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? — Hanover

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc.

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc. — Michael

You continue to refuse to acknowledge the difference between the measurement and the thing measured. T1 and T2 are points designated by the measurer, therefore a feature of the measurement. The measurement is the difference between T1 and T2. However, the thing measured is the passage of time which occurs. Your confusion is due to your refusal to acknowledge a distinction between the measurement (the specified number of seconds) and the thing measured (the passage of time). You've been insisting that the thing measured is a number of seconds, rather than recognizing that seconds is the measurement, not the thing which is measured. And so I gave up trying to explain to you the difference.

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. — noAxioms

No one said free will has infinite capacity? Obviously we are limited by the circumstances we are in. But limitations are not absolutely either. So free will has it's own niche, to act according to a judgement of the circumstances.

In the circumstances you describe, an appeal to Newton would not help the poor soul, but a radio call to someone inside the spaceship, to please shoot me a line, might help. That demonstrates the benefit of free will, allowing one to act according to a judgement of the circumstances. And. it demonstrates how free will could actually get the person back to the space ship, in contrast to your suggestion of asking Newton to help, which of course, would be useless.

However, the thing measured is the passage of time which occurs. — Metaphysician Undercover

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it.

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc. — Michael

The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B.

The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B. — Hanover

You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points.

That seems to be begging the question.

Best I can do is point you to something like quantum spacetime and quantum gravity.

There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false.

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity.

Let me start by saying my previous post here was poorly written. Now,

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. — fishfry

There is no ∞-th item of a series. [...] But 1 is the limit, it's not a member of the sequence. — fishfry

The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence $a_{n}=1-0.5^{n}$ converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ∞-th item. Cool.

The issue that I see is:
1 – if we admit that time is infinitely divisible;
2 – and we admit that $a_{n}=1-0.5^{n}$ gives us the lenght covered by Achilles in the Zeno Walk at each step;
the walk only finishes if it accomplishes an infinite amount of steps. Right?

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed?; if so, is there not a corresponding state for the mechanism when the full time elapses?
In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run?

I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts.

Does that make sense?

1 – Is that also the case for non-standard analysis and arithmetic?



–––––

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

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

But there is that for an advantage of continuous time over discrete.

–––––

They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false. — Michael

Same here. General relativity is suspected to break down at high enough energies or small enough scales — where the quantum effects can't be ignored —, "like" Newtonian theory breaks down when v or Gm/R become large enough.

You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points. — Michael

I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right?

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

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

I don't think you can have it both ways.

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it. — Michael

I'm not quite sure what you are saying. Do you think that the passage of time occurs when we can't measure it? Analogously (if that's a word), if we can't measure the location or momentum of an object, it doesn't have them? Does that mean that it doesn't exist?
There are two philosophies that I can think of that would justify those views. One of them is Logical Positivism, which was developed precisely to justify both Quantum Mechanics and General Relativity. The other is Bishop Berkeley's idealism. Which do you hold?

There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false. — Michael

That means you think it is possible that space-time is continuous at the quantum level. Interesting. But I suppose it fits with your acceptance of continuous space-time in mathematics.
The empirical evidence for your position is the empirical fact that we can't measure very small units of time or space. I'm not sure that constitutes convincing empirical one way or the other. Or have I got something wrong?

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

Which ones do you have in mind? You mentioned the problems with a converging series. But that's a mathematical problem, not an empirical one. How does empirically non-continuous space and time solve those issues?

I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right? — Hanover

I don't think the question makes sense, but you'll have to ask a physicist who knows more about quantum gravity to explain it. I can only point out to you that there are physical theories that take spacetime to be discrete.

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it. — Michael

Yes, I agree with that. But, there is no "seconds" inherent in that passage of time, nor does it appear like there are any natural points for division within that passage of time, which appears to us to be absolutely continuous. This is why we assume principles which allow for infinite divisibility of time, because we see no reason for any real restrictions to its division. Therefore we tend to believe that we can simply insert a point (T1) at any random place, and another point (T2) at another random place, and determine the amount of time that has passed between those two arbitrarily assigned points.

Now, in your opening post in the thread, you concluded a "metaphysically necessary smallest period of time", and you used reference to the empirically based principle "60 seconds will pass" to support this conclusion. Therefore you've exposed inconsistency between two empirically based principles. The one principle being the assumption that the passing of time is continuous, as it appears, and the consequent principle that we can arbitrarily insert points, and divide it in absolutely any way that we please. The other principle being that "60 seconds will pass". There is inconsistency because the former leads to the example of the stairway to hell in the op, in which there is always more steps, and more time to pass, before 60 seconds can pass.

