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... — Relativist

Good source. It says that the limit is approached as the input approaches the specified value.
This means that the limit isn't reached at some finite point in the series, exemplified by the comment:
"This means that the value of the function f can be made arbitrarily close to L, by choosing x sufficiently close to c"
The approaching goes on while x is still at some finite step.

Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete.

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? — Metaphysician Undercover

Read carefully. I didn't say that.

I said
1) that discussion of the question above and your personal beliefs in the matter is off topic
and
2), that you [cannot / choose not to] understand what others mean when they presume what Michael conveyed better than I could:

That we coin the term “X” to refer to some Y isn’t that Y depends on us referring to it using the term “X”. This is where you fail to make a use-mention distinction. — Michael

You (M-U) seem to either not be able to separate "X" and Y, or you refuse to communicate with those that do.

And do you believe that...

My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X".

That my dog is named "Bella" depends on me. That Bella exists and eats and sleeps does not depend on me.

That this period of time is named "60 seconds" depends on us. That 60 seconds pass does not depend on us.

You don't seem to understand how reference works.

"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. — fishfry

I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation.

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. — Ludwig V

The problem, as displayed in my reply to fishfry above, is that each of the two incompatible ways will use the same terms. The same terms will then have incompatible meanings as you demonstrate with "ad infinitum", and the natural tendency for human beings not to take the time required to detect such differences, leads to equivocation and the appearance of paradox.

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. — Ludwig V

That's right, and the issue with Achilles and the tortoise has extra complexities which are often overlooked. Achilles and the tortoise are both moving. At any point in time, t1, Achilles is at a location and the tortoise is at a location. At t2, Achilles reaches the location where the tortoise was at t1. But the tortoise has moved to a new location. At t3 Achilles reaches the location where the tortoise was at t2, the tortoise has moved to a new location. And so on. In this case, therefore, it is not only impossible for Achilles to overtake the tortoise, it is also impossible for Achilles to catch up to the tortoise, so it appears like the tortoise will necessarily beat Achilles to the finish line.

We cannot describe the tortoise's position as a simple limit to Achilles' position, because the tortoise is already moving at a constant velocity, and no matter how fast Achilles accelerates he cannot catch up to the tortoise. This is the problem of acceleration, which demonstrates the fundamental incompatibility between distinct rest frames. Einstein attempted to bridge this incompatibility by stipulating the speed of light as the limit, (therefore absolute rest frame) in his special theory of relativity.

Read carefully. I didn't say that. — noAxioms

That's why I was asking, "are you saying...". You seemed to be saying what I asked about, but I was not sure. Now you are making it clear that you were saying something else. So I now understand that you were attacking a straw man effigy of me, and that's why I couldn't understand you.

2), that you [cannot / choose not to] understand what others mean when they presume what Michael conveyed better than I could: — noAxioms

If I don't understand, then I don't understand. I believe that I said that I don't understand why someone would make an assumption which to me is so clearly false. So to explain why someone would make such an assumption, would require an explanation as to how this obviously false assumption makes some sense to the person. The quoted passage, in which you say Michael makes sense of his perspective, makes no sense to me, because it is a straw man representation of what I am arguing. I told Michael that I didn't understand how this was relevant, and he needed to explain its relevance. I now fully understand that it was a straw man, and that's why I couldn't apprehend the relevance. Michael did not understand what I was arguing, therefore produced a straw man representation of me, and I could not understand what Michael was accusing me of, because he did not understand me and tried to defend against a straw man of me.

You (M-U) seem to either not be able to separate "X" and Y, or you refuse to communicate with those that do. — noAxioms

I have no problem distinguishing between the sign and what is signified. The problem here is that Michael refused to recognize the distinction between when the thing signified is a real physical object, and when the thing signified is a mental fabrication, a fiction, or an ideal. So Michael was talking about ideals, principles of measurement, like a day, and a second, as if these mental fabrications have some sort of existence independent from human minds, in some Platonic realm or something like that.

