

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots$

Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity.

...

Suppose we switch off a lamp. After 1 minute we switch it on. After ½ a minute more we switch it off again, ¼ on, ⅛ off, and so on. Summing each of these times gives rise to an infinite geometric series that converges to 2 minutes, after which time the entire supertask has been completed.

I have been arguing that it is a non sequitur to argue that because the sum of an infinite series can be finite then supertasks are metaphysically possible. The lack of a final or a first task entails that supertasks are metaphysically impossible. I think this is obvious if we consider the supertask of having counted down from infinity, and this is true of having counted up to infinity as well.

We can also consider a regressive version of Thomson's lamp; the lamp was off after 2 minutes, on after 1 minute, off after 30 seconds, on after 15 seconds, etc. We can sum such an infinite series, but such a supertask is metaphysically impossible to even start.

Totally agree, but I'm not aware of anybody claiming a proof that supertasks are possible. Maybe I missed it — noAxioms

You've got this backward. Some supertasks are coherent and consistent, therefore logically logically possible. In this case, that is the proof that they are "possible". If someone wants to insist that they are impossible then a poof is required.

If I understood the OP, the walker spends arbitrarily small amounts of time on each step, 1/2 second, 1/4 second, etc. That violates the known laws of physics. So it's not a physical situation. It's a cognitive error to think we're contrasting math to physics. There is no physics in this problem. — fishfry

I dealt with this already. If you restrict the meaning of "physical" to that which abides by the law of physics, then every aspect of what we would call "the physical world" which violates the laws of physics, dark energy, dark matter, for example, and freely willed acts of human beings, would not be a part of the "physical" world.

But a physical thing must obey the known laws of physics. — fishfry

That's not true at all. It does not correctly represent how we use the word "physical". "Physical" has the wider application than "physics". We use "physical" to refer to all bodily things, and "physics" is the term used to refer to the field of study which takes these bodily things as its subject. Therefore the extent to which physical things "obey the known laws of physics" is dependent on the extent of human knowledge. If the knowledge of physics is incomplete, imperfect, or fallible in anyway, then there will be things which do not obey the laws of physics. Your claim "a physical thing must obey the known laws of physics" implies that the known laws of physics represents all possible movements of things. Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false.

Sorry, what? Given me an example of something that violates Newton's laws, unless it's an object large enough, small enough, or going fast enough to be subject to quantum or relativistic effects. — fishfry

I gave you an example. A human body moving by freely willed acts violates Newton's first law.

"Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia."

There is no such "external force" which causes the freely willed movements of the human body. We might create the illusion that the violation can be avoided by saying that the immaterial soul acts as the "force" which moves that body, but then we have an even bigger problem to account for the reality of that assumed force, which is an "internal force". Therefore Newton's first law has no provision for internal forces, and anytime such forces act on bodies, there is a violation of Newton's laws.

That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature. — fishfry

If you understand this, then you ought to understand that being physical in no way means that the thing which is physical must obey the laws of physics. It is not the case that we only call a thing "physical" if it obeys the laws of physics, the inverse is the case. We label things as "physical" then we apply physics, and attempt to produce the laws which describe the motions of those things. Physical things only obey the laws of physics to the extent that the laws of physics have been perfected.

Ok. Scary that you and I are thinking along the same lines. What is your point here with respect to the subject of the thread? — fishfry

Ok, now we're getting somewhere. The point, in relation to the "paradox" of the thread is as follows. There are two incompatible scenarios referenced in the op. Icarus descending the stairs must pass an infinite number of steps at an ever increasing velocity because each step represents an increment of time which we allow the continuum to be divided into. In the described scenario, 60 seconds of time will not pass, because Icarus will always have more steps to cover first, due to the fact that our basic axioms of time allow for this infinite divisibility. The contrary, and incompatible scenario is that 60 seconds passes. This claim is supported by our empirical evidence, experience, observation, and our general knowledge of the way that time passes in the world.

What I believe, is that the first step to understanding this sort of paradox is to see that these two are truly incompatible, instead of attempting to establish some sort of bridge between them. The bridging of the incompatibility only obscures the problem and doesn't allow us to analyze it properly. Michael takes this first step with a similar example of the counter , but I think he also jumps too far ahead with his conclusion that there must be restrictions to the divisibility of time. I say he "jumps to a conclusion", because he automatically assumes that the empirical representation, the conventional way of measuring time with clocks and imposed units is correct, and so he dismisses, based on what I call a prejudice, the infinite divisibility of time in Icarus' steps, and the counter example.

I insist that we cannot make that "jump to a conclusion". We need to analyze both of the two incompatible representations separately and determine the faults which would allow us to prove one, or both, to be incorrect. So, as I've argued above, we cannot simply assume that the way of empirical science is the correct way because empirical science is known to be fallible. And, if we look at the conventional way of measuring time, we see that all the units are fundamentally arbitrary. They are based in repetitive motions without distinct points of separation, and the points of division are arbitrarily assigned. That we can proceed to any level, long or short, with these arbitrary divisions actually supports the idea of infinite divisibility. Nevertheless, we also observe that time keeps rolling along, despite our arbitrary divisions of it into arbitrary units. This aspect, "that time keeps rolling along", is what forces us to reject the infinite divisibility signified by Icarus' stairway to hell, and conclude as Michael did, that there must be limitations to the divisibility of time.