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

Since you dismiss general relativity as probably false, then there is no need to maintain "spacetime". When we analyze space and time separately, then one might be discrete, and the other continuous. Logically, motion, which is a change of spatial location (place) requires the passing of time. We cannot conceive of a change in place without time passing because that implies the thing is in two different places at the same time. However, when time is separated from the constraints of spatial change we can conceive of time passing without spatial change. This allows that spatial change occurs as discrete 'quantum leaps', position at T1, to position at T2, without any spatial continuity between them. Between T1 and T2 there would be time passing, but no spatial change until that time has passed. That passage of time during which spatial change does not occur, is justified by activity at a deeper level, non-spatial, or immaterial activity, which determines the relationship between the spatial positions at T1 and the spatial positions at T2.

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

I discussed that in my post, but you quoted the bit at the bottom which abandons the chessboard model in favor of quantum mechanics, calling the former model a naïve

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?

None, but there's also no evidence that it is there when not being measured. It's all about measurement and not about discreetness.

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? — Hanover

In that frame, it took time 1 to get from T-1 to T-2. That's pretty obvious, no? In natural units, that's light speed.

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.

If the answer is zero, then T-2 is no-t when it is at L-2.
In computer jargon, what you are describing is 'jaggies', the tendency of 'straight' lines to appear jagged when displayed on say your computer screen, a discreet array. An object that moves fast (faster than one L per T) will either be at multiple locations at the same time, or it will skip all the locations between and only be at one location per time.
I've played a game with the latter physics. I could get my ship to go super fast and go straight for the enemy blocking my way. If I did it right, I would be in front of him at one time unit, and beyond him the next time unit, apparently passing right through without collision because there was no time 1.5 where I was where he was.

More problems with that model: If the particle is moving at 0.7 per time unit, it is never at a location in space except every 10 time units where you find it 7 units from where it was before. It can't be anyplace between since it is never at a space location at the same time as a time quanta. This is silly. You probably need to fill in the dots between, but then the motion is erratic rather than sporatic.

I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right? — Hanover

No, they're 0,1 from each other, which isn't zero. One of the coordinates is different.

Anyway, you seem to see the sorts of contradictions that arise from such a naive model. If space and time is discreet, quantum mechanics describes it far better than the chessboard model.

the walk only finishes if it accomplishes an infinite amount of steps. Right? — Lionino

Right

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed?

By completing all the steps. This is not a contradiction.

if so, is there not a corresponding state for the mechanism when the full time elapses?

Not any more than there is a last natural number. I'm presuming you're talking about the state of something like the lamp. The state of Achilles is easy: He's where the tortoise is.

I don't see a problem until the premise of a last step is introduced, which is by definition contradictory.

given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, — Michael

They're both incomplete, just like
Newtonian mechanics was incomplete, but not false. OK, parts of it were outright false, but it's still taught in (pretty much) any school. GR definitely breaks down at small scales.

No one said free will has infinite capacity? — Metaphysician Undercover

I didn't say infinite capacity. I denied that your free will has any capacity at all, since even the most trivial capacity would get you back to your ship 2 meters away, even if not quickly.

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

The spaceship example shows this to be nonsense. It would be a revolution indeed if anybody could do that.

a radio call to someone inside the spaceship, to please shoot me a line, might help. That demonstrates the benefit of free will — Metaphysician Undercover

Free will isn't necessary to do any of that. A robot has the same capacity to make such a call, and robots by definition lack it. This is also utterly off topic to this discussion, but I took the easy bait anyway.

I didn't say infinite capacity. I denied that your free will has any capacity at all, since even the most trivial capacity would get you back to your ship 2 meters away, even if not quickly. — noAxioms

I told you how the person gets back to the ship using free will. That's one point for free will, zero for you.

A robot has the same capacity to make such a call, and robots by definition lack it. — noAxioms

A robot cannot decide whether or not to make the call, a person can. The person could decide not to, if perhaps the release of the tether was intentional. Two for free will, zero for you.

This is also utterly off topic to this discussion, but I took the easy bait anyway. — noAxioms

It's not off topic, because there is an issue of what is "physically possible", and whether physical possibility" is limited by the laws of physics. My argument is that there is a number of physical activities such as the effects of dark matter and dark energy, which violate the laws of physics. Furthermore, free will violates Newton's first law, and it causes physical movements. Therefore physical possibility is not limited by the laws of physics.

