Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it.Yes ok, so the coach can turn into a pumpkin and the lamp can turn into a plate of spaghetti. Are you agreeing with me on that point? — fishfry
It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.I was making my point about mathematical convergent sequences. Don't know whether it strictly applies to walking. — fishfry
Yes. But I have an obstinate feeling that that fact is a reductio of the process that generated it. So I'm not questioning what you say, but rather what we make of it. — Ludwig V
It may be a bad habit to think of applications of a mathematical process. But that's what's going on with the infinite staircase. So it might be relevant to that.
3 minutes ago — Ludwig V
I agree with you. It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined.That's one of the cognitive traps of the lamp problem. IMO the final state is simply not defined by the premises of the problem, AND there is no solution that makes the sequence continuous, therefore spaghetti is as sensible as anything else. And I've convinced myself that this is the solution to the problem. — fishfry
That's the way ω is defined, isn't it? Although I'm not sure what you mean by "continuous" there.The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous. — fishfry
I said I had no problem with any of that.You believe in limits, you said so. And if you believe even in the very basics of set theory, in the principle that I can always union two sets, then I can adjoin 1 to {1/2, 1/3, 1/4, 1/5, ...} to create the set {1/2, 1/3, 1/4, 1/5, ..., 1}.
It's such a commonplace example, yet you claim to not believe it? — fishfry
Treating infinity as a number, something you didn't do in your unionized set aboveOr what is your objection, exactly?
Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2}It's an infinite sequence. I stuck the number 1 on the end.
It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element.The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set?
OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it.It's a perfect description of what's going on. And it's a revealing and insightful way to conceptualize the final state of a supertask. Which is why I'm mentioning it so often in this thread.
Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion.In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length.
It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,.Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes.
But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved).But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.
Spot on, yes.IMO the final state is simply not defined by the premises of the problem, — fishfry
Yea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere.A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael
I agree with you. — Ludwig V
It suits my approach well, in that the existence of the problem is a result of the way it is defined, or not defined. — Ludwig V
The walker is on step one, the walker is on step two, etc. So if we define the final state to be that the walker is at the bottom of the stairs, that definition has the virtue of making the walker's sequence continuous.
— fishfry
That's the way ω is defined, isn't it? Although I'm not sure what you mean by "continuous" there.
I still feel uncomfortable, because it does get to the bottom of the stairs by placing a foot on each of the stairs, in sequence. But that's exactly the hypnotism of the way the problem is defined. And if an infinite physical staircase is the scenario, then anything goes.. — Ludwig V
Well between the two of you I have no idea what a supertask is anymore.
— fishfry
A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael
I said I had no problem with any of that.
Is it a belief thing, like it is some kind of religious proposition or something? "Hey, I'm going rogue here and will suspend belief that 7 is a factor of 35". — noAxioms
Treating infinity as a number, something you didn't do in your unionized set above — noAxioms
It's an infinite sequence. I stuck the number 1 on the end.
Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2} — noAxioms
The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set?
It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element. — noAxioms
OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it. — noAxioms
Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion. — noAxioms
Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes.
It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,. — noAxioms
But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved). — noAxioms
A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time."
— Michael
Yea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere. — noAxioms
Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition? — fishfry
* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction. — fishfry
Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?
— fishfry
If supertasks are impossible and motion is possible then motion isn't a supertask. — Michael
* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction.
— fishfry
By definition supertasks are non-terminating processes, therefore you've gone wrong somewhere if you conclude that they can terminate after 2N seconds. — Michael
Also I think the clearest example I gave was that of having counted down from infinity. We can assert (explaining what happened in reverse) that I recited 0 after 60 seconds, recited 1 after 30 seconds, recited 2 after 15 seconds, recited 3 after 7.5 seconds, etc., and we can say that we can sum an infinite series with terms that match the described (and implied) time intervals, but it doesn't then follow that we can have counted down from infinity; we can't even start such a count. — Michael
The mathematics is evidently a non sequitur — Michael
, and it's a non sequitur in the case of having counted up to infinity as well. — Michael
I don't know what you mean that supertasks are nonterminating by definition. — fishfry
You did lose me when you said that counting 0, 1, 2, ... is "counting down from infinity." I did not understand that example when you gave it earlier. Mathematically, the ordered set <1, 2, 3, ...> exists, all at once. Its counting is completed the moment it's invoked into existence by the axiom of infinity. — fishfry
I don't know what you mean that supertasks are nonterminating by definition.
