60 seconds can pass without anyone measuring it. — Michael

That's like saying today would be April 29 even if there was never any human beings to determine this. If you can't understand how this is wrong, I don't know what else to say.

The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it.

The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it. — Relativist

Zeno again?

Say (in some hypothetical world, say current math or future physics) that we have a "sequence of actions" as you say, occurring at times 1/2, 3/4, 7/8, ... seconds.

It's perfectly clear that 1 second can elapse. What on earth is the problem?

You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing.

I can't imagine what you are thinking here, to claim that one second of time can't pass.

I have repeatedly noted in this thread that we can symbolically adjoin a "point at infinity" to any countably infinite sequence, and that's where the limit lives. We can note that 1/2, 3/4, 7/8, ... has the limit 1, which lives in the ordered set {1/2, 3/4, 7/8, ..., 1}.

We can also do the same thing in the integers as 1, 2, 3, 4, ..., $\omega$, where $\omega$ can be thought of as a formal symbol that's greater than every natural number. It also has technical importance as the first transfinite ordinal.

Either way, sequences do "reach" their limit via the limiting process, though the sequence itself does not necessarily attain the limit. It's just semantics.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?

You seem to assign some meaning to the word "event" that I don't understand. — fishfry

Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete.

Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets.

Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete. — Michael

Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof.

Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets. — Michael

I'm not the one advocating for supertasks, yet you keep arguing with me that they are impossible.

I'm not the one advocating for supertasks, yet you keep arguing with me that they are impossible. — fishfry

No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks.

Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof. — fishfry

I've provided arguments, and examples such as Thomson's lamp that shows why. And again, your reference to mathematical sets and mathematical limits does not rebut this.

No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks. — Michael

I gave you a mathematical model that puts your unsupported claims into context.

I've provided arguments, and examples such as Thomson's lamp that shows why. — Michael

Thompson's lamp shows nothing of the sort. I've explained that to you repeatedly as well.

I gave you a mathematical model that puts your unsupported claims into context. — fishfry

And it doesn't address the issue.

If I write the natural numbers in ascending order, one after the other, then this can never complete. To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense, and so supertasks are nonsense.

That we can sum an infinite series just does not prove supertasks. It is clearly a fallacy to apply an infinite series to an infinite succession of acts.

And it doesn't address the issue. — Michael

I asked you to consider a hypothetical world and you pretended I was talking about mathematical sets.

If I write the natural numbers in ascending order, one after the other, then this can never complete. — Michael

Yes, the observable universe is finite. We're agreed on that. How many times are you going to try to convince me of something I've already agreed with many times?

To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense, — Michael

I have not claimed otherwise.

and so supertasks are nonsense. — Michael

According to current physics. That's as far as we can go.

That we can sum an infinite series just does not prove supertasks. — Michael

Nor does it disprove their metaphysical possibility. We just don't know at present.

I have not claimed otherwise. — fishfry

Those who argue that supertasks are possible claim otherwise, and it is them I am arguing against. You're the one who interjected.

Nor does it disprove their metaphysical possibility. We just don't know at present. — fishfry

If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency.

If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency. — Michael

It's physically impossible. I have no idea why you keep claiming it's "metaphysically" impossible or logically incoherent. What's logically incoherent about infinite sets and transfinite ordinals? You just keep repeating the same unsupportable claims. You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics.

What's logically incoherent about infinite sets and transfinite ordinals? — fishfry

I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue.

Here's a definition for you: "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time".

The key parts are "sequence of operations" and "occur sequentially".

As in, I do one thing, then I do another thing, then I do another thing, and so on ad infinitum. It is metaphysically impossible for this to end. If it ends then, by definition, it is not ad infinitum.

I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue. — Michael

A discussion can't continue when you keep making unsubstantiated, evidence-free claims.

I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. Yes it's speculative, but nobody is saying it's "metaphysically impossible" or "logically incoherent."

https://en.wikipedia.org/wiki/Eternal_inflation

Here's a definition for you: "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time".

The key parts are "sequence of operations" and "occur sequentially". — Michael

Please stop embarrassing yourself.

I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. Yes it's speculative, but nobody is saying it's "metaphysically impossible" or "logically incoherent." — fishfry

Which has no bearing on what I'm arguing.

Which has no bearing on what I'm arguing. — Michael

You are not arguing, you're repeating your lack of argument. I'll let you have the last word, you are incapable of rational discussion.

