Ok I hadn't seen that before. Whatever shows at the end (if that even makes sense) it's certainly finite, since you're adding up finitely many finite numbers then resetting to 0. — fishfry

The counter resets to 0 after 9. It will only ever show the digits 0-9 — Michael

I misunderstood the question to mean it kept counting into infinity.

That is, the sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ...

That sequence doesn't converge. — fishfry

True. It seems to be Thompson's lamp but with 10 different states instead of 2 (on and off). In which case the solution of the ball skipping on the table does not work immediately. But the issue does seem to be missing limits as well.

I would propose a parametric curve on the ball path, and, for fantasy sake, by whatever mechanism, the plate knows at what part of the parabola the ball is at, defining the counter. As time goes on, the revolution gets smaller and smaller. Eventually the ball will completely rest on the table, which is 0:



Preliminarly this seems like a solution.

I don't see how that follows at all. No mathematical thought experiment can determine the nature of reality. We can use math to model Euclidean geometry and non-Euclidean geometry, but math can never tell is which is true of the physical world. You can use math to model and approximate, but it is never metaphysically conclusive. — fishfry

Agreed:

So concluding something about the nature of time from thought experiments seems to put the horses behind the chariot or maybe to be analogous to ontological arguments, where we conclude something about the world by relying our own, perhaps mistaken, human intuitions. — Lionino

-

By the way the Thompson's lamp sequence is 1, 0, 1, 0, 1, 0, ... and that doesn't converge either. — fishfry

Yes, but check the solution at https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp

But then I am interested in a counter that would indeed count to infinity when it gets to 30 seconds. Then I wonder, what would it show at the 60th second?

But then I am interested in a counter that would indeed count to infinity — Lionino

Assume the counter counts to infinity. After 30 (or 60) seconds, what is the first digit of the number it shows?

If it does count to infinity, I am not sure if it would show any natural number :sweat:

But the counter only shows the standard 0-9 digits. At no point does it switch from showing some natural number to simply showing the ∞ symbol.

To repeat what I said earlier: with these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.

Supertasks are illogical. Time cannot be infinitely divisible.

In which case I would say the counter is missing a key feature for the mission. Perhaps it would become an infinitely long counter showing an infinitely long line of 9s or 1s or 3s (does it even make any difference?). The counter has to be only metaphysically possible, not physically possible, and we already have the possibility of infinity as an assumption. But that is for 30s, I have no clue what would follow from 60s. If our mathematical descriptions are representative of quantities in a continuous, infinitely divisible space-time, perhaps we would be dealing with transfinite numbers.

You seem to take issue with that first paragraph, but your reasoning against it doesn't make any sense. Unless the universe ceases to exist then 60 seconds is going to pass. The passage of time does not depend on the counter. — Michael

In logic we must follow the premises regardless of truth or falsity. Your example makes premises which describe a machine doing what has been called a "supertask". We have no premises to say that a supertask is impossible, only the premises which describe an instance of doing it, therefore demonstrating the logical possibility of a supertask.

Now, you introduce another premise, "Unless the universe ceases to exist then 60 seconds is going to pass". This premise contradicts what is implied by the others which describe the supertask.

So, what we have is a contradiction, without the information required to resolve the contradiction. In your reply to fishfry, you simply choose "60 seconds is going to pass", and conclude "supertasks are not possible". But this choice is made without the required argument, it simply reveals your prejudice.

we already have the possibility of infinity as an assumption — Lionino

And that assumption entails a contradiction, proving the assumption false.

Now, you introduce another premise, "Unless the universe ceases to exist then 60 seconds is going to pass". This premise contradicts what is implied by the others which describe the supertask. — Metaphysician Undercover

No it doesn't.

And that assumption entails a contradiction — Michael

What contradiction?

What contradiction? — Lionino

That the counter doesn't show 0 and doesn't show 1 and doesn't show 2 and doesn't show 3 and doesn't show 4 and doesn't show 5 and doesn't show 6 and doesn't show 7 and doesn't show 8 and doesn't show 9 even though it must show exactly one of them.

Perhaps it would become an infinitely long counter showing an infinitely long line of 9s — Lionino

And since the counter is infinitely long, there is no first digit.

And what about rejecting the premise of the counter being apt for the task? You've designed a counter that is metaphysically constrained in such a way that it cannot perform the metaphysical task given in the way you want it to perform. What if you employ a counter that can show ∞?

Let's move away from numbers as that is clearly causing some confusion.

After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

What colour is the square when this supertask completes (after 60 seconds)?

After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on. — Michael

That seems to be a Thompson's lamp with 3 states rather than 2.