Now the issue is difficult because we do not find naturally existing points of divisibility within the passage of time, and all empirical evidence points to a continuum, and the continuum is understood to be infinitely divisible. So the other option, that of empirical science is also incorrect. Both of the incompatible ways of representing time are incorrect. What is evident therefore, is that time is not a true continuum, in the sense of infinitely divisible, and it must have true, or real limitations to its divisibility. This implies real points within the passage of time, which restrict the way that it ought to be divided. The conventional way of representing time does not provide any real points of divisibility.

"Real divisibility" is not well treated by mathematicians. The general overarching principle in math, is that any number may be divided in any way, infinite divisibility. However, in the reality of the physical universe we see that any time we attempt to divide something there is real limitations which restrict the way that the thing may be divided. Furthermore, different types of things are limited in different ways. This implies that different rules of division must be applied to different types of things, which further implies that mathematics requires a multitude of different rules of division to properly correspond with the divisibility of the physical world. Without the appropriate rules of divisibility, perfection in the laws of physics is impossible, and things such as "internal forces" will always be violating the laws of physics.

quote="fishfry;900943"]The walker spends ever smaller amounts of time on each step, and that eventually violates the Planck scale.[/quote]

The Planck limitations are just as arbitrary as the rest, being based in other arbitrary divisions and limitations such as the speed of light. The Planck units are not derived from any real points of divisibility in time.

The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1 — fishfry

No, the point of the puzzle is to demonstrate that the sum is always less than one, and that the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum. The assumption that the sum is equivalent to one is what creates the paradox.

the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity. — fishfry

I hope you're joking, but based on our previous discussions, I think you truly believe this. What a strangely sheltered world you must live in, under your idealistic umbrella.

The premises violate the known laws of physics... — fishfry

Exactly, and since we know that many physical things commonly violate the laws of physics, the fact that the premises are logically consistent and that they violate the laws of physics, indicates that we need to take a closer look at the laws of physics.

Modern math is incoherent. Is it possible that you simply haven't learned to appreciate its coherence? — fishfry

No, I've read thoroughly many fundamental axioms, and found clear incoherencies, which I've shared in this forum. Many people accept premises and axioms because they are "the convention", so they do not proceed with the due diligence to determine whether there is inconsistency between them. Then, they proceed to utilize them because they are extremely useful. Problem would only arise under specific conditions which would be avoided, or a workaround developed for. So it's not a matter of learning to "appreciate its coherence", I've already learned to appreciate its usefulness, facility, and convenience. But I think that you are mistaken to think that facility necessarily implies coherency.

I don't see why not. The whole point of the puzzle is to sum 1/2 + 1/4 + ... = 1, and then to ask what is the final state. — fishfry

The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process. — fishfry

OK. I remembered WIttgenstein's oracular remark that death is not a part of life. My concern that the limit is not generated by the defining formula isn't the problem I thought it might be.

I've convinced myself both ways. On the one hand we can't physically count all the natural numbers, because there aren't enough atoms in the observable universe. We're finite creatures.
On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot. — fishfry

I don't really believe in "possible" without qualification. There's logically possible, (is mathematically possible the same or something different? Does is apply here?), physically possible, and a range of others, such as legally possible. So what kind of possibility is a supertask?

A lamp that cycles in arbitrarily small amounts of time is not physical. A staircase that we occupy for arbitrarily small intervals of time is not physical. So trying to use physical reasoning is counterproductive and confusing. That's my objection to all these kinds of puzzles. People say there's a conflict between the math and the physics ... but as i see it, there's no physics either. — fishfry

So your reply is that it is neither. It suggests a combination of physical and mathematical rules which is incoherent but generates an illusion. That's why

It may "lead" somewhere but there's no law that constrains the final state. It may be discontinuous, like Cinderella's coach that's a coach at 1/2, 1/4, 1/8, ... seconds before midnight, then becomes a coach at midnight. That's why it's perfectly possible that the lamp becomes a pumpkin after 1 second. — fishfry

But then you say

On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot

Obviously, as each stage gets smaller, I will complete it more quickly. But still, it will take some period of time, and the final step looks out of reach. That looks like a combination of physical and mathematical rules.
It isn't a real problem because I can analyze the task in a different way. I can complete the first yard, the second yard.... When I have completed 1760 yards, I have completed the task. But the supertasks seem not to permit that kind of analysis. Is that the issue?

Can we not count the intervals starting with 1 — ToothyMaw

No. In the dichotomy scenario, there is no first step to which that number can be assigned.

To count a set means to place it into bijection with: — fishfry

OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known.


I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox.

Depends on the exchange rate. — fishfry

I also would hate to have to talk about the poor kilometerage that Bob's truck gets.

It [the even-oddness of ω]is neither, and who's asking such a thing? — fishfry

The lamp scenario asks it, which is why the comment was relevant.

Some supertasks are coherent and consistent, therefore logically logically possible. In this case, that is the proof that they are "possible" — Metaphysician Undercover

I think the person to whom I was replying was suggesting that somebody had asserted a proof that a physical supertask was possible. But I did not recall anybody posting such an assertion.