My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X". — noAxioms

I'll repeat, just so that you'll quit with this utterly ridiculous straw man accusation you are hitting me with. I have no problem whatsoever with the separation between the symbol and the thing represented by the symbol. That is not at all what is at issue here, so you are simply making a straw man representation of what I am arguing.

The problem here is with the nature of the thing represented. Do you understand that sometimes what is represented by a symbol is a real physical object, and sometimes what is represented by a symbol is an ideal, such as a mathematical object? And do you understand that these two types of objects are completely different, and need to be understood in completely different ways. My claim is that it is highly doubtful that an ideal such as "a second" has any existence independent from the human minds which I believe fabricate them, and propagate them through educational processes.

Michael has been arguing that "a second" has some sort of real independent existence, just like a physical object which we can point to, instead of being an ideal which is propagated by human minds, as I claim. So, I've asked him to show me such a thing as "a second". All he has provided is a definition referring to some ideal state (ground state) of a cesium atom. Since the definition refers to an ideal state (ground state), rather than any real existing physical object, it seems very clear that "a second" is an ideal, not an independent physical thing.

That this period of time is named "60 seconds" depends on us. That 60 seconds pass does not depend on us.

You don't seem to understand how reference works. — Michael

This is not a simple issue of reference, it is an issue of the type of thing which is referred to. In the example here "seconds" is an ideal, just like any other principle of measurement, metre, foot, degree Celsius, circle, triangle, and number in general. They are all ideal, and are therefore dependent on the human minds which employ them, unless you assume some sort of Platonic realm for the independent existence of such ideals.

So, we say "time is passing", just like we say "the earth is spinning around the sun", and we refer to real independent things with those phrases, "time", "the earth", "the sun". However, when we go to measure those things, we employ principles, which are ideals, and have no such independent existence. So, when we say "60 seconds has passed", "60 seconds" refers to an ideal which has been applied to measure the independent passing of time. The 60 units of seconds do not pass, time passes, and "60 seconds" refers to the ideal which is employed.

For example, we could say "the earth circles the sun". In this case, "circles" refers to the ideal which is employed in measuring the earth's motions. We know that a circle is an ideal figure, and that true circles do not have any real existence independently from human minds, because pi, which is the essential feature of a circle, is an indefinite, irrational number. So the motion of the earth isn't really a circle, that's just the ideal which is referred to in that statement. Likewise, in the statement "60 seconds" has passed, "seconds" is just the ideal which is referred to, as the principle employed in the act of measurement. And we find these ideals in all forms of measurements, metres, degrees of temperature, frequencies, etc..

We cannot describe the tortoise's position as a simple limit to Achilles' position, because the tortoise is already moving at a constant velocity, and no matter how fast Achilles accelerates he cannot catch up to the tortoise. This is the problem of acceleration, which demonstrates the fundamental incompatibility between distinct rest frames. Einstein attempted to bridge this incompatibility by stipulating the speed of light as the limit, (therefore absolute rest frame) in his special theory of relativity. — Metaphysician Undercover

I do agree with you that people seem not to understand the meaning of limit in this context. Many of them seem to think that calculus solves the problem, though it clearly doesn't.
As to special relativity, I'm not going to argue with you. Most people neglect acceleration, on the grounds that including it won't make any material difference. Does Einstein's theory tell us when Achilles will overtake the tortoise? I thought that both started at the same time, and therefore in the same rest frame. When Achilles overtakes the tortoise, won't they both be in the same rest frame?
Neglecting acceleration, let's say Achilles gives the tortoise a head start of 100 units of length and that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds.

There are similar paradoxes that don't involve two moving elements:-

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on. — Wikipedia

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. — fishfry

You explained your interpretation:

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 — fishfry

So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number.

Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1

Your blurred analysis conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers.

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. — fishfry

The relativity thing was more of a refinement and had little practical value for some time. Newtonian physics put men on the moon well over a half century later.
QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?

QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is.

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 it by placing its elements into order-bijection with themselves. 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.

What is this 'the former'? The physical activity of making a declaration? There's definitely some abstraction going on there, as there is with any deliberate activity.
The latter seems to be the expression of a rule that maps the two halves of the bijection, which seems to be about as physical of an activity as was the declaration.

if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation?