To which the same solution applies (image missing):

I would propose a parametric curve on the ball path, and, for fantasy sake, by whatever mechanism, the plate knows at what part of the parabola the ball is at, defining the counter. As time goes on, the revolution gets smaller and smaller. Eventually the ball will completely rest on the table, which is 0: — Lionino

Just replace 0 and 1 and 2 with white, red, and blue. The square starts as white, so it finishes as white as the ball rests still on the plate.

by whatever mechanism, the plate knows at what part of the parabola the ball is at, — Lionino

This is just a meaningless hand-wavy rationalisation and is inconsistent with the specific timing intervals:

Red after 30 seconds, blue after another 15 seconds, white after another 7.5 seconds, etc.

Each bounce of the ball is the timing interval, e.g. when it first hits the plate it turns red, when it hits the plate a second time it turns blue, when it hits the plate a third time it turns white, etc.

The simplest answer is that supertasks are illogical. It is metaphysically impossible for an infinite sequence of events to be completed in a finite amount of time.

This is just a meaningless hand-wavy rationalisation and is inconsistent with the specific timing intervals: — Michael

Sensors. Touching the sensor means the colour is white.

We can come up with other mechanisms that can better represent this exponential acceleration. Which I didn't do:
1 – for simplicity sake
2 – to use the same example as the SEP

Your "solution" doesn't work, as shown by this alternative:

The ball bounces at a rate such that it first strikes the panel after 30 seconds, then again after a further 15 seconds, then again after a further 7.5 seconds, and so on.

Each time the ball strikes the panel the colour of the panel changes, rotating through white, red, and blue.

What colour is the panel when the ball comes to a rest?

Of course the solution doesn't work when you change the mechanism to be exactly like Thompson's lamp without the limit.

Likewise, Earman and Norton's solution doesn't work if you remove the limit (falling ball).

The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit.

The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. — Lionino

That's precisely the point. The lamp turning on and off and the square changing colours are each examples of an infinite sequence of events. If you claim that it is possible for an infinite sequence of events to complete then you should be able to determine the completed state of the lamp/square. If you cannot determine the completed state of the lamp/square then I will reject your claim that it is possible for an infinite sequence of events to complete.

Of course the solution doesn't work when you change the mechanism to be exactly like Thompson's lamp without the limit.

Likewise, Earman and Norton's solution doesn't work if you remove the limit (falling ball).

My example keeps the falling ball so I haven't "removed the limit".

I don't even understand what the supposed paradox is. — fishfry

The paradox is this:

1.The bottom of the stairs is reached at the 1 minute mark.
2.Reaching the bottom of the stairs entails taking a final step.
3. Therefore there is a final step
4.The steps are countably infinite (1:1 with the natural numbers)
5. There is no final (largest) natural number.
6.Therefore there is no final step

#3 & #6 are a contradiction.

ZENO'S PARADOX
Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. — keystone

There are no empirical differences, agree. Presentism is the movie reel being played (a sort of literal analogy of the moving spotlight version of presentism). The reel by itself is eternalism (even if it still represents a preferred frame, which eternalists typically deny). The photo is just a frame, and not even that, since it is just a mental state since nothing in the present can be detected. If the state is all there is, then all memories are false and do not constitute evidence of anything.
The film analogy is discreet by nature, but doesn't have to be if the 'frames' are stacked instead of arranged side by side.

I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state.

Reconciling general relativity with presentism is quite challenging.

There is a way to disprove GR, but it is similar to proving/disproving an afterlife: You cannot report the findings in a journal. Both premises of SR contradict presentism, so different premises must be used to take that stance. This has been done, but the theory was generalized about a century after GR came out. It necessarily denies things like black holes and the big bang.

Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.

I beg to differ, but again, the addition of a premise of a preferred moment has nothing to do with the validity of Zeno's assertions. He makes no mention of the present in any of them. If you disagree, then you need to say how the additional premise interferes with Zeno's logic.

Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.

Not sure of the difference. If I cut a string, I don't get points, I get shorter strings.

While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said “My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.”

You cannot directly observe a particle in a superposition state

You can under some interpretations.

I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points)

I don't think QM states are like points. The analogy is going way off track it seems.

I believe you are discussing whether time is discrete or continuous.

It's one of the things I'm discussing. Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"

If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation.

In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted).

Necessary only if the first premise is to be accepted.

I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.

Yet again, one's interpretation of time isn't relevant to the above analysis.

I explicitly wrote abstract string.

Fine, Then it's a mathematical line segment.

let’s say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.

You're going to have to spell out exactly how an eternalist stance makes a difference here. All I see is an assertion that it makes a difference, but I don't see how.