— fishfry
Tasks are performed ad infinitum. I never stop counting. There's always another number to count. — Michael
I'm talking about reciting the numbers. So imagine someone reciting the natural numbers up to infinity. Now imagine that process in reverse. That's what I mean by someone counting down from infinity. — Michael
It is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted down from infinity. It is impossible, even in principle, to start such a count. The maths doesn't change this. — Michael
Did I not move you, surprise you, convince you, that if you count 1, 2, 3, ... successively halving the time intervals, that you will indeed count every single natural number in finite time? If not, why not? — fishfry
But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1> — fishfry
It's easy, I'll do it right here on a public Internet forum.
1, 15/16, 7/8, 3/4, 1/2. Done.
That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is. — fishfry
Because it begs the question. — Michael
That's not counting down from infinity. — Michael
I go 1 at 60, 2 at 30, etc.
Name the first number that I fail to count
Third time I'm asking you the question.
This is a standard inductive argument. If it's impossible to name the first natural number at which a property fails to hold, the property must hold for all natural numbers.
Please give this argument some thought. — fishfry
In your opinion. But you have no proof or evidence. On the contrary, the mathematics is clear. — fishfry
But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1> — fishfry
It begs the question. Your premise is necessarily false. Such a supertask is impossible, even in principle, to start. — Michael
You just listed five rational numbers and are claiming that this is proof of you reciting all the natural numbers in descending order? — Michael
You're talking nonsense. — Michael
What number do you recite after 1? — Michael
7/8 will do just fine. I necessarily had to jump over all but finitely members of the sequence. — fishfry
No, we're reciting the numbers in descending order. It's impossible to do, even in principle. The fact that we can assert that I recite the first number in N seconds and the second number in N/2 seconds and the third number in N/4 seconds, and so on ad infinitum, and the fact that the sum of this infinite series is 2N, doesn't then entail that the supertask is possible.
That we can sum such an infinite series is a red herring. — Michael
No you haven't. Your premise begs the question and simply asserts that all the natural numbers have been recited within 60 seconds. — Michael
So do I. There's a paradox about agreement, that it is the purpose, but also the end, of the discussion. So people tend to focus on disagreements.I love when people agree with me. It happens so seldom around here — fishfry
I found that discussion very helpful.And in fact we have a name for that. In ortdinal theory, an ordinal with a predecessor is a successor ordinal. And an ordinal without a predecessor is a limit ordinal. So your intuitions are spot on. — fishfry
Have I understood right, that 0 means "walker is not on the step", and that "the step" means "the step that is relevant at this point" - which could be 10, or 2,436? So 0 would be appropriate if the walker is on the floor from which the staircase starts (up or down)But in the staircase problem, if 1 is "walker is on the step" and 0 otherwise, then we have the sequence 1, 1, 1, 1, ... which has the limit 1. So 1, the walker is on the step, is the natural state at the end of the sequence. — fishfry
I don't like that way of putting it, at least in the paradoxes. Doesn't the arrow paradox kick in when you set off in the.reverse direction? Or perhaps you are just thinking of the numbers as members of a set, not of what the number might be measuring. I suppose that's what "ordinal" means?That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is. — fishfry
I confess to not knowing the answer to Zeno. It's a clever argument. Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But Michael objects to that, for reasons I don't yet understand. — fishfry
Michael's way of putting the point is, IMO, a bit dramatic. The boring truth for me, is that the supertask exists as a result of the way that you think of the task. If you think of it differently, it isn't a supertask. It's not about reality, but about how you apply mathematics to reality.Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask? I don't understand this point. — fishfry
Well, if you insist on describing things in that way .... I'm not sure what you mean by "model". I think of what we are doing as applying a process of measuring and counting to space - or not actually to space itself, but to objects in space. A geometrical point has no dimensions at all. So it is easy to see how we can pass infinitely many points in a finite time. (I'm not quite sure how this would apply to numbers, but they do not have any dimensions either.) This doesn't apply to the paradoxes we are considering, which involve measurable lengths, but it may help to think of them differently.Not to mention that, if we take the real numbers as a model of space, we pass through uncountably many points in finite time. That's another mystery. — fishfry
That's all very well. But it also takes us back to the question what this "operation" actually is. If you think of it as an action that takes a measurable amount of time, you can't, by definition. When we perform a calculation, that is an action in physical time. But a mathematical operation isn't quite like that, and somewhere in that is the answer (possibly).A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael
Yes, but you didn't speak all the natural numbers, and indeed, if induction means what I think it means, your argument avoids the need to deal with each natural number in turn and sequence.Name the first one that's not. It's a trivial exercise to identify the exact time at which each natural number is spoken. "1" is spoken at 60, "2" at 90, "3" at 105, "4" at 112.5, and so forth.