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". — Ludwig V

Which includes magic lightbulbs and staircases? I'm open minded, I don't think I can predict the future. Even a few hundred years ago nobody could imagine the science and technology of today.

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. — Ludwig V

Yes ok

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

Not too strenuously. As I mentioned I don't place as much metaphysical import on these puzzles.

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. — Ludwig V

Have mostly been talking about the lightbulb. Haven't talked about Achilles or Zeno.

But it's not the natural numbers that are parallel. It's the natural numbers augmented by the point at infinity. That's my conceptual setting for these problems.

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. — Ludwig V

There's an order-isomorphism between the ordered sets {1/2, 3/4, 7/8, ..., 1} and {1, 2, 3, ..., $\omega$. One of the virtues of abstraction is that it lets us see that two seemingly different things are really the same, when we only focus on certain attributes. Both these sets are an infinite sequence followed by an extra element. Their order properties are the same. It's no different than playing chess on a board with purple and green squares versus red and black squares. It's the same game with respect to the rules of the game, even though they're different in other respects.

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. — Ludwig V

But if I did apply that frame, then Zeno would have a good point. I did somehow either 1) accomplish infinitely many tasks in finite time; or b) The world's not continuous like the real numbers.

I think Zeno had a very good point, and I don't accept the common wisdom that summing an infinite series solves the problem.

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.[or/quote]

If time is made of instants, then from instant to instant, how do things know what to do next? Where is the momentum and velocity information stores? It's like a computer program where an object has associated with it a data structure containing information about the object. If I shoot an arrow, where is the arrow's data structure stored? I think it's a good question. But I've never really given a lot of thought to the matter. It all seems to work out.
— Ludwig V

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. — Ludwig V

That's a good question too. How do dimensionless points form lines and planes and solids?

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. — Ludwig V

If you agree, I'm happy, because that's the only point I'm making. I've never written more just say less. My only point is that in the lamp and these other problems, we're not defining the state at the limit. Therefore the choice of state is pretty much arbitrary. If we thought about it that way it might be more clear.

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? — Ludwig V

Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings.

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. — Ludwig V

If I can walk from one end of the grocery aisle to the other, I don't see why you can't get down the staircase, infinitely many steps or not.

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. — Ludwig V

It's a thought experiment. There are no infinite staircases.

But I did walk through infinitely many inverse powers of two lengths at the grocery store. I did sum an infinite series in finite time. So there's something interesting going on.

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. — noAxioms

I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute?

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. — noAxioms

Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.

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. — noAxioms

Point is you can define the state at the limit of a sequence to be anything you want. The lamp could turn into a pumpkin too. The premises of the problem don't forbid it.

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. — noAxioms

I looked it up, didn't seem to find a definitive version.

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. — noAxioms

Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not. If you number them 1, 2, 3, ... then the vase is empty at the end, since every ball eventually gets taken out. But if they're not numbered, the vase will have infinitely many balls because you're always adding another 9. Is that about right?

The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence an=1−0.5n
...
converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ∞-th item. Cool. — Lionino

I said no such thing!! If you like, you can think of the limit as being the $\infty$-th item.

That is, if 1/4, 3/4, 7/8, ... are the first, second, third, etc. terms of an infinite sequence with limit 1, then 1 may be sensibly taken as the $\infty$-th item, or as I've been calling it, the item at $\omega$, which is traditional in this context.

So I believe I've been trying to get across the opposite of what you thought I said. There is an $\infty$-th item, namely the limit of the sequence.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.

The issue that I see is:
1 – if we admit that time is infinitely divisible;
2 – and we admit that an=1−0.5n
[bad markup omitted]
gives us the lenght covered by Achilles in the Zeno Walk at each step;
the walk only finishes if it accomplishes an infinite amount of steps. Right? — Lionino

I think trouble ensues when you try to apply abstract math to the physical world. I certainly can walk across the room, clearly accomplishing infinitely many Zeno-steps in finite time. I have no explanation nor does anyone else. The common explanation that calculus lets us sum an infinite series, I reject. Because that's only a mathematical exercise and has no evidentiary support in known physics.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? — Lionino

No.