Additionally, none of the paradoxes explicitly rule out (experience of each task) as a possible solution.

It takes some minimum time to explicitly comprehend/experience a step in a series of steps. Hence the explicit experience of each step of a supertask cannot be completed in finite time.

If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see.

Hence needing to see them being irrelevant.

The paradox is this:

1.The bottom of the stairs is reached at the 1 minute mark.
2.Reaching the bottom of the stairs entails taking a final step.
3. Therefore there is a final step
4.The steps are countably infinite (1:1 with the natural numbers)
5. There is no final (largest) natural number.
6.Therefore there is no final step

#3 & #6 are a contradiction. — Relativist

Thanks for clarifying that for me.

I don't see a paradox. All I see is a lack of understanding of mathematical limits.

Consider the sequence 1/2, 3/4, 7/8, 15/16, ...

Mathematically, this sequence as a limit of 1.

The sequence never "reaches" 1; nor is there a last step. Neither of these statements is controversial once you understand what a limit is. Sadly, most people have never taken calculus; and most students who take calculus never really learn what a limit is. The subject isn't taught properly till a math major class in real analysis. So almost everyone in the world is ignorant of the mathematical theory of limits, and is therefore vulnerable to confusions about "reaching" and "last steps."

It's perfectly clear that if you start at 1 and move leftward on the number line, you necessarily skip over all but finitely many elements of the sequence, so that it's always only finitely many steps back from 1 to the start of the sequence.

When you dress the story up with fictional staircases and physics-violating lamps, people get confused.

But there is no confusion. 1 is the limit of the sequence, but the sequence never "reaches" 1 nor is there a last step. The definition of a limit is logically rigorous and unambiguous. The fictional staircases and lightbulbs only have the purpose of confusing people.

Finally, we can consider the sequence 1, 2, 3, ... which never reaches infinity nor does it have a last step. But we can place an arbitrary symbol at the end, usually called $\omega$, so that the sequence looks like this:

1, 2, 3, 4, ..., $\omega$

Once again there is no "reaching" and no last step, but it's mathematically legitimate to say that $\omega$ is the limit of the sequence. And we see that if you start at $\omega$ and take any step back, you will land on a natural number, and it's always only finitely many steps backward from $\omega$ to 1.

In fact these two augmented sequences 1/2, 3/4, 7/8, ..., 1 and 1, 2, 3, ..., $\omega$ are order-isomorphic.

So there's just no paradox. There is only taking perfectly well-understood mathematical facts and dressing them up with physics-contradicting staircases and lightbulbs so as to confuse people.

In the case of the lamp, we have a sequence 0, 1, 0, 1, ... that has no limit. No matter what you define as the final state of the lamp (the state at $\omega$), you can't make the sequence continuous. So I say the lamp turns into a pumpkin at midnight, just as Cinderella's coach did. Since the lamp is entirely physical, and its switching circuitry violates the known laws of physics, that's as sensible as any other solution.

The staircase story has a perfectly natural solution, though. At each step, the walker is present on that step. So the corresponding sequence is 1, 1, 1, 1, ... So if we define the limiting state is 1, we have made the walker's sequence continuous. That's a natural solution.

We could say that the limiting state of the walker is "not downstairs," but that would make his path discontinuous. There's a clear preference for the continuous solution.

There's no way to make 0, 1, 0, 1, ... so the pumpkin is as reasonable as anything else.

I would propose a parametric curve on the ball path, and, for fantasy sake, by whatever mechanism, the plate knows at what part of the parabola the ball is at, defining the counter. As time goes on, the revolution gets smaller and smaller. Eventually the ball will completely rest on the table, which is 0: — Lionino

Cute. In this case I agree that it's natural, in the sense of preferring continuity, to say that the final (ie limiting) state is resting on the table.

Yes, but check the solution at https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp — Lionino

Will check it out, thanks.

So there's just no paradox. There is only taking perfectly well-understood mathematical facts and dressing them up with physics-contradicting staircases and lightbulbs so as to confuse people. — fishfry

:up: Amen :roll:

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction. — Michael

This is an interesting argument. I have some issues with it.

First, I should note that infinite divisibility is a weak condition. The rational numbers are infinitely divisible, but they are not a continuum. They are full of holes, such as the point where sqrt(2) should be.

The question often argued is whether physical spacetime is a continuum in the sense of the uncountably infinite, Cauchy-complete real numbers. But you have strengthened the claim to saying it's not even a countably infinite non-continuum like the rationals. So even if you're right, your claim is too strong to be right. That's a meta-argument, not an argument. But claiming spacetime isn't even like the rationals is much stronger than claiming it's not like the reals. [This is all beside the point, but I wanted to make the point that infinite divisibility is not enough to make something a continuum].