No argument here.

So bottom line it's clear to me that we can't count the integers physically

Depends on what you mean by count, and especially countable, since plenty of equivocation is going on in this topic.
If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them.

but we can easily count them mathematically

Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics?

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.

That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things.
I never made the claim that a supertask is physically possible. I simply followed through with it as a premise, which, unless falsified, can be physically true 'as far as we know'.

Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one.

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.

We split the atom, you know. That was regarded as a metaphysical impossibility once too.

QM does very much suggest the discreetness of matter, but Zeno's premise doesn't rely on the continuity of matter. It works best with a single fundamental particle moving through continuous space and time, and overtaking another such particle.

The next shift just may well incorporate some notion of infinitary set theory; in which case actual supertasks may be on the table.

They were never off the table since current physics doesn't forbid them. Maybe future physics will for instance quantize either space or time (I can think of some obvious ways to drive that to contradiction). Future findings take things off the table, not put new ones on. The initial state of physics is "I know nothing so anything is possible'.

I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist

Heh, despite the detractor standing on an obvious example of such a geometry.

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.

Octonians shows signs of this sort of revolution.

eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending.

Actually, the big bang theory already does that much.
Yes, I know about eternal inflation, and something like it seems necessary for reasons I gave in my prior post.

Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

It is a mistake to talk about 'time creating these other universe'. Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one.

All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses.

And two, the many-world interpretation of quantum physics.

That's the type III.

In Everett's many-world's interpretation, an observation causes the thing to be in both states.[/quote]Ouch. Is that a quote? It did not match any google search.
Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism.

In some other universe I didn't write this. I know it sounds like bullshit,

I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it.

These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists.

Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable.
I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear.

Well I can walk a mile

Ah, local boy. I am more used to interacting with those who walk a km. There's more of em.

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. — fishfry

That wording implies a sort of meaningful simultaneity that just doesn't exist.

And suppose that in the first bubble universe, somebody says "1".

The universes in eternal inflation theory are not countable.

Yes, each step in a supertask can and does have a serial number. That's what countably infinite means.

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. — Michael

You're not going to get past step 10 at best. I just takes longer than the step duration to recite a syllable. I don't think this is your point, but it's a poor wording due to this. Yes, step 13 has a defined duration at known start and stop times. The duration simply isn't long enough to recite anything.

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",

No. It means 'without final step'. You're apparently equivocating "without end" to mean that the process is incomplete after any amount of time.

What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

There we go with the finite definition again.
"The sequence of operations ends" means that "all operations in the sequence are performed".

This is a great example of the endless repetition of assertions/bad-definitions I'm seeing in this topic. Surely you know this answer is coming from me.

C2. P1 or P2 is false.

C3. P2 is necessarily true.

C4. Therefore, P1 is necessarily false. — Michael

Is it? You take supertasks to mean that time is discrete instead of continuous, meaning there is a smallest amount of time. If that is so, P2 is necessarily true according to mathematical theorems, but it is not representative of reality and especially not of time.

Relevant:

On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy. From Black’s argument one can see that the Zeno Dichotomy cannot be completed in the first sense. But it can be completed in the second. The two meanings for the word “complete” happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.

Hermann Weyl (1949, §2.7) suggested that if one admits that the Zeno race is possible, then one should equally admit that it is possible for a machine to carry out an infinite number of tasks in finite time. However, one difference between the Zeno run and a machine is that the Zeno run is continuous, while the tasks carried out by a machine are typically discrete. This led Grünbaum (1969) to consider the “staccato” version of the Zeno run, in which Achilles pauses for successively shorter times at each interval.

"The sequence of operations ends" means that "all operations in the sequence are performed". — noAxioms

The operations in the sequence occur one after the other, so all operations are performed only if some final operation is performed.

The logic of consecutive tasks is different to the logic of concurrent tasks. Your account equivocates.

If I never stop counting then … I never stop counting, and if I never stop counting then at no time have I ever counted every number.

As a regressive version of the argument, rather than me speeding up as I recite the numbers up to infinity let’s say that I slow down as I recite the numbers down from infinity.