If not, how is the walk ever completed? — Lionino

In math? Via the standard limiting process. In physics? I don't know, I'm not a physicist. But the physicists don't know either. They don't regard it as a meaningful question.

; if so, is there not a corresponding state for the mechanism when the full time elapses? — Lionino

Nobody knows the answer to any of these questions.

In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run? — Lionino

The justification is purely mathematical. Physics doesn't support these notions since we can't reason below the Planck length.

I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts. — Lionino

The metaphysical meaning is perfectly clear within the math. I don't know how it works in the physical worldl

Does that make sense? — Lionino

The math is clear. The physics is unknown. But motion is commonplace.

1 – Is that also the case for non-standard analysis and arithmetic? — Lionino

.999... = 1 is a theorem of nonstandard analysis. I don't see how it could help. I don't know if anyone's thought about applying NSA to these puzzles.

If you agree, I'm happy, because that's the only point I'm making. I've never written more just say less. My only point is that in the lamp and these other problems, we're not defining the state at the limit. Therefore the choice of state is pretty much arbitrary. If we thought about it that way it might be more clear. — fishfry

:smile:

If I can walk from one end of the grocery aisle to the other, I don't see why you can't get down the staircase, infinitely many steps or not. — fishfry

But I did walk through infinitely many inverse powers of two lengths at the grocery store. I did sum an infinite series in finite time. — fishfry

I find this very confusing. I take your point about abstraction. But I find that abstraction can create confusion, because it persuades us to focus on similarities and neglect differences. My reaction here is to pay attention to the difference between these kinds of infinite series. It's not meant to contradict the abstraction.

It's a thought experiment. There are no infinite staircases. — fishfry

Exactly. So it isn't about physics. But it isn't about mathematics either. So it seems to me an exercise in imagination, and that provides a magic wand.

Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings. — fishfry

Deep? or Deepity? (RIP Dennett)

How do dimensionless points form lines and planes and solids? — fishfry

Yes. Euclid (or Euclidean geometry at least) starts from a foundation - axioms and definitions. But they are an extension of our common sense processes of measuring things. (You can understand more accurate and less accurate measurements.) Extend this without limit - Hey Presto! dimensionless points! That is, to understand what a point is, you have to start from lines and planes and solids and our practice of measuring them and establishing locations. I find that quite satisfying. Start with the practical world, generate a mathematics, take it back to the practical world. (Yes, I do think that actual practice in the real world is more fundamental than logic.)
Once you define geometrical points in that context, there is no difficulty about passing or crossing an infinite number of points. (But the converging series does not consist of points, but of lengths, which are components.)

So there's something interesting going on. — fishfry

My supervisor used to say that when he got really excited, which was not often.

I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute? — fishfry

No. Nobody seem to have suggested that was possible. It simply isn't a supertask.

Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.

Yes, I mean that, and it's not a mystery to me. If spacetime is continuous, then it's an example of a physical supertask and there's no contradiction in it.

The lamp could turn into a pumpkin too.

No, the lamp changes things. It introduces a contradiction by attempting to measure a nonexistent thing. That in itself is fine, but the output of a non-measurement is undefined.

I looked up [Bernadete's Paradox of the God], didn't seem to find a definitive version.

Nicely stated by Michael in reply 30, top post of page 2 if you get 30 per page like I do.

Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not.

It's important to the demonstration of the jar being empty, so yes, it makes a difference.

If you number them 1, 2, 3, ... then the vase is empty at the end, since every ball eventually gets taken out. But if they're not numbered, the vase will have infinitely many balls because you're always adding another 9. Is that about right?

The outcome seems undefined if they're not numbered since no bijection can be assigned, They don't have to have a number written on them, they just need to be idenfifed, perhaps by placing them in order in the jar, which is a 1-ball wide linear pipe where you remove them from the bottom.

It nicely illustrates that ∞*9 is not larger than ∞, and so there's no reason to suggest that the jar shouldn't be empty after the completion of the supertask. Again, it seems that any argument against this relies on a fallacious assumption of a last step that sooo many people are making in this topic.

So I believe I've been trying to get across the opposite of what you thought I said. There is an ∞-th item, namely the limit of the sequence. — fishfry

That can't be a step, since every step in a supertask is followed by more steps, and that one isn't. I have a hard time with this ∞-th step.