Can we not count the intervals starting with 1
— ToothyMaw
No. In the dichotomy scenario, there is no first step to which that number can be assigned. — noAxioms

Really? What if one were to begin by summing each interval as represented by a bijective function like f(n) = 60/2^n where n is a number in the natural numbers representing the cardinality of a set like N = {30, 15, 15/2}? Does that not include a first step? And would that sum not eventually terminate given a smallest sliver of time exists or continue indefinitely given time is infinitely divisible?

a set like N = {30, 15, 15/2}? Does that not include a first step? — ToothyMaw

Yes, that series has a first step, but not a last one. You can number the steps in the series if you start at the big steps. Similarly, you can number the dichotomy steps in reverse order, since the big steps are at the end.

And would that sum not eventually terminate given a smallest sliver of time exists

If there's a smallest sliver of time, there is no bijection with the set of natural numbers since there are only a finite number of steps.

or continue indefinitely given time is infinitely divisible?

'Continue indefinitely' is a phrase implying 'for all time', yet all the steps are taken after only a minute, so even if time is infinitely divisible, the series completes in short order.

See here:

As Salmon (1998) has pointed out, much of the mystery of Zeno’s walk is dissolved given the modern definition of a limit. This provides a precise sense in which the following sum converges:

Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. — Michael

I am not sure why you think it's necessary to point that out to me.

A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. — Michael

I took the class. I read a couple of different books, not that particular one. But I'm not sure why you are taking the time to explain to me the basics of convergent infinite series in real analysis.

From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. — Michael

Well no, not really, because a convergent infinite series does not have a temporal component. There's no notion of "add the next thing then add the next thing ..." Rather, the sum of the series is 1 in a single moment if you will. Rather, there are no moments at all. 1/2 + 1/4 + ... = 1 in the same sense that 1 + 1 = 2. They are two expressions that mean the same thing.

We can contrast convergent infinite series with loops in a programming language. Loops are executed in time, consume energy, and produce heat in the computational substrate. Mathematical series don't do any of that. Mathematical operations happen atemporally. They are, they don't do.

When you try to put it into a physical context, that's where the confusion comes in.

]Suppose we switch off a lamp. After 1 minute we switch it on. After ½ a minute more we switch it off again, ¼ on, ⅛ off, and so on. Summing each of these times gives rise to an infinite geometric series that converges to 2 minutes, after which time the entire supertask has been completed. — Michael

No physical lamp can switch that fast, so there's nothing physical about this thought experiment.

Consider a function f defined on the ordered set {1/2, 3/4, 7/8, ..., 1}. This is a perfectly well defined set of rational numbers. Suppose f(1/2) = 0, f(3/4) = 1, f(7/8) = 0, and so forth; and f(1) = a plate of spaghetti.

That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight.

That is literally the answer to the Thompson's lamp puzzle.

I have been arguing that it is a non sequitur to argue that because the sum of an infinite series can be finite then supertasks are metaphysically possible. — Michael

You have not so argued. You have so claimed. You have not provided any proof or even evidence that a supertask is "metaphysically impossible." Maybe it is, maybe it isn't. The Planck scale and the physics of switching circuits preclude the existence of the lamp, according to current physics. A century or two from now, who can say? In particular, how can you personally know that future physics won't let us peer below the Planck scale?

The lack of a final or a first task entails that supertasks are metaphysically impossible. — Michael

I just showed you the final state. A supertask is a function on the set {1/2, 3/4, 7/8, ..., 1}. The final state of Cinderella's coach is pumpkin. The final state of Thompson's lamp is plate of spaghetti.

I do not see the problem. Neither the lamp nor the coach are physical entities. This is purely an abstract thought experiment, and my solution is mathematically sound.

I think this is obvious — Michael

That's not a proof. If you had a proof you'd give it, instead of simply claiming how obvious it all is to you that something is "metaphysically impossible." How do you know what's metaphysically possible? None of us are given to know the ultimate nature of the world. Not currently, anyway, and maybe never.

if we consider the supertask of having counted down from infinity, and this is true of having counted up to infinity as well. — Michael

Bit of a subject change, not sure where you're going with this.

We can also consider a regressive version of Thomson's lamp; the lamp was off after 2 minutes, on after 1 minute, off after 30 seconds, on after 15 seconds, etc. We can sum such an infinite series, but such a supertask is metaphysically impossible to even start. — Michael

I think you must have your own private meaning for "metaphysically impossible," because after all our conversation, you have not convinced me of your point. I think literal, physically instantiated supertasks may or may not turn out to be possible. I base my opinion on the long history of revolutions in scientific understanding. The earth turned out to revolve around the sun. We split the atom. We evolved from more primitive animals. Many things formerly thought to be "metaphysically impossible" turned out to be not only possible, but true.

What makes you think you can see the indefinitely far away scientific future?

To count a set means to place it into bijection with:
— fishfry
OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known. — noAxioms

If I stand in a parking lot and call out "one, two, three, ..." and keep going, I can never count all the natural numbers.

In math, I can say, Let {1, 2, 3, ...} be the set of natural numbers whose existence is guaranteed by the axiom of infinity. Now I've counted the natural numbers.

I see a difference in those scenarios. I suppose someone could say that my conceptualization of {1, 2, 3, ...} is a physical process in my brain. Is that what yu mean that there's no difference between mathematical and physical counting?

I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox. — noAxioms

I'm probably not in a position to differ. Clearly we can walk from one place to another. Maybe that's a supertask. I don't know.

I also would hate to have to talk about the poor kilometerage that Bob's truck gets. — noAxioms

Do British people talk about kilometerage? I've actually never heard that usage but I suppose it makse sense.

