I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called. — noAxioms

Bernadete's Paradox of the Gods:

A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.

It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.

I think it's the crux of Zeno's paradox that the mathematics of an infinite series fails to address. The solution, similar to my proposed solution above, is that movement is not infinitely divisible (either because space is discrete or because movement within continuous space is discrete).

The solution, similar to my proposed solution above, is that movement is not infinitely divisible (either because space is discrete or because movement within continuous space is discrete). — Michael

I'm not yet comprehending this to my liking. To my current understanding, an infinite series is the very thing which makes something otherwise perfectly continuous discrete. It's, for one example, the difference between a perfect circle and an apeirogon with equal sides: the first is perfectly continuous, the second discrete.

Due to this, I've so far always assumed the resolution to Zeno's paradoxes is that movement is not infinitely divisible precisely because it is perfectly continuous while it occurs. Such that it's our imposed conceptualizations of measurement upon an otherwise immeasurable process which makes Zeno's paradoxes possible.

If movement is continuous then an object in motion passes through every ${1\over{n}}m$ marker in sequential order, but there is no first ${1\over{n}}m$ marker, so this is a contradiction.

If movement is continuous then an object in motion passes through every 1nm marker in sequential order, but there is no first 1nm marker, so this is a contradiction. — Michael

To the best of my understanding, not within process philosophy.

I’ll first try to better explain my own current stance:

It's the the very marker you address that I take to be the conceptual measurement imposed: In process theory, there is no beginning nor any permanent thinghood, only continuous becoming. Like with quantum mechanics, wherein everything is a wave till measured. Whenever we measure, we quantify (and vice versa): one given, quantitative parts of that one given, multiple whole givens, and so forth. But the movement of whatever we quantify remains purely continuous, wave-like in this very limited sense.

We thereby quantify there being one arrow that is being projected. Likewise we quantify there being one target it is going to penetrate. Yet, as per process theory, both are otherwise merely processes of becoming themselves, that are forever in flux in manners devoid of any absolute beginning. We furthermore empirically know (this by imposing measurement/quantity) when the quantified arrow first starts its motion toward the target, we know that if travels through air via certain placements in space, and that it eventually hits the target whereupon the arrow stops its motion. But when we then try to quantify this very (here, by analogy, wave like) process of the arrows motion what we end up with are quanta of space that appear to be infinite in number. These, in turn, then facilitate Zeno’s paradox of the arrow.

I don’t know how to address this properly with the arrow paradox, so I’ll use Achillies and the tortoise instead:

Here suppose motion occurring in a finitely divisible (hence quantized) space. For the sake of argument, say this finitely divisible space from point A to point B has only ten divisions. The tortoise is at the fifth division of this space while Achilles is at the second division of this space—both moving toward the tenth division of space. How would conceiving of space in such finitely quantized manner change Zeno’s paradox so as to allow Achilles to catch up to the tortoise?

----

I so far can’t apprehend a coherent way demonstrating logically (non-empirically) that Achillies can catch up with the tortoise in such a scenario.

And this because at the very least physical motion (if not also any psychological change) seems to me to be completely continuous in its ontological nature. This again, as per core concepts of process theory. A continuous change which we measure/quantize and thereby impose upon the notion of fixed beginnings and fixed thinghood—which, in an ontology of flux, don’t in fact occur.

I acknowledge this train of thought deviates from the thread’s intent. But, to cut things short, I don’t yet understand how a finite quantization of space or of motion is interpreted as resolving Zeno’s paradoxes (as per my example above)—this given what we empirically know to be (Achilles can catch up with the tortoise).

On second thought, scratch the example I just gave of Achilles and the tortoise in finite quanta of space. While I still see deeper problems with such interpretation of motion, I've realized it can be addressed mathematically via ratios - and I don't want to get into debates regarding the nature of time and space. It was a case of me talking before thinking. I won't delete my previous post, though.

