after we finish pushing the button — Michael
THERE IS NO AFTER WE FINISH PUSHING THE BUTTON!! — Fire Ologist
They claim that it is possible to have completed an infinite succession of tasks in finite time. — Michael
I am not going to add confusion and complexity to this by starting to discuss “super tasks”. — Fire Ologist
Is this a joke? — Michael
The concept of the “super task” is not essential to anything I’m saying. — Fire Ologist
And supertasks didn’t come up until later in the post and really another way of incorrectly claiming there is anything relevant to the lamp problem at two minutes. — Fire Ologist
So why do you disagree with the other things I’m saying? — Fire Ologist
In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. — Fire Ologist
I’ve addressed all of these premises before. There is no half until after there is a whole. You don’t travel half a distance first then travel the second half and thereby complete the whole. To call a distance “half” you first call another distance “whole” and then cut it in half. The whole always comes first. So when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. — Fire Ologist
When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. But I think that Wikipedia also puts it in a reasonably clear fashion.In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable, — Fire Ologist
Though I would have said "even though the counting may never finish due to an infinite number of elements."Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements — Wikipedia
I read somewhere that Hilbert never discussed his hotel after the casual mention of it in a paper, even though it provoked enormous discussion. I'm pretty sure he invented it only to help people realize what infinity means. All these cases play in the border country between the mathematical and the physical; they are entirely imaginary (not in the sense that they are possible, but only in the sense we can imagine impossibilities). Consequently, the normal rules of possibility and impossibility are suspended and people think the fact that they can in some sense imagine them means that they are, in some sense, possible.If this is a mathematics conversation then why are we ever referring to stairs, lamps, hotels, switches, starting lines at races?? — Fire Ologist
I agree with that and for that reason think that to say that the conclusion or limit of the set can be anything at all is misleading. In a convergent series, specifying the limit is essential to defining the series. But that doesn't mean that the function that generates the set can generate it's own limit. In fact, if it could, it wouldn't be an infinite set.when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. — Fire Ologist
Sensors are placed after 100m, 150m, 175m, and so on. — Michael
So you never finish placing the sensors. — Fire Ologist
st at their locations — Michael
In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable — Fire Ologist
When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. — Ludwig V
There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts.Anyway, this is just a specialist term. It doesn't have to "make sense" to be cogent and useful. — SophistiCat
I don't disagree. But half the problem, for us ordinary folk, is understanding that procedure, especially if, as in this case, it can't actually be carried out. The difficulty is understanding the difference between "and so on" as laziness, when it could be carried out, but one is too lazy or busy to actually do so, and "and so on" in the context of a mathematical induction, when it can't. In the background, I understand, there are people who have doubts about the validity of mathematical induction.you only need to establish a procedure of how you would do it, or even just prove that such a procedure exists. — SophistiCat
I think that's too simple. It's about the applied math. The issue is about applying the math to physical space (and time). After all, there is no problem about applying ordinary arithmetic to these situations.Locations are in physical space. This isn’t a math problem yet. — Fire Ologist
Sometimes, I am so slow I cannot believe it. The answer to the question is available, if only you would apply ordinary arithmetic to the problem. All the paradox proves is that an analysis in terms of a convergent series does not apply to the question.When we reach the 200m finish line, what distance and what time is displayed on the screen? — Michael
It is as well not to confuse the conclusion you want to draw from the analysis with the point of the thought experiment. After all, Zeno did not draw your conclusion from it. Nor do I.The thought experiment is only to examine the internal consistency of continuous space and time, not the practicality of carrying out the experiment. — Michael
There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts. — Ludwig V
In the background, I understand, there are people who have doubts about the validity of mathematical induction — Ludwig V
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