It [the even-oddness of ω]is neither, and who's asking such a thing?
— fishfry
The lamp scenario asks it, which is why the comment was relevant. — noAxioms

It's not a physical lamp, since no physical circuit could switch that fast. Therefore it's an imaginary lamp. Its state is defined at each of the times 1/2, 3/4, 7/8, ... But its state is not defined at 1. Therefore we may define its state as 1 as becoming a plate of spaghetti.

It's just like Cinderella's coach. It's a coach at midnight minus 1/2, midnight minus 1/4, etc. At at exactly midnight, it turns into a coach.

The Planck-scale defying lamp circuit is every bit as fictional as Cinderella's coach. Since the state at 1 is not defined, I'm free to define it as a plate of spaghetti. That's the solution to the lamp problem.

The reason it's as sensible as any other solution is that there is no final state that makes the lamp function continuous.

If, for example, the lamp is on at 1, on at 3/4, on at 7/8, and so forth, then we could still define the state at 1 as a plate of spaghetti; but if we defined the lamp to be on at 1, that would have the virtue of making the lamp function continuous. Continuous functions are to be preferred.

But no possible value for the final state of the lamp makes the problem continuous. Therefore any old arbitrarily solution will do just as well as any other.

As far as I'm concerned, that solves the problem. Until, of course, some future genius not yet born figures out how to implement a switching circuit that makes the lamp physical.

so we can't have counted up to infinity because there is no last number — Michael

That is exactly the Zeno Walk.

From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. One might only doubt whether or not the standard topology of the real numbers provides the appropriate notion of convergence in this supertask.

Your objection is that:

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence.

And the problem is

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.

We may not like how this train of thought goes, and we might settle for the more intuitive and less troublesome metaphysics, but the possibility of either remains, especially when human minds have issues wrestling with the infinity concept.

But its state is not defined at 1 — fishfry

However, how do you arrive at that conclusion? The two options that I can think of is by admitting that the sum of an infinite series is an approximation instead of the exact value, or by casting some doubt on the idea of an ∞-th item of a series. The latter seems to cause more problems than solve them for me. Did you use a different reasoning?

OK. I remembered WIttgenstein's oracular remark that death is not a part of life. My concern that the limit is not generated by the defining formula isn't the problem I thought it might be. — Ludwig V

Jeez that's kind of creepy ... true, I suppose. Death is the least upper bound of the open set of life.

I don't really believe in "possible" without qualification. There's logically possible, (is mathematically possible the same or something different? Does is apply here?), physically possible, and a range of others, such as legally possible. So what kind of possibility is a supertask? — Ludwig V

In the future, if physics ever figures out how to work with physically instantiated infinities, supertasks might be possible. Way too soon to know.

So your reply is that it is neither. It suggests a combination of physical and mathematical rules which is incoherent but generates an illusion. — Ludwig V

I just mentioned that I could argue it either way.

But then you say
On the other hand, supertasks are possible, because I can walk a mile, meaning I walked 1/2 a mile, 1/4 mile, dot dot dot
Obviously, as each stage gets smaller, I will complete it more quickly. But still, it will take some period of time, and the final step looks out of reach. That looks like a combination of physical and mathematical rules. — Ludwig V

How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series. Except for the fact that nobody knows if spacetime below the Planck length is accurately modeled by the mathematical real numbers. Maybe it's not. We just don't know. In any event, I don't know.

It isn't a real problem because I can analyze the task in a different way. I can complete the first yard, the second yard.... When I have completed 1760 yards, I have completed the task. But the supertasks seem not to permit that kind of analysis. Is that the issue? — Ludwig V

I don't think it's much of a problem. It doesn't keep me up at night. I just saw a Youtube video of an interview of Graham Priest, a famous philosopher. He thinks Zeno and other paradoxes are important. a lot of people think they're important.

I think some of Zeno's other paradoxes are more interesting. When you shoot an arrow, it's motionless in an instant. How does it know where to go next, and at what speed? I think that's a more interesting puzzle. Where are velocity and momentum "recorded?" How does the arrow know what to do next?

However, how do you arrive at that conclusion? — Lionino

By the conditions of the problem. Is this about the lamp? The problem says it's on at 1/2, on at 3/4, off at 7/8, etc.. The problem itself doesn't define the state at 1. So I'm free to define the state at 1 any way I like. Because the problem itself leaves that information unspecfified.

The two options that I can think of is by admitting that the sum of an infinite series is an approximation instead of the exact value, — Lionino

No. The mathematics is pristine. 1/2 + 1/4 + 1/8 + ... = 1 in the same sense that 1 + 1 = 2. Two names for the same thing. May be used interchangeably. Exactly equal. Denote exactly the same real number.

or by casting some doubt on the idea of an ∞-th item of a series. — Lionino

There is no ∞-th item of a series. There is the limit of a series (or sequence, I think you may mean here). That's not the same thing. In the sequence 1/2, 3/4, 7/8, 15/16, ... there is no last element. The sequence has a limit of 1. But 1 is the limit, it's not a member of the sequence.

The latter seems to cause more problems than solve them for me. Did you use a different reasoning? — Lionino

Not sure what you mean. Reasoning in terms of what? The final lamp state is not defined. I can arbitrarily define any function at a point where it's not defined, especially when there's no natural reason (such as continuity) to prefer one limit state over another.

