60 seconds can pass without anyone measuring it. — Michael
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
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. — Michael
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. — fishfry
Metaphysically impossible? Repeating a claim ad infinitum is neither evidence nor proof. — 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. — Michael
I've provided arguments, and examples such as Thomson's lamp that shows why. — Michael
I gave you a mathematical model that puts your unsupported claims into context. — fishfry
And it doesn't address the issue. — Michael
If I write the natural numbers in ascending order, one after the other, then this can never complete. — Michael
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
and so supertasks are nonsense. — Michael
That we can sum an infinite series just does not prove supertasks. — Michael
I have not claimed otherwise. — fishfry
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. — Michael
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
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
You can count the natural numbers by placing them into bijective correspondence with themselves. This is the standard meaning of counting in mathematics. — fishfry
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
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.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
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does?You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly? — 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.I agree that it's impossible to do infinitely many physical thinks in finite time according to present physics. — fishfry
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
You defined the second task as a non-supertask, requiring infinite time. That's why not.There is a bijection yes. It does not imply that both or neither completes.
— noAxioms
Why not?
Exactly so.That's like saying today would be April 29 even if there was never any human beings to determine this. — Metaphysician Undercover
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.I'm not the one advocating for supertasks — 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 would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. — fishfry
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
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....like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms
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
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
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
By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask. — noAxioms
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
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
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
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
I also think you are misinterpreting the meaning of limit. — Relativist
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
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
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. — Metaphysician Undercover
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
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
Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. — Metaphysician Undercover
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
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
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
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".Given that ad infinitum means "without end" — Michael
Quite so. That's why these puzzles are not simply mathematical and why I can't just walk away from them.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
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.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
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.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
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