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On July 18 2013 09:10 DoubleReed wrote:Show nested quote +On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure.
Yep. You are completely correct.
One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject.
In a math "debate", a general rule that I observe is the following: If you're debating about technical details, then you're talking to a crank (and probably getting nowhere). If you're debating about philosophy, it's not necessarily apparent that you're talking to a crank.
Appeals to ignorance are also very common, as you've just experience. For example. people love saying that we don't understand infinity. There are some things in math that we don't understand, infinity is not one of them. Infinity is a rigorously defined and well-understood concept. With knowledge from a high school or 1st or 2nd year math course, pretty much any perceived problems or hole in our human knowledge of math that one would think of (other than famous unsolved problems), isn't actually a problem nor a hole, but rather a personal lack of knowledge in math.
On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory.
This is correct. The distinction is between countable and uncountable. As an example to back up your fact that "Sum(Measure(An)) = Measure(Union(An))" doesn't work when n is an element of an uncountable set, we can use the uncountable set [0,infinity) and set An as the independent events "a Brownian motion hits 3 at time n". Then the LHS = 0 and the RHS = 1.
Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact:
If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly.
Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.
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On July 18 2013 19:42 paralleluniverse wrote:Show nested quote +On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject.
Sure, though I fail to see how pure math is interesting given the topic. Math can describe any number of universes, its the job of physics to see which one is ours.
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On July 18 2013 19:50 Snusmumriken wrote:Show nested quote +On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. Sure, though I fail to see how pure math is interesting given the topic. Math can describe any number of universes, its the job of physics to see which one is ours.
The one where 2+2=4 is ours.
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On July 18 2013 19:50 Snusmumriken wrote:Show nested quote +On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. Sure, though I fail to see how pure math is interesting given the topic. Math can describe any number of universes, its the job of physics to see which one is ours.
You've got it the other way around. The job of physics isn't to see what math describes our universe, it's to use or develop math to describe our universe.
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On July 18 2013 20:21 paralleluniverse wrote:Show nested quote +On July 18 2013 19:50 Snusmumriken wrote:On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. Sure, though I fail to see how pure math is interesting given the topic. Math can describe any number of universes, its the job of physics to see which one is ours. You've got it the other way around. The job of physics isn't to see what math describes our universe, it's to use or develop math to describe our universe.
Agreed. Our math is only applicable to our own universe.
+ Show Spoiler +
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On July 18 2013 08:33 Shiori wrote:Show nested quote +On July 18 2013 08:06 wherebugsgo wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This'll be my last post on this subject since I regret joining the discussion in the first place. Too much miscommunication going on. The reason I mentioned infinite vs finite sets is because Shiori said pick n (some natural number) and produce that many trials of a coin being flipped. Then what's the probability you get a heads afterward. So on one hand he's saying take a finite number of trials and on the other hand he's saying take an infinite number of trials. Clearly the probabilities are going to be different in those two cases. I was addressing the finite case since that was what he first suggested. No, that isn't what I said, but I admit I worded it poorly initially. I said you have a probability function f(n) which gives you the probability of there being at least 1 heads for n trials. If n = 1, then f(n) = 1/2. If n=2, then f(n)=1/4. The limit of this function for n--> infinity is zero, which implies that the probability of a random coin flip simulator which runs forever has a zero probability of never getting heads. The point is that the probability of never getting heads is zero by definition, but it's also logically possible. That's all these examples are meant to show. There's no paradox, because probability zero doesn't mean "can't happen." I'm not sure why you're so resistant to this idea, because all it means is that the mathematical definition of probability zero means a particular thing. I attempted to explain things by moving over to the extended real line, where x/infinity = 0 for any x, because the extended real line is used in measure theory, and because it might make the example easier to intuitively understand. At this point, I'm not sure whether you reject that probability zero things do occur, or just that these particular examples are wrong. That's why I brought up the idea of randomly selecting a real number on any interval. The probability of some particular number being generated is exactly zero, and yet obviously some real number would be generated by construction. That's all I've been trying to say. Whether or not you could actually carry out any of these things in practice is largely irrelevant to the definition of the phrase "almost surely," which is the point of exercise. Also, you randomly deciding to start flaming me certainly didn't help matters, particularly when you took me stating that I'm not a high schooler as an accusation that you're a high schooler. Like what the fuck, man?