This is not true. Perhaps you are reading a different account of the story than I did, which is the one on wiki, which says simply:
"In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead". The 2nd bolded part is the non-sequitur, and the first bolded part follows from the 2nd. None of it makes the assertion you claim. The non-sequitur makes the argument invalid. There are ways (such as with the light switch) that make it seem more paradoxical. — noAxioms

Sorry no Axioms, I can't follow what you are saying, perhaps you could spell out your supposed "non-sequitur" in a clear explanation, instead of simply asserting it. In Zeno's paradox, the tortoise is given a head start.

Same non-sequitur. It is not true that Icarus always has more steps to take, only that he does while still on a step, but the time to complete all the remaining steps always fits in the time remaining in his minute. — noAxioms

That Icarus always has more steps to take is the valid conclusion from the premises. Yes, "the time to complete all the remaining steps always fits in the time remaining in his minute", as you say. However, the remaining steps are indefinite. Likewise the amount of divisions which can be made to the remaining time are indefinite. Therefore Icarus' minute is never completed, and he never completes his steps. That is the conclusion we must make.

However, the OP concludes that the minute passes, and the bottom reached. The OP therefore relies on a sense of physical reality, that a minute must pass, which is outside of the premises. That's why it's not a valid conclusion. There is nothing within the premises to indicate that a minute must pass, and everything indicates that a minute will never pass.

OK, which premise then is false in the Zeno case? The statement is really short. One premise that I see: "the pursuer must first reach the point whence the pursued started", which seems pretty true to me. — noAxioms

The false premise for Zeno is that each distance, and each time period will always be divisible. That's the problem points out. Think of the way a runner actually runs. One foot hits the ground, then the next foot hits the ground some distance further ahead. The runner does not cover every inch of ground in between, the motion of the feet in contact with the ground, takes place in increments.

Nice thought experiment!

There's no paradox in the sense of a contradiction. It just seems weird.

When we analyse it closely, we see that all the set-up does is establish the existence of an infinite descending staircase, and a location of Icarus on the staircase at every point in the time interval [0,60). The curved closing parenthesis there means that that does NOT include time 60 seconds. That is crucial!

So the set-up says nothing about where Icarus is at 60 seconds, or any time after that. We need to add additional assumptions/axioms/specifications in order to say anything about that. Those won't be able to meaningfully say "at the bottom of the staircase", as the above story does, because there's no such place.

If it were up to me to specify, then I'd choose to say that at 60 seconds and later Icarus is back at the top. That's a purely aesthetic choice based on it being the least arbitrary place out of those mentioned, and I like to avoid arbitrariness where possible. But we could just as well assume that at 60 seconds he will be on step 3. Whatever step we assume, it will not generate a contradiction.

Similarly with the Thomson's Lamp case. When we ask "is the lamp on or off at one minute" we are asking for something that the set-up doesn't give us enough information to answer. The setup tells us whether the lamp is on or off at every instant in [0,60) and tells us nothing about whether it is on or off at 60 or later. We cannot infer whether it would be on or off at 60 because we know nothing about the physics of the world in question, which must be enormously different from that of our own, in order to allow complete switching of a finite-sized lamp in infinitesimally small time periods. I expect we could invent some physical rules to support either an on or an off assumption.

Hi andrewk. It's good to see you, been a while. Doing well? I hope.

Thanks MU. yes, all well here. I rarely post now, leaning towards expressing my musings about the world through music rather than words. But I periodically lurk, and I couldn't resist the temptation of a juicy infinitish thought experiment.

ZENO'S PARADOX


Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs.


Zeno Non-sequitur fallacy - I agree that the conclusion doesn't logically follow from the premise, but that's only true if you interpret the paradox from a fresh perspective. From the traditional understanding of motion, the conclusion indeed seems to follow logically. This is precisely Zeno's point.


I wrote the following in a different thread but it's relevant here. Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zeno’s famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present (I use God here not to push a religious view, but for simplicity). The discreteness that are looking for is not in space but in measurement/observation. In other words, God's fundamentally cannot watch everything. This actually should come as no surprise since the Quantum Zeno Effect demonstrates that an observed system cannot evolve.

Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM and the importance of observation/measurement.

STAIRCASE PARADOX


A minute cannot pass - This scenario involves an infinitely large object (the staircase), an infinitely complex task (traversing the entire staircase), and the passage of one minute. You're suggesting that the issue lies in the impossibility of a minute passing? It seems you may have labeled the most logical and uncontroversial element in the paradox as illogical. If you think the problem has to do with Icarus's steps then frame your solution in that context.
No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. Could it be that supertasks are impossible?
restricting ourselves to the physical world - The physical world is not the only realm that exists; there's also the abstract world, which operates under its own set of rules. For instance, in an abstract world I can define, it's perfectly valid to set the speed of light at 100 m/s. This isn't incorrect—it's simply a different premise. However, I do believe in a kind of symmetry where truths in the physical world often find parallels in the abstract world.


This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it?


Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1.


Non-standard numbers-I'm certain you're a strong mathematician, but I also feel like you're overcomplicating things. This reminds me of an Einstein quote: “If you can't explain it to a six year old, you don't understand it yourself.”

,
Only a potential infinity-My purpose in presenting this paradox is to underscore the problems associated with the concept of actual infinity.


Thomson's Lamp-Indeed, the Staircase Paradox shares significant similarities with Thomson's Lamp Paradox, particularly in that both scenarios lead to states considered invalid by conventional logic after one minute has elapsed. In the Staircase Paradox, we are left unsatisfied by claims that the staircase either exists or does not exist. Similarly, in Thomson's Lamp Paradox, we find it unsatisfactory to definitively say whether the lamp is on or off. The difference is that, supertasks aside, Thomson's Lamp is a critique of infinite series whereas the Staircase Paradox is a critique of N.


Trip from 0 to 1-I don't get it.


[0,60)-Your point is valid, for brevity I didn't explicitly state that the first instant he passes the stairs he arrives on the ground. However, as the poem indicates, my view is that at that instant, he actually arrives at a singularity, similar to what one might encounter at the center of a classical black hole.

PARADOX OF THE GODS


As Michael noted, your barrier paradox is Bernadete's Paradox of the Gods. I find this paradox intriguing. In the realm of physics, I think quantum tunneling offers a solution to this issue.

Nice exposition of Zeno!

Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. — keystone

[...]

The discreteness that ↪Metaphysician Undercover
↪Michael
are looking for is not in space but in measurement/observation. — keystone

Yes, I’m in agreement with you as to the non-discreteness of space.

Keystone
Fair enough re: potential vs actual infinity. Will continue reading the thread hoping to learn. smile

Still wrestling with how there can be a last step or a first step in infinite steps probably because it appears arbitrary to impose mathematical limits to the concept of "infinity". Can understand the "human" need to do so, hence limiting infinite steps to a "mere" 'series of'.

I accidentally wrote this on the wrong thread so I'm moving it over here. I have some thoughts that may be of interest.

The staircase problem is called an omega-sequence paradox, a paradox that involves counting 1, 2, 3, ... and doing something at each step, then expecting the behavior to be defined in the limit. The answer to all those paradoxes is that you haven't defined what happens at the limit. You've told me what Thompson's lamp does at every finite $n \in \mathbb N$, but you have not told me what it does at the limit. Therefore the lamp could be on, it could be off, or it could have turned into the Mormon Tabernacle Choir. You haven't specified the behavior at the limit, so it can be anything you like.

There's a mathematical name for the upward limit of the natural numbers. It's $\omega$, lower-case Greek omega. You can think of it as a formal symbol that is greater than every other natural number, but that does appear in the sequence, as follows:

$0, 1, 2, 3, 4, \dots, \omega$.

You can think of $\omega$ as a "point at infinity." Or from a formalist view, it doesn't mean anything. It's just a symbol that satisfies $n \lt \omega$ for any natural number $n$, as well as all the usual meanings for 47 < googolplex and so forth.