Now to the argument.

"If time is infinitely divisible then supertasks are possible."

By this I take it that you mean that if we take, say, the rationals in the unit interval to model one second of time, we could do something in [0,1/2) and something else in [1/2, 3/4) and so forth, and thereby do infinitely many things in one second, which is the definition of a supertask. Have I got your argument right?

So yes, I agree that if time is dense -- that's the math term for the property that there's always a third thing between any two distinct things -- then supertasks are possible. I'd never thought of that argument before and it's pretty good. Although for all we know, there could be some law of nature that the smaller the time interval, the longer things take to happen, wrecking your supertask. You can't rule that out. Just like objects gaining mass as their velocity approaches the speed of light. Strained analogy but I hope you see what I'm getting at.

"Supertasks entail a contradiction."

What contradiction is that? You just convinced me that if time is like the rational numbers (dense but full of holes) supertasks are possible. Then you claim supertasks entail a contradiction, but I'm not sure what contradiction that is.

So your argument's incomplete here, and if you did explain this elsewhere in the thread, I apologize for having missed it.

I have another concern, which is that in our current theory of physics, we can not reason sensibly about intervals of time below the Planck time. So you are making an argument that can never, even in theory (pending the next revolution in physics) be observed, measured, or confirmed by experiment.

That's what we call speculation. Like the cosmological theory of eternal inflation, in which the universe had a definite beginning but exists infinitely far into the future. That's not physics, that's mathematical metaphysics. Science fiction with equations. I do think your idea has a problem in this area. You can't actually reason below Planck scale.

You can argue that reality allows for the possibility of contradictions if you want, but most of us would say that it is reasonable to assert that it doesn't. — Michael

On the contrary, most people would agree that life is full of contradictions. I love you and I hate you. We should clean up the environment but that raises the cost of energy for the poor. (Ok that's a tradeoff and not a logical contradiction, but it's still a situation where two virtues are in conflict). I am large, I contain multitudes. (That diet's not working). The electron is a particle. It's a wave. No, it's an excitation in a quantum field. That's the latest attempt to resolve the contradictions in physics.

As we go through our daily lives we are faced with one contradiction after another. And when we study physics, we see contradictions and impossibilities at the most fundamental nature of reality.

I do not believe you can convince me that nature isn't self-contradictory. Why shouldn't it be? What law of nature says that nature must satisfy Aristotelian logic?

tl;dr: Well those are my thoughts. Interesting argument though. If time is modeled by the rational numbers, supertasks are possible. I will give that some more thought. But again: why do supertasks entail a contradiction? That's the weak part of the argument I think. That, and the Planck scale issues.

:up: Amen :roll: — jgill

Nice to see you again @jgill, and thanks.

Take the scenario here:

After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

We can sum the geometric series to determine that the limit is 60 seconds. The claim some make is that this then proves that this infinite sequence of events can be completed in 60 seconds.

However, then we ask: what colour is the square when this infinite sequence of events is completed?

As per the setup, the square can only be red, white, or blue, and so the answer must be red, white, or blue. However, as per the setup it will never stay on any particular colour; it will always turn red some time after white, turn blue some time after red, and turn white some time after blue, and so the answer cannot be red, white, or blue. This is a contradiction.

The conclusion, then, is that an infinite sequence of events cannot be completed, and the fact that we can sum the geometric series is a red herring. To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible.

Mathematically, this sequence as a limit of 1.

The sequence never "reaches" 1; nor is there a last step. Neither of these statements is controversial once you understand what a limit is. Sadly, most people have never taken calculus; and most students who take calculus never really learn what a limit is — fishfry

I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term.

However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step?

By definition, a limit is not reached, — Relativist

$f(x)=3x+2$, $f(1)=5$

$\underset{x\to 1}{\mathop{\lim }}\,f(x)=f(1)$

However, the clock does reach 1. At time 1, the stairway descent must have ended — Relativist

Certainly the relationship between time (independent of human control) and physical steps taken over a period of time has ended.

Certainly the relationship between time (independent of human control) and physical steps taken over a period of time has ended. — jgill

That's because the physical steps map to an infinite series in an interval with an open boundary. One can't simply declare there's no final step because the mapping implies there isn't. The taking of steps is a repetitive physical process, and if a physical process ends, there has to be a final step.

if a physical process ends, there has to be a final step. — Relativist

There is no physical process. There's a fictional process that doesn't obey the known laws of physics.

In what sense does anyone think the staircase or the lamp are physical processes?

Question: Cinderella's coach turns into a pumpkin at the stroke of midnight.

Is that transition a physical process? In what world?