You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics. — fishfry

This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc.

Say (in some hypothetical world, say current math or future physics) that we have a "sequence of actions" as you say, occurring at times 1/2, 3/4, 7/8, ... seconds.

It's perfectly clear that 1 second can elapse. What on earth is the problem? — fishfry

P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.

P2. 30 + 15 + 7.5 + ... = 60

C1. The sequence of operations¹ described in P1 ends at 60 seconds without ending on some final natural number.

But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

So C1 is a contradiction. Therefore, as a proof by contradiction:

C2. P1 or P2 is false.

C3. P2 is necessarily true.

C4. Therefore, P1 is necessarily false.

And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum.

_{¹ A happens then (after some non-zero time) B happens then (after some non-zero time) C happens, etc.}

You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing. — fishfry

There is no limiting process in the premises of the op, nor in what is described by . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited.

Agreed.

You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing. — fishfry

You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact.

I also think you are misinterpreting the meaning of limit. This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly? — fishfry

No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does?

I agree that it's impossible to do infinitely many physical thinks in finite time according to present physics. — fishfry

What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?

I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask.

We seem to be talking in circles, with all logic from the 'impossible' side being based on either there being a last infinite number, or on non-sequiturs based on the lack of said last number.

The goal is not unreachable. That simply doesn't follow from arguments based on finite logic, and it is in defiance of modus ponens. It's just necessarily not reached by any specific act in the list.
— Relativist

There is a bijection yes. It does not imply that both or neither completes.
— noAxioms
Why not?

You defined the second task as a non-supertask, requiring infinite time. That's why not.
I can play that game with a finite list of three steps, with the middle step of one task requiring one to make a square circle. It does not follow that the other list of three steps cannot be completed in a short time just because there exists a bijection between the steps of the two tasks.

That's like saying today would be April 29 even if there was never any human beings to determine this. — Metaphysician Undercover

Exactly so.
Your disagreement with views that suggest this is a subject for a different topic. Your displayed lack of comprehension of what the person means when he says things like that is either in total ignorance of the alternatives or a deliberate choice. Being the cynic I am, I always suspect the latter. It's my job as a moderator elsewhere.

I do thank you for verifying my earlier assessment.

I'm not the one advocating for supertasks — fishfry

For the record, I am personally advocating that they have not been shown to be physically impossible. All the 'paradoxes' that result are from inappropriately wielding finite logic in my opinion.
Thomson's lamp is a wonderful example of this, but other examples seem to have more bite.

I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. — fishfry

Does it? It seems to be a more complex model that suggests stupid sizes for 'what is', but not 'actual infinite' more than the standard flat model that comes from the cosmological principle. Yes, I know the page you link mentions 'hypothetically infinite' once. I have a deep respect for the eternal inflation model since something like it is necessary to counter the fine-tuning argument for a purposeful creation.

I agree with Michal that the sort of infinity suggested by eternal inflation is not representative of a supertask. I do realize that some people just deny 'actual infinity' of any kind, but that is not justified, hence is not evidence.

Exactly so.
Your disagreement with views that suggest this is a subject for a different topic. Your displayed lack of comprehension of what the person means when he says things like that is either in total ignorance of the alternatives or a deliberate choice. Being the cynic I am, I always suspect the latter. It's my job as a moderator elsewhere.

I do thank you for verifying my earlier assessment. — noAxioms

Are you saying that you believe that there would still be an April 29, even if there never was any human beings with their time measuring techniques, and dating practises? And do you believe that there would still be "seconds" of time without those human beings who individuate those temporal units in the act of measurement? I don't understand how you can believe that I should accept this as a reasonable alternative. Care to explain?

...like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms

Physics indeed is not exempt from logic. It's logically impossible to reach the 1 minute mark when all steps (even if there are infinitely many of them) fall short of the 1 minute mark.

Calculating the limit does not entail a process that reaches that limit. This is a misinterpretation of the concept of limit.This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L.

What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms

Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. The world told us what new math to use. The world is not constrained by math, nor logic, nor by any historically contingent work of fallible man.

Math and even logic have always been drawn from looking at the world around us. So just as an aside to the main discussion, but responding to this one sentence that caught my eye ... physics IS exempt from math and logic. Meaning that historically, and metaphysically, physics is always ahead of math and logic and drives the development of math and logic.