Ok now i have a question for you or anny of the math wizards in this thread.
if you do this trail an infinite amount of times, how manny times will there be a series where you never get heads?
This should not be a particulary difficult problem to solve and i am curious to the solutions people will come up with.
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^Rephrase the question please, it's written very poorly.
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On July 18 2013 20:42 Rassy wrote:Show nested quote +On July 18 2013 08:33 Shiori wrote:On July 18 2013 08:06 wherebugsgo wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This'll be my last post on this subject since I regret joining the discussion in the first place. Too much miscommunication going on. The reason I mentioned infinite vs finite sets is because Shiori said pick n (some natural number) and produce that many trials of a coin being flipped. Then what's the probability you get a heads afterward. So on one hand he's saying take a finite number of trials and on the other hand he's saying take an infinite number of trials. Clearly the probabilities are going to be different in those two cases. I was addressing the finite case since that was what he first suggested. No, that isn't what I said, but I admit I worded it poorly initially. I said you have a probability function f(n) which gives you the probability of there being at least 1 heads for n trials. If n = 1, then f(n) = 1/2. If n=2, then f(n)=1/4. The limit of this function for n--> infinity is zero, which implies that the probability of a random coin flip simulator which runs forever has a zero probability of never getting heads. The point is that the probability of never getting heads is zero by definition, but it's also logically possible. That's all these examples are meant to show. There's no paradox, because probability zero doesn't mean "can't happen." I'm not sure why you're so resistant to this idea, because all it means is that the mathematical definition of probability zero means a particular thing. I attempted to explain things by moving over to the extended real line, where x/infinity = 0 for any x, because the extended real line is used in measure theory, and because it might make the example easier to intuitively understand. At this point, I'm not sure whether you reject that probability zero things do occur, or just that these particular examples are wrong. That's why I brought up the idea of randomly selecting a real number on any interval. The probability of some particular number being generated is exactly zero, and yet obviously some real number would be generated by construction. That's all I've been trying to say. Whether or not you could actually carry out any of these things in practice is largely irrelevant to the definition of the phrase "almost surely," which is the point of exercise. Also, you randomly deciding to start flaming me certainly didn't help matters, particularly when you took me stating that I'm not a high schooler as an accusation that you're a high schooler. Like what the fuck, man? Ok now i have a question for you or anny of the math wizards in this thread. if you do this trail an infinite amount of times, how manny times will there be a series where you never get heads? This should not be a particulary difficult problem to solve and i am curious to the solutions people will come up with.
It will happen an infinite amount of times (that's the awesomeness of infinity). But it's a smaller infinity than the total amount of runs. That's another cool aspect of infinity.
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On July 18 2013 20:48 Reason wrote:
^Rephrase the question please, it's written very poorly.
One could even say it doesn't mean anything.
Reading about math on an internet forum -_-
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Hmm i thought the question was clear but i shall try rephrase it.
You do a test: you flip a coin an infinite amount of times, the change you will have no heads in this infinite serie of flips is aproaching zero (wich some people here say is equall to zero)
Now you do this test an infinite amount of times,(in other words you flip the coin an infinite*infinite amount of times) what are the odds that you will have at least one infinite series in wich no head will occur.
Its not such a weird question, i think everyone here who has studied 1st year math on university should have seen this question or a similar one when learning about grades of infinity.
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I would say the probability that any given series has no heads is 0 (almost never but not impossible) but there will be an infinite number of them. Tobberoth already answered. The probability that at least one of them will contain no heads is 1, but I'm not sure if it's guaranteed or not.
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@Rassy
Strong law of large numbers says that such trail wont happen even once.
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Ya i think toberoth is right.