The common explanation that calculus lets us sum an infinite series, I reject. Because that's only a mathematical exercise and has no evidentiary support in known physics. — fishfry

The cutting up of the path into infinite steps was already a mathematical exercise. The fact that the physical space can be thus meaningfully cut up is true if the space is continuous. That latter one is the only barrier, since it probably isn't meaningfully, despite all our naïve observations about the nice neat grid of the chessboard.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed — Lionino

As has been stated so many times, by performing all the steps, which happens in finite time no problem. There is a final step only in a finite sequence, so using a finite definition of 'complete' is inapplicable to a non-finite task.

In math? Via the standard limiting process. In physics? I don't know, — fishfry

In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.

Physics doesn't support these notions since we can't reason below the Planck length.

Which is to say that space isn't measurably continuous, so the walk isn't measurably a supertask. I would agree with that.

How do dimensionless points form lines and planes and solids? — fishfry

Mathematics: by not having a last one (or adjacent ones even). Physics: There are no solids.

(But the converging series does not consist of points, but of lengths, which are components.) — Ludwig V

Yes. The latter is a countable set of lengths. The set of points on say a circle is an uncountable set

A robot cannot decide whether or not to make the call, a person can. — Metaphysician Undercover

That's quite the assertion. Above and beyond the usual conservative stance.

The point of my example with the ship was to counter your assertion of Newton forces not being necessary to move and free will being enough. I said you'd need help from Newton. Asking for a line to be thrown to you is you admitting the help from Newton was necessary. That's what the tether is: a way to do it by exerting an external force, since the free will couldn't do it itself.

Which is to say that space isn't measurably continuous, — noAxioms

I take it you are talking about physical space, not mathematical space?

Physics: There are no solids. — noAxioms

But there are 3-dimensional figures in physics, aren't there? It's the solidity that's the problem, isn't it?

The set of points on say a circle is an uncountable set — noAxioms

That's a surprise to me. One can measure or calculate the length of a circumference, can't one? Or is uncountability a consequence of the irrationality of "pi"?

As has been stated so many times, by performing all the steps, which happens in finite time no problem. — noAxioms

Just checking - by "step" do you mean stage of the series? If I am travelling at a steady speed, I will complete more and more steps in a given period of time, and that number (of steps) will approach (but not reach) infinity. Can that really work?

The cutting up of the path into infinite steps was already a mathematical exercise. — noAxioms

So is the cutting up of the path into standard units. It's just a question of choosing the appropriate mathematical calculation for the task at hand.

I said no such thing!! If you like, you can think of the limit as being the ∞-th item. — fishfry

There is an ∞-th item, namely the limit of the sequence.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end. — fishfry

then 1 may be sensibly taken as the ∞-th item, or as I've been calling it, the item at ω — fishfry

Then you say.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1?
— Lionino

No. — fishfry

Is there not a contrast between these two sets of statements?

I think trouble ensues when you try to apply abstract math to the physical world — fishfry

We are applying mathematics not just to this physical world but to any possible world where the physics could be different, and for that we discuss what the mathematics means in the world — as it is necessary that 1+1=2 so that everytime you take one of something and one again you end up with two.

I take it you are talking about physical space, not mathematical space? — Ludwig V

Yes. 'Planck [pretty much anything] is a physical concept, not a mathematical one. In mathematics, there is no number smaller than can be meaningfully discussed.

But there are 3-dimensional figures in physics, aren't there? It's the solidity that's the problem, isn't it?

Sure. A rock, at a given time, is a 3 dimensional thing. A rock, it's entire worldline, is a 4 dimensional thing. Correct. It isn't a solid. You can measure a piece of it at a sort of 4D 'point', an event. The rock worldline consists of a collection of such point events, a huge number, but not infinite. They're not really points since position and momentum cannot be both known, so you can know one or the other or an approximate combination of both.

One can measure or calculate the length of a circumference, can't one? Or is uncountability a consequence of the irrationality of "pi"?

Yes, one can calculate the circumference. No, the irrationality of pi is irrelevant. It could be a line segment of length 1. You know the length, and it isn't irrational, but the segment still consists of an uncountable number of points. There's no part of the segment that isn't a point (or a set of them), and yet points have no size, so no finite number of them can actually fill a nonzero length of that segment.

Just checking - by "step" do you mean stage of the series. If I am travelling at any spead, I will complete more and more steps in a given period of time, and that number (of steps) will approach (but not reach) infinity.

Yes, a step is a finite (nonzero) duration, like the first step is going halfway to the goal. Each step goes half the remaining way to the goal. Those are steps. You complete all the steps by time 1, so the task is then complete. No contradiction so long as we don't reference 'the highest natural number' which doesn't exist.