At the 60 second mark I said “0”, at the 30 second mark I said “1”, at the 15 second mark I said “2”, at the 7.5 second mark I said “3”, etc.

Is it metaphysically possible for such a task to have been performed? No, because there is no first number that I could have started with.

That we can sum an infinite series with terms that match the described (and implied) time intervals is a red herring.

There is a far more fundamental, non-mathematical, logical impossibility with having counted down from infinity, and that very same fundamental, non-mathematical, logical impossibility applies with having counted up to infinity as well. You're being bewitched by maths if you think otherwise.

Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete — noAxioms

Infinity is not reached. You're not considering what it means to be infinite in this context: it means continually dividing the remaining time (prior to the 1-minute mark) in half. Because the remaining time corresponds to a real number line, the process proceeds without ending because the remaining time is infinitely divisible. It's limited by the fact that all points of time that are reached by the process are less than 1 minute- so it is logically impossible for this process to reach the point of time of 1 minute.

The source of confusion is that the clock does hit the 1-minute mark. You are incorrectly interpreting this as implying the the process reaches that point. It can't, because it is logically impossible.

The clock reaching the 1-minute mark implies the process ends, but since the process cannot reach the 1-minute mark, the process must be terminated at some point. Laws of nature would clearly provide a limit to how small we divide the time, but even setting that aside - a stopping point is logically necessary. The math doesn't identify any particular stopping point, but it does imply there has to be one.

The math doesn't identify any particular stopping point, but it does imply there has to be one. — Relativist

An exercise in free will. At each n the cube changes from white to black or vice-versa for time = 1-1/2^n. The clock runs out so you are free to say the process has ended and the final color is black. Your friend can say, no it is white. But you will prevail.

I can explain it very easily. There is two different senses of "limit" being used here. One is a logical "limit" as employed in mathematics, to describe the point where the sequence "converges". And "unlimited" is being used to refer to a real physical boundary which would be place on the process, preventing it from proceeding any further. There is no such "limit" to a process such as that described by the op. The appearance of paradox is the result of equivocation. — Metaphysician Undercover

Mathematicians would just refer to it as an "upper bound."

But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics.

It's only a conceptual thought experiment. And why shouldn't math apply to that?

But anyway, it's an upper bound. If it's a least upper bound, it's a limit.

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

I think a lot of people feel that way.

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. — Michael

The natural numbers do not end, yet they have a successor in the ordinal numbers, namely $\omega$. This is an established mathematical fact.

I regard this as a helpful point of view when analyzing these kind of puzzles. I've explained it as best I can.

"It makes no sense" is not a logical argument. It's only a description of your subjective mental state. Once, violating the parallel postulate or the earth going around the sun or splitting the atom made no sense. You are not making an argument.

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

If you only demonstrated the reductio. All you have is "it makes no sense," and that is not an argument.

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. — Michael

The natural numbers do not end, yet they have a successor in the ordinal numbers, namely ω. This is an established mathematical fact. — fishfry

I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other.

I didn't like ω at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ω is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ω can be used in calculations. So that's that. See the Wikipedia article on this for more details.

But it is also perfectly true that a recitation of the natural numbers cannot end. As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory.

In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right. I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse.

So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number. — Relativist

No. My thinking about limits is extremely precise and perhaps a bit more general than what you're accustomed to. I have never said that a series (or sequence if that's what you mean here) reaches infinity. I would not say that, and I did not say that.

What I said was that there is a mathematical view that sheds light on the subject, and makes it clear in where the limit of a sequence lives. The sequence 1/2, 3/4, 7/8, ... has the limit 1. Of course it never "reaches" 1. But you would have no objection to my putting {1/2, 3/4, ..., 1} into a set together. After all, I am allowed to take unions of sets: and {1/2, 3/4, ...} U {1} = {1/2, 3/4, ..., 1}. So it's a legit set.

Now 1 is in no way a "point at infinity," after all it's just the plain old number 1. And no member of the sequence ever "reaches" it. But it does live there as the limit; as the result of a well-defined limiting process.