Am I understanding your question correctly? I didn't quite understand what you mean by asking if I used different reasoning.

It's like one of those "fill in the missing number" puzzles, like 1, 2, 8, 16. They want you to say 32. But mathematically, you can put in anything you want. If you don't tell me the lamp state at the limit, I can define it any way I want, especially since there's no way to define it in such a way that the sequence attains its limiting value in a continuous manner.

I dealt with this already. If you restrict the meaning of "physical" to that which abides by the law of physics, then every aspect of what we would call "the physical world" which violates the laws of physics, dark energy, dark matter, for example, and freely willed acts of human beings, would not be a part of the "physical" world. — Metaphysician Undercover

Don't be silly. The rotational rate of galaxies is physical, even if our current theory of gravity doesn't explain it. We don't say it's not physical just because we don't have a theory of dark matter or modified gravity yet.

That's not true at all. It does not correctly represent how we use the word "physical". "Physical" has the wider application than "physics". We use "physical" to refer to all bodily things, and "physics" is the term used to refer to the field of study which takes these bodily things as its subject. Therefore the extent to which physical things "obey the known laws of physics" is dependent on the extent of human knowledge. If the knowledge of physics is incomplete, imperfect, or fallible in anyway, then there will be things which do not obey the laws of physics. Your claim "a physical thing must obey the known laws of physics" implies that the known laws of physics represents all possible movements of things. Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false. — Metaphysician Undercover

I think I can't play these word games. If you want to pretend not to know what a physical thing is, I can't argue with you. Bowling balls falling down was a physical phenomenon two thousand years ago even if Aristotle's physics didn't explain it sufficiently.

I gave you an example. A human body moving by freely willed acts violates Newton's first law. — Metaphysician Undercover

How can a human body move by free will? You're the determinist here. You reject randomness. How does this "will" influence the body? Good questions. I don't know. You don't either.

"Newton’s first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia." — Metaphysician Undercover

Sorry what? I didn't say that, is that someone else's quote?

There is no such "external force" which causes the freely willed movements of the human body. We might create the illusion that the violation can be avoided by saying that the immaterial soul acts as the "force" which moves that body, but then we have an even bigger problem to account for the reality of that assumed force, which is an "internal force". Therefore Newton's first law has no provision for internal forces, and anytime such forces act on bodies, there is a violation of Newton's laws. — Metaphysician Undercover

Can you remind me what is the point of all this? I haven't the heart to discuss Newtonian metaphysics.

That's why I included the word "known." I allow that the laws of physics are historically contingent approximations to the laws of nature.
— fishfry

If you understand this, then you ought to understand that being physical in no way means that the thing which is physical must obey the laws of physics. — Metaphysician Undercover

Fine. The speed of the rotating galaxies is physical but we don't have a law of physics that explains it yet.

Can you remind me why we're having this discussion? I think if you wrote less I could respond more deeply. This flood of deepity is a bit much for me.

It is not the case that we only call a thing "physical" if it obeys the laws of physics, the inverse is the case. We label things as "physical" then we apply physics, and attempt to produce the laws which describe the motions of those things. Physical things only obey the laws of physics to the extent that the laws of physics have been perfected. — Metaphysician Undercover

I think if you wrote a shorter post I'd engage. Do you think bowling balls falling down was not physical before Newton? Before Einstein? Before the next genius not yet born?

By your reasoning, nothing at all is physical, since all physical theories are only approximate.

Ok, now we're getting somewhere. The point, in relation to the "paradox" of the thread is as follows. There are two incompatible scenarios referenced in the op. Icarus descending the stairs must pass an infinite number of steps at an ever increasing velocity because each step represents an increment of time which we allow the continuum to be divided into. In the described scenario, 60 seconds of time will not pass, because Icarus will always have more steps to cover first, due to the fact that our basic axioms of time allow for this infinite divisibility. The contrary, and incompatible scenario is that 60 seconds passes. This claim is supported by our empirical evidence, experience, observation, and our general knowledge of the way that time passes in the world. — Metaphysician Undercover

Ok. I haven't engaged with the staircase at all. Can't argue it. But if 60 seconds of time can't pass, how did I walk from the living room to the kitchen for a snack?

What I believe, is that the first step to understanding this sort of paradox is to see that these two are truly incompatible, instead of attempting to establish some sort of bridge between them. The bridging of the incompatibility only obscures the problem and doesn't allow us to analyze it properly. Michael takes this first step with a similar example of the counter ↪Michael, but I think he also jumps too far ahead with his conclusion that there must be restrictions to the divisibility of time. I say he "jumps to a conclusion", because he automatically assumes that the empirical representation, the conventional way of measuring time with clocks and imposed units is correct, and so he dismisses, based on what I call a prejudice, the infinite divisibility of time in Icarus' steps, and the counter example. — Metaphysician Undercover

I can't argue with you about your analysis of what someone else said. Nor can I argue about the staircase. I haven't really said much in this thread about the staircase. The lamp is much more clear to me.

I insist that we cannot make that "jump to a conclusion". — Metaphysician Undercover

I made it to the kitchen and back. How do you account for that?