So is the cutting up of the path into standard units. It's just a question of choosing the appropriate mathematical calculation for the task at hand.

One must define how the task is divided into steps in order to tell Zeno's story. There are multiple ways to do it, but to be a supertask, the steps need to get arbitrarily small somewhere, and the most simple way to do that is at the beginning or the end of the task. How one abstractly divides the space has no effect on the actual performance of the task. One can argue that all tasks of any kind are supertasks because one can easily divide any finite duration into infinite parts, but the much of the analysis of doing so relies on the mathematics of countable infinities.

So I can go from 0 to 1 and assign a 'step' to every zero-duration point between those limits. That can be done, and can be completed, but since the steps are not countable, it is hard to draw any conclusion from it all.

Then you say. — Lionino

That's me saying something, not fishfry.

I personally don't like the ∞-th step, but it works. The supertask is completed, then the ∞-th step is taken after that. The supertask had all nonzero duration steps, and this additional step has no duration. I don't find it wrong, but I find it needless.

Is there not a contrast between these two sets of statements?

I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there. If fishfry wants to add an addition single step after the supertask completes, that's fine, but it isn't a step of the supertask.

That's me saying something, not fishfry. — noAxioms

Oops :monkey: fixed.

then the ∞-th step is taken after that — noAxioms

The problem I was trying to point out that is that, if we admit a ∞-th step, this step should be associated with a state in one of those mechanisms Michael made up.

I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there — noAxioms

I agree with that too. In the end, I don't think reasoning about infinity gets us anywhere.

The problem I was trying to point out that is that, if we admit a ∞-th step, this step should be associated with a state in one of those mechanisms Michael made up. — Lionino

Michael's mechanisms (some of which he made up) are not resolved by addiing a single step task to the supertask. The supertask reaches 1 when all the steps are completed. It isn't sort of 1, it's there since where else would it be? The arguments against that suggest some sort of 'point immediately adjacent to, and prior to 1', which is contradictory. There are no adjacent points in continuums.

I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there
— noAxioms

But I don't agree that 1 is not reached by the completion of the supertask. Only that 1 is not reached by any step.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end. — fishfry

I think that's all right. When I walk a mile, I start a potentially infinite series of paces. When I have done (approximately) 1,760 of them, I stop. The fact that the 1,760th of them is the last one is, from the point of the view of the sequence, arbitrary, not included in the sequence . The sequence itself could continue, but doesn't.

There's no part of the segment that isn't a point (or a set of them), and yet points have no size, so no finite number of them can actually fill a nonzero length of that segment. — noAxioms

OK. Is that because they have no dimension - are not a part of the line?

The problem I was trying to point out that is that, if we admit a ∞-th step, this step should be associated with a state in one of those mechanisms Michael made up. — Lionino

Because <the infinity symbol> can't be associated with any natural number?

I agree with that too. In the end, I don't think reasoning about infinity gets us anywhere. — Lionino

Then the ultimate paradox is that there seems to be no end to the reasoning.

You complete all the steps by time 1, so the task is then complete. No contradiction so long as we don't reference 'the highest natural number' which doesn't exist. — noAxioms

I don't quite understand. Is the point that the simple arithmetic analysis doesn't reference the highest natural number, so that way of reaching it is OK. It doesn't look like completing all the steps to me - it looks more like jumping over them. But I have travelled over all the spaces.

I find this very confusing. I take your point about abstraction. But I find that abstraction can create confusion, because it persuades us to focus on similarities and neglect differences. My reaction here is to pay attention to the difference between these kinds of infinite series. It's not meant to contradict the abstraction. — Ludwig V

The two sets in question have the same order type, denoted $\omega + 1$. That's mathematically true, and it's all that's relevant to these two examples. I'm not sure what's gained by focussing on the differences. $\omega$ is the limit of 1, 2, 3, ... in exactly the same sense that 1 is the limit of 1/2,, 3/4, 7/8, ..., under the more general topological definition of a limit needed to defined limits among the ordinal numbers.

It's a thought experiment. There are no infinite staircases.
— fishfry
Exactly. So it isn't about physics. But it isn't about mathematics either. So it seems to me an exercise in imagination, and that provides a magic wand. — Ludwig V

Yes ok, so the coach can turn into a pumpkin and the lamp can turn into a plate of spaghetti. Are you agreeing with me on that point?