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On July 18 2013 19:42 paralleluniverse wrote:Show nested quote +On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. In a math "debate", a general rule that I observe is the following: If you're debating about technical details, then you're talking to a crank (and probably getting nowhere). If you're debating about philosophy, it's not necessarily apparent that you're talking to a crank. Appeals to ignorance are also very common, as you've just experience. For example. people love saying that we don't understand infinity. There are some things in math that we don't understand, infinity is not one of them. Infinity is a rigorously defined and well-understood concept. With knowledge from a high school or 1st or 2nd year math course, pretty much any perceived problems or hole in our human knowledge of math that one would think of (other than famous unsolved problems), isn't actually a problem nor a hole, but rather a personal lack of knowledge in math. Show nested quote +On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This is correct. The distinction is between countable and uncountable. As an example to back up your fact that "Sum(Measure(An)) = Measure(Union(An))" doesn't work when n is an element of an uncountable set, we can use the uncountable set [0,infinity) and set An as the independent events "a Brownian motion hits 3 at time n". Then the LHS = 0 and the RHS = 1. Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact: If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.
Now i have a question to parralel universe who seems to be verry sure in his statements.
If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly.
Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.[/QUOTE
Now you pick a number between 0 and 1 an infinite amount of times, what are the odds to pick 0.548 exactly at least once?
If it is 0 exactly as he say, then the answer should be 0
However if it is infinitesimally close to 0, then the odds of picking this number at least once would be 1
No?
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For your first question, I still think the formulation isn't that clear, but if I interpret what you say correctly uzyszkodnik is correct, you can look up Borel's law of 0-1.
The second question is a bit clearer, and the answer is 0.
Edit : and those two questions are of very different nature.
Edit : gosh I'm stupid, I shouldn't talk about proba -_-
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On July 18 2013 21:07 Rassy wrote:Show nested quote +On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. In a math "debate", a general rule that I observe is the following: If you're debating about technical details, then you're talking to a crank (and probably getting nowhere). If you're debating about philosophy, it's not necessarily apparent that you're talking to a crank. Appeals to ignorance are also very common, as you've just experience. For example. people love saying that we don't understand infinity. There are some things in math that we don't understand, infinity is not one of them. Infinity is a rigorously defined and well-understood concept. With knowledge from a high school or 1st or 2nd year math course, pretty much any perceived problems or hole in our human knowledge of math that one would think of (other than famous unsolved problems), isn't actually a problem nor a hole, but rather a personal lack of knowledge in math. On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This is correct. The distinction is between countable and uncountable. As an example to back up your fact that "Sum(Measure(An)) = Measure(Union(An))" doesn't work when n is an element of an uncountable set, we can use the uncountable set [0,infinity) and set An as the independent events "a Brownian motion hits 3 at time n". Then the LHS = 0 and the RHS = 1. Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact: If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0. Now i have a question to parralel universe who seems to be verry sure in his statements. If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.[/QUOTE Now you pick a number between 0 and 1 an infinite amount of times, what are the odds to pick 0.548 exactly at least once? If it is 0 exactly as he say, then the answer should be 0 However if it is infinitesimally close to 0, then the odds of picking this number at least once would be 1 No?
I'll think about it.
Solution here: http://www.teamliquid.net/forum/viewmessage.php?topic_id=419603¤tpage=91#1808
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Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact:
If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly.
Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.
This is really what I was trying to say, in a roundabout way of using examples. I was very poor at communicating it, because probability isn't really my focus in math, and because I'm nothing more than an (competent, I like to think) undergraduate, so thank you very much for making this post (and same with DoubleReed).
While I probably didn't know enough to attempt to convince wherebugsgo in a precise fashion, I find these sorts of debates really helpful at learning aspects of math that I don't usually work with, because there's the opportunity to have someone criticize perceived weaknesses in an argument.
I am very relieved to know that I wasn't wrong about the probability actually being 0 over uncountably infinite possibilities. Thanks muchly.
Are you a mathematician, by the way? May I ask what your specialty is?