I did not "simply assert" all the numbers are spoken. I proved it logically. Induction works in the Peano axioms, I don't even need set theory. — fishfry
Obviously the above is fallacious. It is metaphysically impossible to have recited the natural numbers in descending order. — Michael
The fact that we can sum such an infinite series is irrelevant. And the same is true of your version of the argument. — Michael
That's all very well. But it also takes us back to the question what this "operation" actually is. — Ludwig V
It happens so seldom around here :-) — fishfry
If only I knew what "metaphysically possible" means? — Ludwig V
Agree.strictly with respect to order, they are two different representations of the same ordered set. — fishfry
Are they? Does √ω have meaning? It does for numbers. It's a serious question. I am no expert on how transfinite ordinal numbers are treated. It seems like a different species, like having a set {1, 2, 3, ... , green} which is also a valid set, and countable.Transfinite ordinal numbers are numbers.
Ordering irrelevant. The set supposedly needs to be countable, and it is. Michael's definition of supertask came from wiki, and that definition says it is countable, else it's a hypertask. The SEP definition of supertask omits the 'countable' part and seemingly groups the two categories under one word.Yes, ordered set. I have been casually using the curly braces, but you are absolutely correct. {1/2, 3/4, 7/8, ..., 1} has no order, I could stick the 1 in the middle or at the beginning and it would be the same set, but I'd lose the order that I consider important.
Clearly it isn't a supertask if it is impossible to go only half the remaining distance for some intervals. If that is possible, then it must be a supertask.Yes ok but then ... how is walking across the room by first traversing 1/2, then half of the remaining half, etc., not a supertask?
I take that back. It doesn't violate the bijection. And I spelled it wrong too. So many errors.It violates thebijunction
— noAxioms
That's fine. The rational numbers are both ordered and countable, but they cannot be counted in order.Note that I no longer have an order-preserving bijection.
Sounds like the lamp problem is unsolved. It is still 'undefined'.Ah yes, why am I doing all this?
It solves the lamp problem. The lamp state is a function on <1/2, 3/4, 7/8, ..., 1> defined as "on" at 1/2, "off" at 3/4, "on" at 7/8, and so forth.
But now we see (more clearly, IMO) that the state at 1 is simply undefined. The statement of the problem defines the lamp state at each element of the sequence; but does NOT define the state at the limit.
There is no bottom, and the OP did not suggest a bottom step. He is done, and no stairs are observable. It's mathematical only, but framed with a physical sounding analogy, which makes it fall apart.Note that the staircase is different. The walker is on step 1, on step 2, etc. So the natural, continuous way of completing the sequence is to say that the walker is at the bottom of the stairs.
His assertion isn't justified, I agree.Unless the answer is that we satisfy Zeno and execute a supertask every time we walk across the room. But Michael objects to that, for reasons I don't yet understand.
So much for the postulates of relativity then. I kind of thought we demolished that idea with some simple examples. It seems to be a 'finite automata' model, and the first postulate of SR is really hard (impossbile) to implement with such a model, so a whole new theory is needed to explain pretty much everything if you're going to posit something like that. I haven't read it of course, so any criticism I voice is a strawman at best.Some speculative physicists (at least one, I believe) think the world is a large finite grid
This evaded the question ask. Sure, we all agree that if supertasks are impossible, then supertasks are impossible. He asked how you justify the impossibility of a supertask. All your arguments seem to hinge on a variant that there isn't a largest natural number.Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition?
— fishfry
If supertasks are impossible and motion is possible then motion isn't a supertask. — Michael
The wiki definition you gave made no mention of 'terminate'. If you mean that it doesn't complete, it by definition does in a finite time. If you mean that it has no terminal step, then you're making the mistake I identify just above since the definition does not require one.By definition supertasks are non-terminating processes — Michael
You also wield the term 'ad infinitum', which typically means 'going on forever', which also violates the definition which explicitly requires a finite time to the task You very much do stop counting at time 1. There is at that time not another number, so by counterexample, your assertion that you will never stop counting is false.Tasks are performed ad infinitum. I never stop counting. — Michael
If you mean that it doesn't complete, it by definition does in a finite time. If you mean that it has no terminal step, then you're making the mistake I identify just above since the definition does not require one. — noAxioms
You also wield the term 'ad infinitum', — noAxioms
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