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 [quote="BraveHat"][quote="extro"]First, I'm not clear on what you mean by an abstract law or how it relates to the discussion.[/quote] Is this discussion, by abstract law, I meant any inference that must be made by the positing of an abstract concept, such as, in this case, a certain type of infinite set. In my failed attempt at disproving that that type of infinite set applies to the Universe, I used the "abstract law" that "an element of an infinite set takes up no percentage of the cardinality". The Earth takes up no percentage of space within an infinite Universe. This is an abstract inference, and that was what I mean by "abstract law", but I suppose inference is a better word. [quote="extro"]In the abstract, we can have a 2 dimensional surface that is an infinite flat plane. We can also have a 2 dimensional surface that is like the surface of a sphere, finite, but without borders or any point on it having properties different from another. In the abstract, we have 3 dimensional analogs of each of those 2 dimensional constructs. The are well defined by axioms and rules of inference. Are you saying that if we observe one of these to be the case for our universe, then the other is a physical impossibility in any universe?[/quote] No, here's what I am trying to say. It's [i]possible[/i] that there are observable physical occurrences that are [i]prohibited[/i] by a certain type of infinity (infinite space, infinite time). I [i]thought[/i] that one of those observable physical occurences was "the Earth taking up some percentage of the Universe", but I was wrong. The idea that "the Earth takes up some percentage of the Universe" is prohibited by the concept of infinite space, but it is not observable. "The Earth takes some portion of the Universe" or "The Earth takes some chunk of the Universe" is observable, but is not prohibited by the concept of infinite space. Nevertheless, that does not make it [b]impossible that some observable physical occurrence is prohibited by the concepts of infinite time or infinite space[/b]. Until that is proven impossible, it remains possible. Now space exists within the third dimension, just as in Euclidean geometry, a plane exists within the 2nd dimension, and a line exists within the first dimension. So either there is a higher dimension that [i]contains[/i] space or there isn't. If there is no higher dimension which contains space, then the conclusion that our Universe is of finite space would imply that either [i]every[/i] Universe is of finite space or that our Universe is a sub-Universe of an infinite space Universe. But if our Universe is a sub-Universe of an infinite space Universe, then that means the infinite space Universe is actually our Universe, and our assumption is contradicted. Therefore, [b]if there isn't a higher dimension which contains space, then the conclusion that our Universe is of finite space implies that [i]every[/i] Universe is of finite space.[/b] Now let's suppose that there [i]is[/i] a higher dimension which contains space. Then the objects(s) in that dimension which contain our Universe space within it also contain within it [i]every object within our Universe[/i]. A 1-inch line segment that is contained within a line in Euclidean geometry, for example, would also be contained within any of the planes that contain that line. Likewise, if I observe hair on my knuckles, I need not only conclude that there is hair within [i]this Universe[/i], but also that there is hair within whatever object in a higher dimension [i]contains[/i] this Universe, and within whatever object(s) in an even higher dimension contains object(s) in the next lowest dimension which contain our Universe, and so on and so on. Instantly, by observing my knuckle hair, I can conclude that hair is a part (if only a small part) of all those objects in all those dimensions. Now, we already assume it's [i]possible[/i] that some observable physical occurrence could be prohibited by the concept of infinite space. Still assuming that's possible, then, it's also possible to conclude soundly that [i]any object which contains that observable physical occurrence[/i] is prohibited by the concept of infinite space. This means all objects in higher dimensions which contain the observable physical occurrence would be finite in space. The Universe would be finite in space, and any object containing the Universe would be finite in space because it also contains the observable physical occurrence, and any object containing that object would be finite in space, because it contains within it the observable physical occurrence. (I include as an "object" the dimension itself, if no other object directly contains the object in question). If the occurrence of my knuckle hair were somehow prohibited by the concept of infinite space then all objects in this dimension and in all higher dimensions which contained my knuckle hair would be finite in space. Now if our Universe is an element of a set of many Universes (a multiverse) then whatever object in whatever dimension contains all these Universes, also contains our Universe. And because it's possible to soundly conclude that this object is finite, it is possible to soundly conclude this finite object cannot contain Universes of infinite space, just as it's possible to sound conclude a finite square cannot contain an infinite line or a finite cube cannot contain an infinite plane. Therefore it's [b]possible that concluding soundly that our Universe is of finite space can lead to concluding soundly that every Universe is of finite space.[/b] I can sum up my whole feeling thusly: If my original argument [i]had worked[/i], i.e. if "the Earth is some percentage of the Universe" was a physically observable occurence, then I could have used the same exact argument for the object containing our "Multiverse": "the Earth is some percentage of the object containing the Multiverse". And the "proof" of the object containing the Multiverse being finite in space would be the same. I'd assume the object containing the Mulitverse is infinite in space, and point out the "contradiction" that the Earth can't be some percentage of the object containing the Multiverse, because percentage implies finitude, and it can't be no percentage of the object containing the Multiverse because it's percentage of the object containing the Multiverse is physically observed. And the object containing the Multiverse having been proved finite, would then prove that every Universe is also finite. Am I mad? I need to sleep. I will attempt to explain my other answers later...[/quote]
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extropalopakettle
Posted: Mon Sep 12, 2011 10:21 pm    Post subject: 1

Throwing this on the table: http://en.wikipedia.org/wiki/Planck_length
Nsof
Posted: Mon Sep 12, 2011 9:54 pm    Post subject: 0

If space is continuous then even a finite universe would be uncountable. No?

