Good source. It says that the limit is approached as the input approaches the specified value.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
Read carefully. I didn't say that.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
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
My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X".And do you believe that...
"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
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
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
Read carefully. I didn't say that. — noAxioms
2), that you [cannot / choose not to] understand what others mean when they presume what Michael conveyed better than I could: — noAxioms
You (M-U) seem to either not be able to separate "X" and Y, or you refuse to communicate with those that do. — noAxioms
My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X". — noAxioms
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
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.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
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
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.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
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.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
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.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.
No argument here.if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation?
Depends on what you mean by count, and especially countable, since plenty of equivocation is going on in this topic.So bottom line it's clear to me that we can't count the integers physically
Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics?but we can easily count them mathematically
That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things.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.
They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one.Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.
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.We split the atom, you know. That was regarded as a metaphysical impossibility once too.
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'.The next shift just may well incorporate some notion of infinitary set theory; in which case actual supertasks may be on the table.
Heh, despite the detractor standing on an obvious example of such a geometry.I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist
Octonians shows signs of this sort of revolution.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.
Actually, the big bang theory already does that much.eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending.
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.Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.
That's the type III.And two, the many-world interpretation of quantum physics.
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.In some other universe I didn't write this. I know it sounds like bullshit,
Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable.These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists.
Ah, local boy. I am more used to interacting with those who walk a km. There's more of em.Well I can walk a mile
That wording implies a sort of meaningful simultaneity that just doesn't exist.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
The universes in eternal inflation theory are not countable.And suppose that in the first bubble universe, somebody says "1".
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.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
No. It means 'without final step'. You're apparently equivocating "without end" to mean that the process is incomplete after any amount of time.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",
There we go with the finite definition again.What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?
C2. P1 or P2 is false.
C3. P2 is necessarily true.
C4. Therefore, P1 is necessarily false. — Michael
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
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.Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete — noAxioms
The math doesn't identify any particular stopping point, but it does imply there has to be one. — Relativist
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
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
So I treat supertasks as a reductio ad absurdum against the premise that time is infinitely divisible. — Michael
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
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
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
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
Your blurred analysis — Relativist
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
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
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
But it is also perfectly true that a recitation of the natural numbers cannot end. — Ludwig V
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
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
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
The natural numbers do not end, yet they have a successor in the ordinal numbers, namely . This is an established mathematical fact. — 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")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
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:-This business about actions is what confuses people. — fishfry
What does "after" mean here?By the way, ω is the "point at infinity" after the natural numbers — 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.)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
In that respect, yes. But I can't help thinking about the ways in which they are different.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
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.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
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 two, calculus classes are not designed to teach people how to think about limits in the more general ways that mathematicians sometimes do. — fishfry
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?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
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?Put those together with quasi-physical entities like physics-defying lamps, and you have a recipe for confusion. — fishfry
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
It's only a conceptual thought experiment. And why shouldn't math apply to that? — fishfry
But anyway, it's an upper bound. If it's a least upper bound, it's a limit. — 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.the set {1/2, 3/4, 7/8, ..., 1} — fishfry
Which works until you ask if ω is even or odd.and we inquire about the final state at ω
Totally agree, but I'm not aware of anybody claiming a proof that supertasks are possible. Maybe I missed it.Using mathematics to try to prove that supertasks are possible is a fallacy. — Michael
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.
For this reason, Earman and Norton conclude with Benacerraf that the Thomson lamp is not a matter of paradox but of an incomplete description.
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
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. — Michael
"Tending towards infinity" means counting through the natural numbers - the set is infinite. The process has no end.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. — Relativist
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
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
Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics? — noAxioms
That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things. — noAxioms
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
Octonians shows signs of this sort of revolution. — noAxioms
Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them. — noAxioms
It is a mistake to talk about 'time creating these other universe'. — noAxioms
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
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
And two, the many-world interpretation of quantum physics.
That's the type III. — noAxioms
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
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
Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable. — noAxioms
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
Ah, local boy. I am more used to interacting with those who walk a km. There's more of em. — noAxioms
And suppose that in the first bubble universe, somebody says "1".
The universes in eternal inflation theory are not countable. — noAxioms
Yes, each step in a supertask can and does have a serial number. That's what countably infinite means. — noAxioms
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