But to the main question, the physical/mathematical distinction is important. I can never count all the integers in the physical world (as far as we know -- to be clarified momentarily); but in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count its contents by placing it into order-bijection with itself. That is: The identity map on the natural numbers is an order-preserving bijection that shows that the natural numbers are countable.

The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity. How meat puppets such as ourselves come to have the ability to have such lofty abstract thoughts is a mystery. And if we are physical beings; and if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation? That's another good question.

Perhaps our very thoughts of infinity are nature's way of manifesting infinity in the world.

So bottom line it's clear to me that we can't count the integers physically, but we can easily count them mathematically. And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works. Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so? — noAxioms

Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.

I utterly reject the notion that supertasks are a logical contradiction or metaphysical impossibility. They're only a historically contingent impossibility. We split the atom, you know. That was regarded as a metaphysical impossibility once too.

I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. — noAxioms

I'm not being specific like that. I'm only saying this:

There have been radical paradigm shifts in physics in the past;

There will certainly be radical paradigm shifts in the future; and

The next shift just may well incorporate some notion of physically instantiated infinities or infinitary processes; in which case actual supertasks may be on the table.


I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist; then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.

Mathematical curiosities often become physicists' tools a century or more later. I think it's perfectly possible that physically instantiated infinities may become part of mainstream physics at some point in the future.

I will close with two contemporary examples of where speculative physics is starting to think about infinity.

One, eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending. In this eternal multiverse are many bubble universes; either infinitely many, or at least a very large finite number. Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

And two, the many-world interpretation of quantum physics. Most people have heard of the Copenhagen interpretation, in which observing a thing causes the thing to be in one state or another; whereas before the measurement, it was neither in one state nor the other, but rather a superposition of the two states.

In Everett's many-world's interpretation, an observation causes the thing to be in both states in different universes. The universe splits in two, one in which the thing is in one state, and another universe it's in the other state. In some other universe I didn't write this. I know it sounds like bullshit, but Sean Carroll, a very smart guy and a prominent Youtube physicist (he's a real physicist too) is a big believer. He's recently moved away from mainstream physics, and more into developing a new philosophy of physics that incorporates many-worlds. How many worlds are there? Again this is a little vague, infinitely many or a large finite number.

These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists. Would anyone really bet that they personally can predict the next 200 years of physics?

By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask. — noAxioms

Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved.

Well that's for reading, there's been a lot of back and forth lately and I hope I was able to at least express what I think about all this.

This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc. — Michael

Yes, I agree with you that math and physics use different definitions.

I apologize for getting crabby last night. As I went to bed I was thinking, Why am I snarling at someone about supertasks, I don't even care about supertasks.

You're right, I was not the one you were originally addressing. I jumped in because I was annoyed by your total lack of logic in claiming that supertasks are metaphysically impossible or logical contradictions. I agree with you that supertasks don't exist physically today, but I allow for the possibility of new physics in the future, just as there's always been new physics in the past. I don't think you've supported your metaphysical or logical arguments. That's why I jumped in.

Also it's perfectly clear that I can walk a mile, and I first walked the first half mile, etc., so if someone (not me, really!) wanted to argue that supertasks exist on that basis, I'd say maybe they have a point.

P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.

P2. 30 + 15 + 7.5 + ... = 60

C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.

But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

So C1 is a contradiction. Therefore, as a proof by contradiction:

C2. P1 or P2 is false.

C3. P2 is necessarily true.

C4. Therefore, P1 is necessarily false.

And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum. — Michael

I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers.

But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence.

And suppose that in the first bubble universe, somebody says "1". And in the second bubble universe, somebody says, "2". Dot dot dot. And bubble universe are eternally created, with no upper bound on their number.

Therefore: Under these assumptions, there is no number that doesn't get spoken. And therefore, all the numbers are eventually counted.

You see we don't have to "reach the end," since we can't do that. All we have to do is show that there is no number that never gets counted. Therefore they all do. It's a standard inductive argument. You show something's true for all natural numbers because there can't be a smallest number where it's not true.

I remind you that while eternal inflation is speculative but is taken seriously by a lot of smart people.

Therefore I claim that it is metaphysically possible to physically count the natural numbers; and that no logical contradiction is entailed. I'll grant you that I haven't yet shown how to do it in finite time, and so I have not refuted your point. I'm giving more of a plausibility argument that someday, there might actually be a finite-time supertask. We just don't know. You personally can not know. That's my real point, bottom line.