We need to analyze both of the two incompatible representations separately and determine the faults which would allow us to prove one, or both, to be incorrect. So, as I've argued above, we cannot simply assume that the way of empirical science is the correct way because empirical science is known to be fallible. And, if we look at the conventional way of measuring time, we see that all the units are fundamentally arbitrary. They are based in repetitive motions without distinct points of separation, and the points of division are arbitrarily assigned. That we can proceed to any level, long or short, with these arbitrary divisions actually supports the idea of infinite divisibility. Nevertheless, we also observe that time keeps rolling along, despite our arbitrary divisions of it into arbitrary units. This aspect, "that time keeps rolling along", is what forces us to reject the infinite divisibility signified by Icarus' stairway to hell, and conclude as Michael did, that there must be limitations to the divisibility of time. — Metaphysician Undercover

I actually agree with you that the mathematical real numbers may well not be an accurate representation of the ultimate nature of reality. If that's what you mean about infinite divisibility.

Now the issue is difficult because we do not find naturally existing points of divisibility within the passage of time, and all empirical evidence points to a continuum, and the continuum is understood to be infinitely divisible. So the other option, that of empirical science is also incorrect. Both of the incompatible ways of representing time are incorrect. What is evident therefore, is that time is not a true continuum, in the sense of infinitely divisible, and it must have true, or real limitations to its divisibility. This implies real points within the passage of time, which restrict the way that it ought to be divided. The conventional way of representing time does not provide any real points of divisibility. — Metaphysician Undercover

If you're arguing for a discrete universe, maybe so.

"Real divisibility" is not well treated by mathematicians. — Metaphysician Undercover

It's profoundly and beautifully and logically rigorously treated by mathematicians. I can't imagine what you mean here. You're flat out wrong.

We may not know the ultimate nature of the world, but the mathematical real numbers are treated very well indeed.

The general overarching principle in math, is that any number may be divided in any way, infinite divisibility. — Metaphysician Undercover

Arbitrary, not infinity. I can divide 1 into 1/2, into 1/4, into 1/8. I can divide any finite number of times, and there are infinitely many numbers. but I can't "divide infinitely." That's imprecise and essentially wrong.

However, in the reality of the physical universe we see that any time we attempt to divide something there is real limitations which restrict the way that the thing may be divided. Furthermore, different types of things are limited in different ways. This implies that different rules of division must be applied to different types of things, which further implies that mathematics requires a multitude of different rules of division to properly correspond with the divisibility of the physical world. — Metaphysician Undercover

Not at all. That's the physicists' job. Mathematicians need not concern themselves with the physical world at all.

But mathematicians do have "different rules of division." The rules of division in the integers are very different than the rules of division for the real numbers.

Without the appropriate rules of divisibility, perfection in the laws of physics is impossible, and things such as "internal forces" will always be violating the laws of physics. — Metaphysician Undercover

There can never be perfection in any physical law. You know that. You lost me with internal forces.

The Planck limitations are just as arbitrary as the rest, being based in other arbitrary divisions and limitations such as the speed of light. The Planck units are not derived from any real points of divisibility in time. — Metaphysician Undercover

The math is pretty solid, it's based on Fourier series as I understand it. I think you're a little out over your skis here. But your complaints are about physics. I'm not qualified to help.

No, the point of the puzzle is to demonstrate that the sum is always less than one, and that the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum. — Metaphysician Undercover

You're just typing stuff in. What you wrote isn't true. "the mathematician's practise of making the sum equivalent to one is just an attempt to bridge the gap between two incompatible ways of looking at the theoretical continuum." Not even wrong.

The assumption that the sum is equivalent to one is what creates the paradox. — Metaphysician Undercover

The math is beyond dispute.

the completeness axiom of the real numbers is one of the crowning intellectual achievements of humanity.
— fishfry

I hope you're joking, — Metaphysician Undercover

I've never meant anything more seriously. It was more than 200 years of intellectual struggle from Newton and Leibniz to Weierstrass, Cauchy, Cantor, and Zermelo.

but based on our previous discussions, I think you truly believe this. What a strangely sheltered world you must live in, under your idealistic umbrella. — Metaphysician Undercover

Based on our previous discussions, you're still an ignorant troll. I'm done here. What is wrong with you?

Even if you are determinist and do not agree with free will causation, quantum mechanics clearly demonstrates that your statement is false. — Metaphysician Undercover

I hope you don't mind my saying that your choice of free will as an example was perhaps ill-advised. It's far too contentious to work. Quantum mechanics is a much better choice. But there is the problem that there are many interpretations of it, so it is not clear that it proves what you think it proves.

In the future, if physics ever figures out how to work with physically instantiated infinities, supertasks might be possible. Way too soon to know. — fishfry

I think you are both mistaken to rely on physics to define what one wants to get at in this context. Physics is not only limited by the current state of knowledge, but also by its exclusion of much that one would normally take to be both physical and real. Somewhere near the heart of this is that there is no clear concept that will catch what we might mean by "whatever exists that is not mathematics" or by "whatever applied mathematics is applied to".

Jeez that's kind of creepy ... true, I suppose. Death is the least upper bound of the open set of life. — fishfry

I'm sorry. I didn't mean to gross you out. Perhaps if you think of death as a least upper bound, you'll be able to think of it differently. It is, after all, an everyday and commonplace event - even if, in polite society, we don't like to mention it.

I just mentioned that I could argue it either way. — fishfry

Yes. I was just drawing out the implications. You might disagree.