This is a handy formalism. Now we can solve Thompson's lamp. The problem is that the state of the lamp is not defined at $\omega$. In other words you told me what the state is at every natural number, but not at $\omega$. That literally solves the paradox. It's no different than one of those "complete the sequence" questions. Mathematically, the next number can be anything you like.

In other words: You told me the state of the lamp at every finite number. You did not tell me the state at $\omega$. All confusion about Thompson's lamp is to realize that you just haven't told us the state at $\omega$. And there's no good reason to prefer one answer over the other.

The staircase paradox is a little more interesting, in the sense that you are present at each step 1, 2, 3, ... As before you can still define the behavior at $\omega$ to be either that you are there, or you aren't. But in this case, assuming you are there at $\omega$ is more natural, in the sense that the function that maps $\omega$ to "there or not there," is continuous if you're there.

What I mean is, at each successive step, the state of that step is "you are on it." Now the state at $\omega$ is undefined, but there is a natural way to define it; that is, to assume that your motion is continuous in some sense. So if you are there at every step, you are there at the bottom, the state of step number $\omega$.

So if you believe that your motion down the stairs is "continuous," however you define that in this context, then since continuity preserves limits, you are there at the bottom. But if you can't justify the assumption of continuity, then anything at all might happen at the bottom. You're there, you're not there, you're a sea slug at the bottom of the ocean. Since you haven't specified the value of the function at $\omega$, it can be anything you like.

One more note, and that is that you can indeed count backwards from $\omega$. But as you can see, any step that you take backwards necessarily jumps over all but finitely many numbers; so that it's always a finite number of steps back to zero, even from infinity.

Therefore if you are at the bottom of the stairs, you can just take a tiny tiny step up -- as small as you like -- you will necessarily skip over all but finitely many stair steps, and end up on some natural number like 47 or googolplex. Either way, it's still only finitely many more steps back to the top.

I should say that again, since this comes up so often. Even if you start at the point at infinity, it's always at most a finite number of steps back to zero.

This principle also applies to people making cosmological arguments about the impossibility of an infinite past, because it would take an infinite time to get to the present. Actually that's not true. If you put a point at negative infinity, it's only a finite number of steps to the present.

Finally, I'll mention that $\omega$ as I've used it is just a formal symbol; but in fact can be formally defined and shown to exist within set theory as the first transfinite ordinal number. And once you do that, you can keep on going into the wondrous and mysterious world of the ordinal numbers.

I'll leave you all with just one thought:

It's always only a finite number of steps from infinity back to zero, no matter how small a step you take. That's something a lot of people get wrong.

Fishfry
Is minus one a natural number? And, is zero a natural number? Mathematicians' mathematics is not a strong suit for some. sad smile at one's own ignorance

Is minus one a natural number? And, is zero a natural number? Mathematicians' mathematics is not a strong suit for some. sad smile at one's own ignorance — kazan

Those are great questions.

Is minus one a natural number? — kazan

No. The natural numbers are the positive (or nonnegative, we'll talk about that in a moment) whole numbers like {0?], 1, 2, 3, 4, 5, ...

They're infinite (or endless if you prefer) in one direction.

The entire set of positive, 0, and negative whole numbers is called the integers.

The integers are infinite (or endless) in two directions:

..., -3, -2, -1, 0, 1, 2, 3, ...

The natural numbers are very basic and important, since on the one hand, they seem to be intuitive to every child -- you "just keep adding 1." On the other hand, they go on forever, giving us all a glimpse of infinity.

The integers, however, are much better for doing arithmetic. That's because the integers have "additive inverses." Given the number 5, in the natural numbers there is no number you can add to it to get 0.

But in the integers, there is: namely, -5. We say that the integers have additive inverses. And believe it or not, that happens to be really important in higher math.

So that's a long answer to a simple question but the bottom line is that the integers include the negative whole numbers, and the natural numbers don't.