You cannot know what future physics will allow or conceptualize. That's my whole argument. That's why I say that supertasks violate contemporary physics, Planck scale and all that. But based on the shocking paradigm shifts of the past, there will be shocking paradigm shifts in the future; and physically actualized infinitary processes are as good a candidate as any for what comes next.

I wrote a response to @NoAxioms above in which I laid out my thoughts, it might be of interest ... https://thephilosophyforum.com/discussion/comment/900398

Thanks again for your good cheer in not firing back!

There is no limiting process in the premises of the op, nor in what is described by ↪Relativist . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited. — Metaphysician Undercover

Wow that's deep. Deep and wrong at the same time. That's interesting.

If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer.

I suppose the analogy is color, which is in the eye-brain system of the observer, not in the object or even in the light.

But actually, the limit can be considered part of the sequence. Just as a sequence is a function defined on the natural numbers; a sequence along with its limit can be defined as a function on the natural numbers augmented with a point at infinity, which I've been calling $\omega$.

It's really no different than taking the set {1/2, 3/4, 7/8, ...} and augmenting it with the number 1, to yield the new set {1/2, 3/4, 7/8, ..., 1}. Surely you can see that 1 is a perfectly sensible number on the number line. In many ways it's the ONLY sensible number. All other numbers are derived from it. That and 0. Give me 0 and 1 and I'll build all the numbers anyone needs.

So if that's what you're saying, I find that a very interesting thought. But there is no reason to imbue limits with mysticism. They're very straightforward. They're just the value of a sequence at the augmented point at infinity; which, if you don't like calling it that, is just adding the number 1 to the 1/2, 3/4, ... sequence.

You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact. — Relativist

You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers, we can do something analogous with the natural numbers, and augment them with a symbolic point at infinity $\omega$, so that the augmented natural numbers look like this:

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

Now a sequence is just a function that for each of 1, 2, 3, ..., we assign the value of the sequence, the n-th term. And we can simply assign the limit as the value of the function at $\omega$. It's perfectly legitimate. We can define a function with ANY set as its domains. So a sequence is a function on $\omega$, and a sequence augmented with its limit (or any other value!) is just a function on $\omega + 1$.

This is a key point. I've stated it a number of times recently and I'm not sure I'm getting through. The natural numbers augmented with a point at infinity is a perfectly good domain for a function; and we can use such a function to identify each of the points of a sequence, along with the limit.

I also think you are misinterpreting the meaning of limit. — Relativist

On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth.

This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L. — Relativist

Wiki is not necessarily a good source for mathematical accuracy or subtlety of expression, and in this case they have led you astray.

I hope very much that you will give some thought to what I wrote about defining the limit of a sequence as the value of some function on the naturals augmented with a symbolic point at infinity; or more concisely, as a function on $\omega + 1$. A sequence is just a function on $\omega$, which is a synonym for $\mathbb N$. You can "attach" the limit to the sequence by extending the same function to one on $\omega + 1$ .

I hope this is clear. I find it an extremely clarifying mental model of what's going on with a sequence and its mysterious limit. "Where does the limit live?" I get that it's kind of confusing. The limit lives at the point at infinity stuck to the right end of the natural numbers.

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

That's how people need to learn how to count in order to better understand supertasks and limits.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
— fishfry
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does? — Relativist

Sure, after 1 second. It's perfect obvious from daily existence. When I got up to make a snack I did first walk halfway to the kitchen then halfway again. So now I'm arguing for supertasks. But I could just as well argue against supertasks. So whatever you said, I could probably convince myself to agree with it.

I think the word "reach" is being abused in this conversation. It comes out of badly taught calculus classes, and Wikipedia.

If I am understanding you: You say that if we have a sequence; that if that sequence happens to have a limit, then the limit is not inherent to the sequence, but is rather imposed by the observer. — fishfry

That's not quite what I'm saying. The process described by the op has no limit. That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process

I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass.

Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it.

That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit".

That's not quite what I'm saying. The process described by the op has no limit. — Metaphysician Undercover

Oh I had no idea we were still talking about the OP. This thread's gone way beyond that.

I thought you were making a more general point, that the limit lives in a different kind of conceptual space than the sequence itself, or that the limit was imposed on the sequence by observers.

If I misunderstood then nevermind. I've long forgotten the staircase problem. I don't think I ever actually understood it.

That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process — Metaphysician Undercover

Well 1/2 + 1/4 + 1/8 + ... is a well known convergent sequence. It converges to 1. And surely we've all experience one second going by. So that's the paradox, right?