The sequence is endless, and there's an extra point that's defined to be strictly greater than all the others. We can't get to the limit by successors, but we can get there by a limiting process. — fishfry

Yes. In the context of the Achilles problem that's fine and I understand that you are treating that and the natural numbers as parallel. It's not clear to me that it really works. It makes sense to say that "1" limits "1/2, 1/4, ..." But I'm not at all sure that it makes sense to say that <omega> limits the sequence of natural numbers. "+1" adds to the previous value. "<divide by 2>" reduces from the previous value. The parallel is not complete. There are differences as well as similarities.

How can it be out of reach? I went to the supermarket today. I walked from one end of the aisle to the other. I reached the end. I did indeed evidently sum a convergent infinite series. — fishfry

Did you "get to the limit by successors" or "get there by a limiting process"? I don't think so. You are just not applying that frame to your trip.

I think some of Zeno's other paradoxes are more interesting. When you shoot an arrow, it's motionless in an instant. How does it know where to go next, and at what speed? I think that's a more interesting puzzle. Where are velocity and momentum "recorded?" How does the arrow know what to do next? — fishfry

I've met other mathematicians who agree that Achilles is not interesting. But I'm fascinated that you think the arrow is interesting. I don't. Starting is a boundary condition and so not part of the temporal sequence, any more than the boundary of my garden is a patch of land. End of problem.

But this may be interesting in the context of what we are talking about. A geometrical point does not occupy any space. It is dimensionless. One could say it is infinitely small. But it is obvious that there is no problem about passing an infinite number of them. It is a question of how you think about them. This is not quite the same as Zeno's problem, but it is close.

That is a perfectly sensible answer to the question, "What is the state at the limit?" It's perfectly sensible because the conditions of the problem don't specify the value at the limit. And since the lamp is not physical, it can turn into anything we like at the limit. It's no different than Cinderella's coach, which is a coach at 1/2 second before midnight, 1/4 second before midnight, and so on, and turns into a coach at midnight. — fishfry

I agree with that.
Perhaps then, these problems are not mathematical and not physical, but imaginary - a thought experiment. (The Cinderella example shows that we can easily imagine physically impossible events) That suggests what you seem to be saying - that there are no rules. (Which is why I posited another infinite staircase going up). But if there are no rules, what is the experiment meant to show? The only restriction I can think of is that it needs to be logically self-consistent - and the infinite staircase is certainly that. I guess the weak spot in the supertask is the application of a time limit. However, I also want to say that I cannot imagine an endless staircase, only one that has not ended yet - once I've imagined that, I can wave my hand and say, that is actually an infinite staircase.

a set like N = {30, 15, 15/2}? Does that not include a first step?
— ToothyMaw
Yes, that series has a first step, but not a last one. You can number the steps in the series if you start at the big steps. Similarly, you can number the dichotomy steps in reverse order, since the big steps are at the end. — noAxioms

Okay, so it is possible to have a first step. If I could, say, produce an equation based on the one in my earlier post that could calculate the last time interval given a smallest stipulated chunk of time, would that be a valid final step in the summation?

And would that sum not eventually terminate given a smallest sliver of time exists
If there's a smallest sliver of time, there is no bijection with the set of natural numbers since there are only a finite number of steps.

or continue indefinitely given time is infinitely divisible?
'Continue indefinitely' is a phrase implying 'for all time', yet all the steps are taken after only a minute, so even if time is infinitely divisible, the series completes in short order. — noAxioms

That was sloppy thinking and use of language on my part. Sorry.

How can a human body move by free will? — fishfry

I think, and then I do. The "force" which moves me comes from within me, and therefore cannot be described by Newton's conceptions of force.

The rotational rate of galaxies is physical, even if our current theory of gravity doesn't explain it.

...

The speed of the rotating galaxies is physical — fishfry

"Speed", and "rate" are measurements derived from comparing things. This is explained in the theory of relativity, and by that theory such things are dependent on the frame of reference. A measurement does not exist without the act which measures. I see that this is indicative of your way of thinking, when you say that by referring to the axiom of infinity you can count all the natural numbers. This is a new fangled sort of doing by proxy, where the assertion (here called an "axiom") "I have done X" means that X has been done. That is the same sort of mistake which Michael was making in insisting that measurements like seconds and days exist without being measured. I referred Michael to Wittgenstein's "standard metre" example.

I hope you don't mind my saying that your choice of free will as an example was perhaps ill-advised. It's far too contentious to work. Quantum mechanics is a much better choice. But there is the problem that there are many interpretations of it, so it is not clear that it proves what you think it proves. — Ludwig V

I do not pretend to be providing a proof when I provide an example. However, I'll take your advise and refer to quantum mechanics if I'm asked to provide examples of how it is that a measurement cannot exist without an act which measures.

I think the person to whom I was replying was suggesting that somebody had asserted a proof that a physical supertask was possible. But I did not recall anybody posting such an assertion. — noAxioms

The use of "physical" in this thread has gotten so ambiguous, that equivocation abounds everywhere.

. In other words, at a sufficiently small scale, when an object (esp. particle) moves from A to B it does so without passing any half-way point. Your use of the phrase "quantum jump" is fitting. — Michael

How much time elapses from travel to point a to point b and where is the object located during that time lapse?

Does the object leave existence between a and b and if it does, what maintains its identity during that interval?

How much time elapses from travel to point a to point b and where is the object located during that time lapse?

Does the object leave existence between a and b and if it does, what maintains its identity during that interval? — Hanover

That’s a question for physicists to answer.