And, is zero a natural number? — kazan

For some reason this question attracts a lot of controversy. People like to argue about it.

But I am here to tell you that it doesn't matter. Why is this?

Well, what's important about the natural numbers is their order. You have a linear, discrete procession of things. There is a first thing, then a next thing, then a next thing, and so on, forever.

If you call the first thing 0 or you call the first thing 1, it doesn't make any difference at all to the order structure of the sequence of "first, next, next, next, ..."

Suppose your friend thinks 0 is a natural number and you believe the natural numbers start with 1.

Then you just invent a new numbering system in which 1 means 0, 2 means 1, 3 means 2, and so forth.

You can tell your friend that you are starting with 0, but you just call it 1. And what they call 1, you call 2.

So you are right, and they are right. Why? Because the only interesting thing about the natural numbers is that (1) there's a first one; and (2) there's always a next one.

Those two rules generate the natural numbers, no matter what you call them!!

I hope this is taken to heart by someone. The answer to whether 0 is a natural number or not is that it absolutely doesn't matter. Just tell people what convention you're using and they'll be fine with it.

Now you might ask, what about arithmetic? 0 + anything = anything, but if you don't include 0 you don't have a number with that property.

And that's what I mentioned earlier. If you want to do arithmetic, the natural numbers are lousy anyway, they don't have additive inverses. So if you're doing arithmetic, you'll be working in the integers, not the naturals.

That's why it doesn't matter if 0 is a natural number. If you only care about order and "nextness," it doesn't matter what you call the first element.

And if you want to do arithmetic, you'll be using the integers anyway, which include zero.

Hope this was helpful. And again, these were great questions. Surprisingly deep. Thinking about various number systems and their properties like order or additive inverses is the basis of higher math.

Fishfry
Yes,very helpful.Thank you for taking the time.
Which begs the question, (smile) how, if it's possible, would "the lower case omega" concept of "upper (lower?) limit" be applied to all integers? Surely, if it's possible,this could be useful in some areas of mathematics ( besides arithmetic ).
Sorry,don't mean to hijack this thread.

Yes,very helpful.Thank you for taking the time.
Which begs the question, (smile) how, if it's possible, would "the lower case omega" concept of "upper (lower?) limit" be applied to all integers? Surely, if it's possible,this could be useful in some areas of mathematics ( besides arithmetic ). — kazan

Another good question. Yes we could put "negative omega" at the leftward infinite end of the integers. There's not much use for it. The interesting aspect of omega is to keep going with the "add 1" game to get a whole infinite structure of higher ordinals continuing "to the right," in the positive direction. There's nothing new of interest that happens if you do the same thing on the left, it would just be a mirror image of the ordinal structure on the right.

This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it? — keystone

I don't see a reason to think that a person will reach the bottom of the infinite staircase, ever. You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts.

"There was an old woman. She was only 2 years old, and really just a baby." There's my paradox.

Fishfry
Fair enough.
Nothing new of interest, comes to mind. Apart from adding negative l c omega with (to?) (positive) l c omega and getting the same answer as subtracting them,(still in the realms of arithmetic,) presumably zero?
There that pesky zero again. Thanks again.

You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus

As far as I understand about paradoxes, that's precisely what a paradox is about. It is a self-contradictory statement, but arrest our attention. The aim of this thread (or purpose of @keystone) is not to reach a conclusion, but to result in persistent contradiction between interdependent elements: The staircase being endless and reaching the bottom of it in just a minute.

It is clearly a paradox.

To explain this more deeply, @Michael and @noAxioms wrote very interesting posts using maths and logic.

Sorry Fishfry,
Never in memory, has "pure" mathematics been of such interest as now. Feel like you've open a window and there's a gale blowing in, here.
Had more questions about l c omega, but will give them further thought first. And catch up with you elsewhere and later. Lounge perhaps in a few days?