I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass. — Metaphysician Undercover

It has not been productive in the past for us to discuss mathematics, and your misunderstanding of limits is not my job to fix. I gave at the office. Nothing personal but you know we have been down this road. I sort of get what you're saying but mostly not. "The process carries on, unlimited, even though there's a limit." I haven't the keystrokes to untangle the myriad conceptual difficulties with that statement, and the beliefs and mindset behind it; even if I had the inclination. I hope you'll forgive me, and understand.

Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. — Metaphysician Undercover

LOL. And the meaning of Moby Dick is only because of what we all determined the symbols to mean. Man and His Symbols, Jung. Yes we are symbolic beasts.

But within the sphere of math, the definition of a limit is as objective as can be. We lay down a definition, you know the business with epsilon and L, and we confirm that the sum converges; just as in the sphere of the English language, Moby Dick is a story about a bunch of guys who go whaling and it mostly doesn't end well.

To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it. — Metaphysician Undercover

I'm sorry, I can't really talk about the staircase problem specifically, I never paid much attention to it at the beginning. I mostly got interested in this thread when other issues were introduced. But mathematicians are very good at determining limits, and the one in question is perfectly well known to everyone who ever took a year of calculus. You might take a look at the Wiki page on limits.

That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit". — Metaphysician Undercover

You don't need any esoteric "principles other than those stated in the description of the process" to determine the sum of a geometric series as a particular limit.

I think "reciting natural numbers" is a red herring, because it's perfectly clear that there are only finitely many atoms in the observable universe, and that we can't physically count all the natural numbers. — fishfry

Then rather than recite the natural numbers I recite the digits 0 - 9, or the colours of the rainbow, on repeat ad infinitum.

It makes no sense to claim that my endless recitation can end, or that when it does end it doesn't end on one of the items being recited – let alone that it can end in finite time.

So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible.

Given that ad infinitum means "without end" — Michael

Yes, it does. But there is a small but significant mistranslation there. I have no problem with saying that "infinite" means "endless", but "ad" does not mean "without". It means "to".
So there are two different ways of thinking about infinity embedded here. One thinks of infinity as a destination, which, paradoxically, cannot, by definition, be reached. The other doesn't, but rather denies that there is a destination. At first sight, one wants to say that the second is correct.
But how do we know that the operation "+1" generates a sequence without end? In one way, it doesn't seem absurd to think that it might be done. After all, given that "+1" can be defined, the result of every step is determined (fixed) - the whole sequence is always already there, for us to inspect. But that seems a mistake; we can't survey the whole sequence and notice that there is no end.
Well, there is the proof that there can be no largest (natural) number. We can prove lots of other things, as well. We don't have to survey the whole sequence to do any of that. It really is magical, and yet inescapably logical. (Poetry? Perhaps. But this is about how we think about things, so it is also philosophy.)

So it seems that we are locked into two incompatible ways of thinking about infinity. One as if it were a sequence which stretches away for ever. The other as a succession of operations which can be continued for ever. (Two metaphors - one of space, one of time.) I'm not suggesting it needs to be resolved, just that we are subject to confusion and need to think carefully, but also recognize that our normal ways of thinking here will need to be adapted and changed.

Well I can walk a mile, and I first walked the first half mile, and so forth, so it's a matter of everyday observation that supertasks exist. That would be an argument for supertasks. Zeno really is a puzzler. I don't think the riddle's really been solved. — fishfry

Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them.

The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass. — Metaphysician Undercover

Yes, quite so. But it follows that applying the calculus to Achilles doesn't demonstrate that Achilles will overtake the tortoise. I think that only ordinary arithmetic can do that.

Then rather than recite the natural numbers I recite the digits 0 - 9 on repeat ad infinitum.
It makes no sense to claim that I can finish repeating the digits ad infinitum, or that when I do I don't finish on one of those digits.
This is an issue of logic and nothing to do with what is physically possible. — Michael

Does it make any sense to claim that you can repeat the digits ad infinitum? All you can do is repeat the digits again and perhaps promise or resolve to repeat them again after that.

The truths of mathematics and logic are timelessly true, aren't they? There is no change in that world. We frame them in the present tense, but it is, grammatically speaking, the timeless present, not the present that is preceded by the past and followed by the future. We speak of logical operations, but what does that mean? Their results are always already fixed - determinate. So when we carry them out - nothing in logic changes.