I have suggested this mathematical model as a thought aid to these kinds of paradoxes. If you find it helpful all to the good, but if not, that's ok too. I find it helpful.

For what it's worth, in math, the natural numbers have an upward limit, called $\omega$, that plays the same role for the sequence 1, 2, 3, ... that the number 1 is for the sequence 1/2, 3/4, ...

It's the limit. It's more general notion of limit, one that allows us to reason about a "point at infinity." Which is exactly what these puzzles are about. That's why it's a handy framework for thinking about these kinds of puzzles.

You have a sequence that's defined (on/off, on a step, whatever) at each member of a convergent sequence; and you want to speculate on the definition at the limit. $\omega$ is exactly what you need; or rather, a set called $\omega + 1$, which is like the set {1/2, 3/4, ,,,, 1}. It's a set that contains an entire infinite sequence and its limit. It's exactly what we need to analyze these problems.

If it helps, here's the Wiki page on ordinals, at least so that you know they're a real thing. You can "keep counting past the natural numbers," and you get some very cool mathematical structures. Ordinals find application in proof theory and mathematical logic.

Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. — Relativist

I'm going to defer talking about supertasks today, had enough for a while.

There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1 — Relativist

Please reread what I wrote. This is not on topic if you understand what I'm saying.

Your blurred analysis — Relativist

I'm doing my best to fit you with a sharper pair of mathematical eyeglasses to unblur your vision ... but you keep making a spectacle of yourself!!

conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers. — Relativist

You might consider using words like "reach" and "approach" with precision. They are not part of the mathematical definition of a limit. They're casual everyday synonyms that you are allowing to confuse you.

I've watched this debate for a long time - though I don't claim to have understood all of it. But I think those two quotes show that you are talking past each other. — Ludwig V

He'll come around :-)

I didn't like ω at all, when it was first mentioned. I'm still nowhere near understanding it. But the question whether a mathematical symbol like ω is real and a number is simply whether it can be used in calculations. That's why we now accept that 1 and 0 are numbers and calculus and non-Euclidean geometries. ω can be used in calculations. So that's that. See the Wikipedia article on this for more details. — Ludwig V

This paragraph gratified me. If you are struggling to understand my posts then I'm getting through to at least one person. My talk about $\omega$ is something most people haven't seen, but the ideas aren't that hard. For what it's worth there's a Wiki page on ordinal numbers. The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up.

You have a great insight that what makes a mathematical concept real is, in the end, its utility. Sometimes not even to anything practical, but just to math itself. We want to solve the equation x + 5 = 0 so we invent negative numbers. That kind of thing. In that sense, the ordinals exist.

But another way to think about it is that it's just an interesting new move in a game. As if you were learning chess and they told you how the knight moves. You don't say, "Wait, knights slay dragons and rescue damsels, they don't move like that." Rather, you just accept the rules of the game. You can think of ordinals like that. Just accept them, work with them, and at some point they become real to you. Just like the moves of the chess pieces any other formal game.

But I have tried to give a very concrete, down to earth example of how this works.

Suppose that we have the sequence 1/2, 3/4, 7/8, ... It converges to 1.

Now we can certainly form the set {1/2, 3/4, 7/8, ..., 1}. It's just some points in the closed unit interval.

But it gives us a model, or an example, of a set that contains an entire infinite sequence that "never ends" blah blah blah, and also contains its limit.

If you believe in the set {1/2, 3/4, 7/8, ..., 1}, then you should have no trouble at all believing in the set

{1, 2, 3, ..., $\omega$}. That's also just a set that contains an entire infinite sequence, along with its limit. We typically don't encounter this concept in the math curriculum that most people see, but it's perfectly standard once you go a little further. Also a lot of people have seen the extended real numbers with $\pm \infty$ and nobody complains about that, or do they?

It's true that the distances are different inside the two sets. But in terms of order, the two sets are exactly the same: an infinite series, along with its limit.

Anyway, this framework is very handy for understanding supertask type problems. That's why I'm mentioning it.

So if you don't like $\omega$, that' s no problem. Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence.