Fannel Jesus

No paradox with the old lady.
Just different periods of the past in her life.
Won't a better paradox be if set in the present tense.
Just a suggestion. smile

Fair enough.
Nothing new of interest, comes to mind. Apart from adding negative l c omega with (to?) (positive) l c omega and getting the same answer as subtracting them,(still in the realms of arithmetic,) presumably zero? — kazan

Better not to try to subtract the endpoint infinities from each other, The result is undefined.

Never in memory, has "pure" mathematics been of such interest as now. Feel like you've open a window and there's a gale blowing in, here. — kazan

Thanks much.

Had more questions about l c omega, but will give them further thought first. And catch up with you elsewhere and later. Lounge perhaps in a few days? — kazan

I hear all the cool kids hang out in the lounge these days.

As far as I understand about paradoxes, that's precisely what a paradox is about. It is a self-contradictory statement, but arrest our attention. — javi2541997

Yes, but usually where the contradiction occurs exactly is supposed to be *non-obvious*. "He went down some endless stairs, and reached the end". It's not non-obvious where the contradiction is. It's immediately obvious.

Because it's obvious, it's not so much a paradox as it is just a plain contradiction.

Here's what Wikipedia says about paradoxies

A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation.[1][2] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.

`despite apparently valid reasoning from true or apparently true premises` - that's key! The premises and steps in reasoning have to make some kind of sense.

The premise that you can reach the end of an endless staircase doesn't apparently look like valid reasoning or true premises to me.

Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs. — keystone

I suggested that movement was discrete, not that space was discrete. 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.

Similarly with the Thomson's Lamp case. When we ask "is the lamp on or off at one minute" we are asking for something that the set-up doesn't give us enough information to answer. The setup tells us whether the lamp is on or off at every instant in [0,60) and tells us nothing about whether it is on or off at 60 or later. We cannot infer whether it would be on or off at 60 because we know nothing about the physics of the world in question, which must be enormously different from that of our own, in order to allow complete switching of a finite-sized lamp in infinitesimally small time periods. I expect we could invent some physical rules to support either an on or an off assumption. — andrewk

I don't think the physics is relevant. The question can be asked of any universe with any physical laws. The thought experiment is entirely metaphysical.

Repeating my specific example:

After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.

What digit does the counter show after 60 seconds?

The issue we have is that if there is no smallest unit of time then the counter is metaphysically possible, but this entails a paradox as the answer to what the counter shows after 60 seconds is undefined yet the counter will show something after 60 seconds. Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible.

We could replace the counter with some supernatural deity capable of keeping such a count if it makes things easier to consider (similar in kind to Benardete's Paradox of the Gods).

`despite apparently valid reasoning from true or apparently true premises` - that's key! The premises and steps in reasoning have to make some kind of sense. — flannel jesus

I agree! There has to be at least some kind of sense on the premises. Yet, there are, among these, a large variety of paradoxes of a logical nature. A basic pattern of a paradox is having a way of reasoning. Right?

Well, following the paradox within this OP, we can conclude there is a bit of reasoning. For example: @fishfry used the reason pretty well in this comment: https://thephilosophyforum.com/discussion/comment/898761

He even states:

The staircase problem is called an omega-sequence paradox, a paradox that involves counting 1, 2, 3, ... and doing something at each step, then expecting the behavior to be defined in the limit.

Sadly, I am not good enough at maths and logic, so I can't post valid or interesting comments regarding this paradox. What I try to defend is that what @keystone wrote is actually a paradox. Maybe it has its flaws, or he was inspired by other paradoxes which were quoted in the comments above. But it there is still a paradox.

The answer to all those paradoxes is that you haven't defined what happens at the limit. — fishfry

I think this is a misrepresentation. The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false.

And note that this only considers progressive interpretations of these paradoxes (i.e. how they can complete). Regressive interpretations (i.e. how they can start) must also be considered. I don't think mathematical limits are relevant to these at all.

That comment doesn't justify why the person should reach the end of the stair case.