If I stand in a parking lot and call out "one, two, three, ..." and keep going .. — fishfry

OK, that other meaning of 'count'.

I think we're talking past each other. When asked for the difference between a mathematical and physical supertask, you seem to focus on two different definitions of countable: The assignment of a bijection, and calling or writing down each of the numbers.

I'm talking about a physical supertask as described by Zeno, which arguably has countably (first definition) steps performed in finite time. Nobody is posited to vocalize the number of each step as it is performed.

It's just like Cinderella's coach. It's a coach at midnight minus 1/2, midnight minus 1/4, etc. At at exactly midnight, it turns into a coach.

Bit off on the lore. It turns into a pumpkin, and at the 12th stroke, where presumably midnight is the first stroke, but I googled that and could not find an official ruling on the topic.

The Planck-scale defying lamp circuit is every bit as fictional as Cinderella's coach. Since the state at 1 is not defined, I'm free to define it as a plate of spaghetti. That's the solution to the lamp problem.

No argument. That seems to be a valid way out of most attempts to assign a count to the nonexistent last/first step, or to simply assert the necessity of the nonexistent thing.

I like Bernadete's Paradox of the Gods because it doesn't make those mistakes, and thus seems very much paradoxical since motion seems prevented by a nonexistent barrier.

For educational purposes concerning how infinity works, I like Littlewood-Ross Paradox because it is even more unintuitive, but actually not paradoxical at all since it doesn't break any of the above rules. It shows a linear series (effectively 9+9+9+...) being zero after the completion of every step.

If I could, say, produce an equation based on the one in my earlier post that could calculate the last time interval given a smallest stipulated chunk of time, would that be a valid final step in the summation? — ToothyMaw

If you stopped the summation there, then yes, there would be a final step, but it wouldn't have infinite steps defined then. It wouldn't be a supertask.

And would that sum not eventually terminate given a smallest sliver of time exists

If there's a smallest quanta of time, then there can be no physical supertasks.

I think, and then I do. The "force" which moves me comes from within me, and therefore cannot be described by Newton's conceptions of force. — Metaphysician Undercover

LOL. Tell that to the guy stranded 2 meters from his space ship without a tether. No amount of free will is going to get you back to it. You're going to need a little help from Newton.

The use of "physical" in this thread has gotten so ambiguous, that equivocation abounds everywhere. — Metaphysician Undercover

Yea, I noticed.

How much time elapses from travel to point a to point b and where is the object located during that time lapse? — Hanover

I'll attempt this. Michael talks about motion from A to B without there being a between. This can happen two ways.
1) Space is quantized. There simply isn't a location halfway between A and B. For a slow particle, this might mean that it spend quite a bit of time at A, and then suddenly is at B. That seems rather contradictory since one might ask what changes during all those times when the thing was at A. If it is at A twice, it is stationary and has no obvious state anywhere to go to B after some nonzero time.

2) Time is quantized, which is troublesome for fast particles. You have time 1 where the thing is at A, and time 2 where it is at B, quite a ways away. There are valid locations between A and B that the particle never visited since the time it should have been there is nonexistent.

All the above sort of presumes a naïve finite-automata sort of view of a quantized space and or time. It presumes a particle has a location (A, B) at a given state and time. Well that presumption was pretty much thrown out of the window with quantum theory.
A more realistic answer to your question comes from there. It says that you measure the particle at A, and later at B (maybe hours later). Where was the particle between those times? If not measured, it doesn't have a location. It does exist, but needs to be measured to have a location or (not and) a momentum.

what maintains its identity during that interval?

Physics has no concept of identity of anything. It is a human convention, a pure abstraction. Any given convention seems falsifiable by certain examples.

That’s a question for physicists to answer. — Michael

Why were you qualified to talk about it before but now I have to find a physicist on a Saturday morning to answer these follow up questions?

Some things can be dismissed on logical grounds, like the notion of continuous motion and the infinitely divisible half-way points an object in motion must then move through.

But one cannot use armchair philosophy to determine the smallest unit of space/time/movement.

It says that you measure the particle at A, and later at B (maybe hours later). Where was the particle between those times? If not measured, it doesn't have a location. It does exist, but needs to be measured to have a location or (not and) a momentum. — noAxioms

Assuming at the most microscopic level, the object is on an 8x8 chessboard. The pawn moves from e2 to e3. There is no e2.1 or other smaller increments in this finite world. At T1 it's at e2 and T30 it's at e3. The assumption is that at some point in time, it was no where while transitioning (moving?) from e2 to e3.

What empirical evidence is there that observations have been made of there being no object for some length of time and then it suddenly reappearing?

Assuming at the most microscopic level, the object is on an 8x8 chessboard. The pawn moves from e2 to e3. There is no e2.1 or other smaller increments in this finite world. At T1 it's at e2 and T30 it's at e3. The assumption is that at some point in time, it was no where while transitioning (moving?) from e2 to e3. — Hanover

We can't examine this at the macroscopic scale. At whatever the smallest scale is: at Time1 it's at Location1 and at Time2 it's at Location2. There's no intermediate Time1.5 where it doesn't exist or Location1.5 that it moves through.

If it's at L-1 at T-1 and L-2 at T-2, how long did it take to get from L-1 to L-2? If the answer is 0, then it was at L-1 and L-2 at the same time because if T-2 minus T-1 = 0, then T-1 = T-2.