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On July 18 2013 22:53 paralleluniverse wrote:Show nested quote +On July 18 2013 21:07 Rassy wrote:On July 18 2013 19:42 paralleluniverse wrote:On July 18 2013 09:10 DoubleReed wrote:On July 18 2013 08:43 yOngKIN wrote:On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. measurable only in terms of our limited huma knowledge of math and physics Uhh... no. Measurable as in Lebesgue Measure. Yep. You are completely correct. One thing I've learned is that people like to debate math on the internet and get it all wrong (like this guy), when in fact math is almost never up for debate. Especially, when it's well-established, hundreds of years old math, like measure theory. Math debates are often the most futile because people substitute their intuition with unrelenting zeal in place of mathematical rigor. Particularly in topics like measure theory, which produces counter-intuitive results to those who haven't learned the subject. In a math "debate", a general rule that I observe is the following: If you're debating about technical details, then you're talking to a crank (and probably getting nowhere). If you're debating about philosophy, it's not necessarily apparent that you're talking to a crank. Appeals to ignorance are also very common, as you've just experience. For example. people love saying that we don't understand infinity. There are some things in math that we don't understand, infinity is not one of them. Infinity is a rigorously defined and well-understood concept. With knowledge from a high school or 1st or 2nd year math course, pretty much any perceived problems or hole in our human knowledge of math that one would think of (other than famous unsolved problems), isn't actually a problem nor a hole, but rather a personal lack of knowledge in math. On July 18 2013 07:42 DoubleReed wrote:
What are you guys talking about?
Finite vs Infinite shouldn't matter for those problems. Uncountable and countable are the only restrictions on such sets. There is no additivity of an uncountable number of sets.
If A and B are disjoint (and measurable), Measure(A) + Measure(B) = Measure(A U B)
You can also do this for many sets. Sum(Measure(An)) = Measure(Union(An)) where n is finite (just stretching the previous statement to multiple sets). So {An} is a finite sequence of disjoint, measurable sets.
You can also do this if n is a countably infinite set, so {An} is a countably infinite sequence of disjoint, measurable sets.
But you can't do that if n is an uncountably infinite set. That doesn't work. You can't pretend that it does, and there are plenty of easy exceptions.
I'm confused because it seems like you should be differentiating between countable and uncountable, rather than finite and infinite. This is measure theory. So just use measure theory. This is correct. The distinction is between countable and uncountable. As an example to back up your fact that "Sum(Measure(An)) = Measure(Union(An))" doesn't work when n is an element of an uncountable set, we can use the uncountable set [0,infinity) and set An as the independent events "a Brownian motion hits 3 at time n". Then the LHS = 0 and the RHS = 1. Also, no I don't understand what they're arguing about either. But in a probably futile attempt to resolve it, let me state the following fact: If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0. Now i have a question to parralel universe who seems to be verry sure in his statements. If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly. Not "approximately 0", or "infinitesimally close to 0", or "1/infinity", or "approaches 0", or "0 in the limit", or whatever. It's simply 0.[/QUOTE Now you pick a number between 0 and 1 an infinite amount of times, what are the odds to pick 0.548 exactly at least once? If it is 0 exactly as he say, then the answer should be 0 However if it is infinitesimally close to 0, then the odds of picking this number at least once would be 1 No? I'll think about it.
I'm sure you'll correct me if I'm wrong, but:
If every real number in [0,1] has equal probability of selection, then the probability of randomly selecting any particular number between [0,1] (e.g. 0.548 exactly), is 0 exactly.
Now you pick a number between 0 and 1 an infinite amount of times, what are the odds to pick 0.548 exactly at least once?
If it is 0 exactly as he say, then the answer should be 0
However if it is infinitesimally close to 0, then the odds of picking this number at least once would be 1
The bolded parts are the kicker. No finite state machine can give every real number an equal probability of selection. For instance, no FSM is capable of picking any fraction of Pi that lies in the 0-1 range.
Since there are countably infinite possible FSMs, it seems to me that the number of possible divisions of the 0-1 range should also be treated as countably infinite. So it strikes me that the following is true:
The probability of eventually randomly selecting any particular real number between [0,1] is 0 exactly.
The probability of eventually randomly selecting 0.548 is 1 exactly.
Because by specifying an actual real number you've demonstrated it to be one of the countably infinite set FSMs can generate.
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EPT