Does that help?

But it is also perfectly true that a recitation of the natural numbers cannot end. — Ludwig V

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers.

And if we can't imagine that, we can certainly imagine {1/2, 3/4, 7/8, ..., 1}. There's nothing mysterious about that. An entire infinite sequence is in there, along with an extra point. It's a legitimate set.

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work. They are containers for infinite collections of things.

By the way, $\omega$ is the "point at infinity" after the natural numbers. And $\omega + 1$ is the name for the set {1, 2, 3, 4, ..., $\omega$}.

$\omega + 1$ is the natural setting for all supertask puzzles. We have the state at each natural number, and we inquire about the final state at $\omega$.

That's why I like $\omega + 1$ as a mental model for these kinds of problems.

As I said earlier, it is remarkable that we can prove it. Yet we cannot distinguish between a sequence of actions that has not yet ended from one that is endless by following the steps of the sequence. So we are already in strange territory. — Ludwig V

This business about actions is what confuses people. They set up scenarios that violate the laws of physics, like the lamp that switches in arbitrarily small intervals of time, and then they try to use physical reasoning about them. Then they get confused.

In the way I'm describing this, you may think that the difference is between the abstract world (domain) of mathematics and another world, which might be called physical, though I don't think that is right. — Ludwig V

Well yes, you are correct to feel that it's not quite right. Because there is nothing physical about the lamp or the staircase. So it's a category error to try to use everyday reasoning about the physical world. That's why people get confused.

I'm very puzzled about what is going on here, but I'm pretty sure that it is more about how one thinks about the world than any multiverse. — Ludwig V

I think it all comes down the fact that calculus classes care about computation and not theory. That, and the fact that we don't know the ultimate nature of the world, and there's are good reasons to think it's not anything like the mathematical real numbers.

So on the one hand, the continuity of the world is an open question. And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion.

The natural numbers do not end, yet they have a successor in the ordinal numbers, namely . This is an established mathematical fact. — fishfry

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

See here.

The page itself isn't all that enlightening, but it does at least show that the ordinal numbers really are a thing in math, I'm not just making it all up. — fishfry

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

This business about actions is what confuses people. — fishfry

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

By the way, ω is the "point at infinity" after the natural numbers — fishfry

What does "after" mean here?

If you want to think about the sequence 1/2, 3/4, 7/8, ... "never ending," that's fine. Yet we can still toss the entire sequence into a set, and then we can toss in the number 1. That's how sets work — fishfry

Yes, but it seems to me that this is not literally true, because numbers aren't objects and a set isn't a basket. (I'm not looking for some sort of reductionist verificationism or empiricism here.)

Just think about {1/2, 3/4, 7/8, ..., 1}. It's the exact same set, with respect to what we care about, namely the property of being an infinite sequence followed by one extra term that occurs after the sequence. — fishfry

In that respect, yes. But I can't help thinking about the ways in which they are different.

That's a confusing way to think about it. It "ends" in the sense that we can conceptualize all of the natural numbers, along with one extra thing after the natural numbers. — fishfry

Yes. But it doesn't end in the sense that we can't count from any given natural number up to the end of the sequence.

And two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. — fishfry

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

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

Fair enough. That coincides with my intuition that supertasks are not possible. But given that they are not physically possible either, can I conclude that they are not possible at all?

Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion. — fishfry

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

But you talk about a "real physical boundary." Here you imagine that the staircase is physical. It's not. The conditions of the problem violate known laws of physics. — fishfry

I really don't see how there could be a staircase which is not physical. That really makes not sense. However, just like in the case of the word "determine", we need to allow for two senses of "physical". You seem to be saying that to be physical requires that the thing referred to must obey the laws of physics. But the classic definition of "physical" is "of the body". And when a body moves itself, as in the case of a freely willed action, that body violates Newton's first law. Therefore we have to allow for a sense of "physical" which refers to things which are known to violate the laws of physics, like human beings with freely willed actions.