Aha. You'd have to ask those who care so much. I think they only show that underspecified problems can have arbitrary answers. But others see deeper meanings.
— fishfry
Deep? or Deepity? (RIP Dennett) — Ludwig V

RIP.

Yes. Euclid (or Euclidean geometry at least) starts from a foundation - axioms and definitions. But they are an extension of our common sense processes of measuring things. (You can understand more accurate and less accurate measurements.) Extend this without limit - Hey Presto! dimensionless points! That is, to understand what a point is, you have to start from lines and planes and solids and our practice of measuring them and establishing locations. I find that quite satisfying. Start with the practical world, generate a mathematics, take it back to the practical world. (Yes, I do think that actual practice in the real world is more fundamental than logic.)
Once you define geometrical points in that context, there is no difficulty about passing or crossing an infinite number of points. (But the converging series does not consist of points, but of lengths, which are components.) — Ludwig V

Ok

So there's something interesting going on.
— fishfry
My supervisor used to say that when he got really excited, which was not often. — Ludwig V

[/quote]

yes

I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute?
— fishfry
No. Nobody seem to have suggested that was possible. It simply isn't a supertask. — noAxioms

Ok. @Michael has been using that as an example of a supertask so I can't say. I haven't studied them much.

Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.
Yes, I mean that, and it's not a mystery to me. If spacetime is continuous, then it's an example of a physical supertask and there's no contradiction in it. — noAxioms

Ok. Perhaps you and @Michael could hash this out. He thinks supertasks are metaphysically impossible, and you think they're everyday occurrences. I'm agnostic on the matter except to say that I don't think they're metaphysically impossible, whether they're physically possible or not.

No, the lamp changes things. It introduces a contradiction by attempting to measure a nonexistent thing. That in itself is fine, but the output of a non-measurement is undefined. — noAxioms

The state of the lamp is defined at each of the times 1/2, 3/4, 7/8, ... but it's not defined at 1.

Like any other function defined at some elements of a set but not others, I am free to define it any way I like.

I looked up [Bernadete's Paradox of the God], didn't seem to find a definitive version.
Nicely stated by Michael in reply 30, top post of page 2 if you get 30 per page like I do. — noAxioms

Thanks I'll check it out.

Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not.
It's important to the demonstration of the jar being empty, so yes, it makes a difference. — noAxioms

Something went wrong with the quoting when I quoted your post. Anyway ... yes the ping pong balls. I have no opinion about that one.

The outcome seems undefined if they're not numbered since no bijection can be assigned, They don't have to have a number written on them, they just need to be idenfifed, perhaps by placing them in order in the jar, which is a 1-ball wide linear pipe where you remove them from the bottom. — noAxioms

ok

It nicely illustrates that ∞*9 is not larger than ∞, and so there's no reason to suggest that the jar shouldn't be empty after the completion of the supertask. Again, it seems that any argument against this relies on a fallacious assumption of a last step that sooo many people are making in this topic. — noAxioms

I'll agree that the subject of omega sequence paradoxes is full of fallacious assumptions and confused thinking.

So I believe I've been trying to get across the opposite of what you thought I said. There is an ∞-th item, namely the limit of the sequence.
— fishfry
That can't be a step, since every step in a supertask is followed by more steps, and that one isn't. I have a hard time with this ∞-th step. — noAxioms

I say "item" and you change the word to "step," changing my meaning. I agree, it's not a step in a sequence. It's an item in a set.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set. You can think of 0 as the infinitieth item in an ordered set, but not the infinitieth step of a sequence.

The cutting up of the path into infinite steps was already a mathematical exercise. The fact that the physical space can be thus meaningfully cut up is true if the space is continuous. That latter one is the only barrier, since it probably isn't meaningfully, despite all our naïve observations about the nice neat grid of the chessboard. — noAxioms

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length. With our present understanding of the limitations of physics, the question of the ultimate nature of space is metaphysics and not physics.

In math? Via the standard limiting process. In physics? I don't know,
— fishfry
In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation. — noAxioms

Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.

How do dimensionless points form lines and planes and solids?
— fishfry
Mathematics: by not having a last one (or adjacent ones even). — noAxioms

Not sure what you mean. The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.

The point of my example with the ship was to counter your assertion of Newton forces not being necessary to move and free will being enough. I said you'd need help from Newton. Asking for a line to be thrown to you is you admitting the help from Newton was necessary. That's what the tether is: a way to do it by exerting an external force, since the free will couldn't do it itself. — noAxioms

It seems you misunderstood.

I said no such thing!! If you like, you can think of the limit as being the ∞-th item.
— fishfry

There is an ∞-th item, namely the limit of the sequence.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.
— fishfry

then 1 may be sensibly taken as the ∞-th item, or as I've been calling it, the item at ω
— fishfry

Then you say.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1?
— Lionino

No.
— fishfry

Is there not a contrast between these two sets of statements? — Lionino

No. Consider the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ...

It has the limit 0.

We may form the ordered set {1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}. It's a perfectly sensible set.

In this context 0 is the largest element in the set. It's the final "item" if you like. But 0 is not any step in the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, ..., and that sequence has no last step.

Is my use of the words step and item more clear?

We are applying mathematics not just to this physical world but to any possible world where the physics could be different, and for that we discuss what the mathematics means in the world — as it is necessary that 1+1=2 so that everytime you take one of something and one again you end up with two. — Lionino

Ok. But one has to be careful of applying math to the world, this one or any other. Physicists typically model time as a real number, but there's no evidence that time is a continuum as the real numbers are. So math gets applied to physics heuristically or pragmatically, and not metaphysically. We model time using the real numbers because it's handy and gets us results, not because we actually believe time is like the real numbers.

The sequence itself has no last item. But the "augmented sequence," if you call it that, does. We can simply stick the limit at the end.
— fishfry
I think that's all right. When I walk a mile, I start a potentially infinite series of paces. When I have done (approximately) 1,760 of them, I stop. The fact that the 1,760th of them is the last one is, from the point of the view of the sequence, arbitrary, not included in the sequence . The sequence itself could continue, but doesn't. — Ludwig V

I was making my point about mathematical convergent sequences. Don't know whether it strictly applies to walking.

OK. Is that because [points] have no dimension - are not a part of the line? — Ludwig V

They are part of the line. Yes, a point is dimensionless, size zero. Any sum of a finite bunch of zeros is zero. But the number of points on a line segment isn't finite.

Ok. Perhaps you and Michael could hash this out. He thinks supertasks are metaphysically impossible — fishfry

Perhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't.

I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set. — Ludwig V

I have no problem with any that.

You can think of 0 as the infinitieth item, but not the infinitieth step.

OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ∞-1.

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length.

But you can traverse the space of that step, even when well below the Planck length.


In math? Via the standard limiting process. In physics? I don't know,
— fishfry
In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.
— noAxioms

Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.

The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.

So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points.

Ok. Perhaps you and Michael could hash this out. He thinks supertasks are metaphysically impossible
— fishfry
Perhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't. — noAxioms

Well between the two of you I have no idea what a supertask is anymore.

I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion. — noAxioms

I tend to agree with you, that supertasks either (a) may be physically possible via the physics of the future; or (b) are already possible when I go from the living room to the kitchen for a snack, first traversing half the distance, then half of the remaining half, and so forth, and somehow miraculously arriving at my refrigerator. Which keeps things cold in a warm room, in clear violation of the second law of thermodynamics. Truly we live in remarkable times.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set.
— Ludwig V
I have no problem with any that.

You can think of 0 as the infinitieth item, but not the infinitieth step.
OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ∞-1. — noAxioms

You believe in limits, you said so. And if you believe even in the very basics of set theory, in the principle that I can always union two sets, then I can adjoin 1 to {1/2, 1/3, 1/4, 1/5, ...} to create the set {1/2, 1/3, 1/4, 1/5, ..., 1}.

It's such a commonplace example, yet you claim to not believe it? Or what is your objection, exactly? It's an infinite sequence. I stuck the number 1 on the end. The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set? It's a perfect description of what's going on. And it's a revealing and insightful way to conceptualize the final state of a supertask. Which is why I'm mentioning it so often in this thread.

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length.
But you can traverse the space of that step, even when well below the Planck length. — noAxioms

Only mathematically, In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length. Maybe space is continuous. Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes. Maybe something entirely different and not yet imagined is going on. We just don't know.

But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.

In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation. — noAxioms

Well yes, motion is possible. That's one response to Zeno. Not so satisfactory though. Did I complete a supertask when I got up to go to the kitchen for a snack? I have no idea, even though motion through space within an interval of time is an every day occurrence.

The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.
So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points. — noAxioms

Ok. I thought you were claiming supertasks had to related to open intervals.