What is implied here is that the laws of physics are in some way deficient in their capacity for understanding what is "physical" in the sense of "of the body". That's why people commonly accept that there is a distinction between the laws of physics and the laws of nature. The laws of physics are a human creation, intended to represent the laws of nature, that is the goal, as what is attempted. And, so far as the representation is true and accurate, physical things will be observed to obey the laws of physics, but wherever the laws are false or inaccurate, things will be observed as violating the laws of physics. Evidently there are a lot of violations occurring, with anomalies such as dark energy, dark matter, etc., so that we must conclude that the attempt, or goal at representation has not been successful.

It's only a conceptual thought experiment. And why shouldn't math apply to that? — fishfry

Sure, it's a conceptual thought experiment, but the interpretation must follow the description. A staircase is a staircase, which is a described physical thing, just like in Michaels example of the counter, such a counter is a physical object, and in the case of quantum experiments, a photon detector is a physical object. And of course we apply math to such things, but there are limits to what we can do with math when we apply it, depending on the axioms used. The staircase, as a conceptual thought experiment is designed to expose these limits.

But anyway, it's an upper bound. If it's a least upper bound, it's a limit. — fishfry

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

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

However, since our empirical observations never produce a scenario like the staircase, that inconsistency appears to be irrelevant to the application of the mathematics, with those limitations inherent within the axioms. The limitations are there though, and they are inconsistent with what the staircase example demonstrates as logically possible, continuation without limitation. Therefore we can conclude that this type of axiom, completeness axioms, are illogical, incoherent. The real problem is that as much as we can say that the staircase scenario will never occur in our empirical observations, we cannot conclude from this that the incoherency is completely irrelevant. We have not at this point addressed other scenarios where the completeness axioms might mislead us. Therefore the incoherency may be causing problems already, in other places of application.

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

the set {1/2, 3/4, 7/8, ..., 1} — fishfry

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

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

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

and we inquire about the final state at ω

Which works until you ask if ω is even or odd.

Using mathematics to try to prove that supertasks are possible is a fallacy. — Michael

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

The example of a ball skipping on a table to make it change colours seems to be a fusion of Zeno's walk and Thompson's lamp, because 1 – there is no limit, 2 – there no final step.

The time it takes for the ball to finish a revolution and touch the table decreases by half every time. If time is continuous and infinitely divisible, as time approaches 60s, the number of skips goes to infinity, but while the ball is skipping, it does not reach 60 seconds. 60 seconds it exactly when the ball stops skipping, and there is no specified state as to what the table will do when that happens. Thus, the speed in which the table changes colours approaches infinity the closer you get to 60s, but this says nothing about what the table will do at 60s, you might as well say it will turn transparent.

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.

On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy. From Black’s argument one can see that the Zeno Dichotomy cannot be completed in the first sense. But it can be completed in the second. The two meanings for the word “complete” happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.

So the ball keeps skipping and changing the table's colour.

For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description.

I conclude the table and the ball have incomplete description too.

If supertasks had proven their case there would be no debate as to whether time is continuous or discrete, but it doesn't seem to be the case.

Is it metaphysically possible for such a task to have been performed? No, because there is no first number that I could have started with. — Michael

That is the reverse Zeno walk. Achilles starts running but he can't start running because there is no first lenght to be run. But yet we reverse time and Achilles can finish the task. The argument doesn't need to be reversed, it is the same as saying you can't count to infinity because there is no last numeber to be counted. But if we admit that time is infinitely divisible, counting to infinity doesn't seem to amount to a logical impossibility, and so we reverse the time of the task.

This led Grünbaum (1969) to consider the “staccato” version of the Zeno run, in which Achilles pauses for successively shorter times at each interval.

But if we admit that time is infinitely divisible, counting to infinity doesn't seem to amount to a logical impossibility, and so we reverse the time of the task. — Lionino

And that's where you're being deceived by maths. We can't have counted down from infinity because there is no first number and so we can't have counted up to infinity because there is no last number.

The fact that an infinite series can have a finite sum is a red herring in both cases.

And that's where you're being deceived by maths. We can't have counted down from infinity because there is no first number and so we can't have counted up to infinity because there is no last number. — Michael

Can we not count the intervals starting with 1? Would that number not tend towards infinity given time is infinitely divisible or approach a certain value and terminate given a smallest sliver of time exists?

Can we not count the intervals starting with 1? Would that number not tend towards infinity given time is infinitely divisible or approach a certain value and terminate given a smallest sliver of time exists? — ToothyMaw

"Tending towards infinity" means counting through the natural numbers - the set is infinite. The process has no end.

"Tending towards infinity" means counting through the natural numbers - the set is infinite. The process has no end. — Relativist

I know, I'm saying the second part as the alternative to time being infinitely divisible.

Sorry, I overlooked that part.

It's all good. :up:

And you are right anyways; I should have been clearer that I didn't think we would ever actually finish counting the number of intervals given time is infinitely divisible.

The relativity thing was more of a refinement and had little practical value for some time. Newtonian physics put men on the moon well over a half century later.
QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?[/quot]

Relativity more of a refinement? Not a conceptual revolution? I don't think I even need to debate that. In any even it's a side issue. It's clear that the universe doesn't care what mathematics people use. In that sense, the laws of nature are exempt from mathematics. Historically contingent human ideas about the world are always playing catch up to the world itself. But if you disagree that's ok, it's a minor sidepoint of the discussion.
— noAxioms

QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is. — noAxioms

I don't really care much about supertasks and haven't argued that they're coherent or incoherent. I'm mostly trying to clarify some of the bad reasoning around them.

If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them. — noAxioms

Ok. I think some of the quoting got mangled since things I said ended up as part of your post.

But if anyone thinks I can't count all the natural numbers 1, 2, 3, ... by mathematical means, please identify the first one I can't count.

Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics? — noAxioms

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

a) A natural number; or

b) the set of natural numbers, to establish countability; or

c) some ordinal number, if one is a set theorist or logician or proof theorist.

That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things. — noAxioms

Lost me. As far as we know takes into account the great conceptual revolutions of the past, as evidence that there will be more such in the future.

They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one. — noAxioms

Ok. I think I'm a little lost in the quoting and not actually sure what we are talking about here. I'm not strenuously defending whatever ideas you're concerned with.

Octonians shows signs of this sort of revolution. — noAxioms

Well ... ok.

Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them. — noAxioms

But that's exactly my point. If speculative physics is starting to take physically instantiated infinity seriously, then it's perfectly reasonable that in the future, physically instantiated infinity may become a core aspect of physics; in which case supertasks may be on the table.

It is a mistake to talk about 'time creating these other universe'. — noAxioms

Was this for me? I never said any such thing nor quoted anyone else saying it.

Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one. — noAxioms

I'll have to plead ignorance on the question of whether there's a meta-universal time that transcends the bubble universes. Good question though.

All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses. — noAxioms

You are completely agreeing with my point. That if speculative physic already includes infinity, then mainstream physics may include infinity in the future.

And two, the many-world interpretation of quantum physics.
That's the type III. — noAxioms

You are agreeing with my point.

Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism. — noAxioms

Nothing to do with my point, which is that speculative physics already includes infinity, therefore mainstream physics may include infinity after the next scientific revolution.

I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it. — noAxioms

You're agreeing with me again. Why are you typing this stuff in? You've kind of lost me.

Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable. — noAxioms

Ok.

I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear. — noAxioms

Ok.

Ah, local boy. I am more used to interacting with those who walk a km. There's more of em. — noAxioms

Depends on the exchange rate.

And suppose that in the first bubble universe, somebody says "1".
The universes in eternal inflation theory are not countable. — noAxioms

Wow. You have evidence for that? My understanding is that it is an open question in eternal inflation as to the cardinality of the bubbles: finite, countably infinite, or uncountable. But either way my point about reciting the integers stands. I don't actually get the sense that you're engaging with anything I wrote.

Yes, each step in a supertask can and does have a serial number. That's what countably infinite means. — noAxioms

That's not the definition of supertask others are using. But I used the example of bubble universes to illustrate the possibility of counting the natural numbers physically.

Anyway sorry if I got lost in the quoting and didn't really understand some of your responses.