I earlier said that irrationals, as an abstract concept, cannot correlate with physical quantities. that was obviously wrong.
Thok*
Posted: Mon Sep 12, 2011 12:04 am    Post subject: -1

Nsof wrote:
a) a minimal quanta of time,space

This is what I basically meant by "quantum mechanics laughs at us". The problem is that a particle, even when quantized, isn't discrete: it's a probability distribution with a location that only gets realized when we observe it. And quantum mechanics doesn't force abstract particles to live on some fixed grid.

In any case, it's either finite but extremely large or uncountable.
extropalopakettle
Posted: Sun Sep 11, 2011 11:50 pm    Post subject: -2

Hmmm .... this is getting interesting.

If there's an infinite number of particles of matter, and infinite quantized space, and we just consider which locations of space are occupied by matter, then I think we have an uncountable number of possible states.

With a finite number of particles of matter, it's countable.

(gross simplification of course)
Nsof
Posted: Sun Sep 11, 2011 8:20 pm    Post subject: -3

Thok* wrote:
It's a representation of an interval [0,1], and thus uncountable
Why?
I am no expert at physics but if there is
a) a minimal quanta of time,space
b) a finite amount of mass
doesn't that make anything physical at most countable and maybe finite? (if so then obviously some events will have greater than 0 probability of occurring)
extropalopakettle
Posted: Sun Sep 11, 2011 4:00 pm    Post subject: -4

Thok* wrote:
extropalopakettle wrote:
Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking.

It's a good approximation of the case.

And a 0 probability is a good approximation to a very small non-zero probability.

Thok* wrote:
I mean, in some sense we're asking how things work in the universe at a very small scale, at which point quantum mechanics starts to laugh at everybody. But yes, despite actually happen, an exact event has probability 0. We mostly measure ranges of events.

Not sure what you're saying here. "an exact event has probability 0" ... in the real world?

That we mostly measure ranges of events does not argue for the actual events being uncountable.
Thok*
Posted: Sun Sep 11, 2011 3:54 pm    Post subject: -5

extropalopakettle wrote:
Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking.

It's a good approximation of the case.

I mean, in some sense we're asking how things work in the universe at a very small scale, at which point quantum mechanics starts to laugh at everybody. But yes, despite actually happen, an exact event has probability 0. We mostly measure ranges of events.
extropalopakettle
Posted: Sun Sep 11, 2011 3:47 pm    Post subject: -6

Thok* wrote:
extropalopakettle wrote:
Why is that so? I'll accept infinite, but why uncountable?

It's a representation of an interval [0,1], and thus uncountable.

Bravehat suggested that in the real world, there are events that occur that are from an uncountable set of possible outcomes, and thus that possible real world events can have probability 0. Is that actually the case, is what I'm asking.
Thok*
Posted: Sun Sep 11, 2011 12:30 pm    Post subject: -7

extropalopakettle wrote:
Why is that so? I'll accept infinite, but why uncountable?

It's a representation of an interval [0,1], and thus uncountable.

As I said before, the right solution is to accept that addition works well when combining a finite amount of summands, sort of works OK when combining a countable number of summands, and horribly breaks down when combining uncountably many summands. Which is why we switch to defining probability in terms of integration instead.

And then the point is that probability defined that way works as we expect it on intervals, and there are theorems that allows us to consider the probability distributions defined by integrals as infinite limits of the probability distributions defined on finite sets.
extropalopakettle
Posted: Sun Sep 11, 2011 5:42 am    Post subject: -8

BraveHat wrote:
Yes, but my point was that the set of all possible events is uncountable ...

Why is that so? I'll accept infinite, but why uncountable?
BraveHat
Posted: Sun Sep 11, 2011 4:04 am    Post subject: -9

extro wrote:
I wrote:
... couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable.

Infinite, but not uncountable.

Take the simple example of the dart thrown at a number line. Any measurement of its position will have a finite representation (thus part of a countable set of possibilities) and a degree of precision, thus being a specification of some interval in which the dart landed, rather than an exact point, and thus having an associated non-zero probability.

Yes, but my point was that the set of all possible events is uncountable and since any possible event is a member of that uncountable set, it can be considered to have a probability of 0 with respect to that set. Or is that uncountable set one in which probability is ill-defined to begin with?
extropalopakettle
Posted: Sun Sep 11, 2011 3:35 am    Post subject: -10

BraveHat wrote:
... couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable.

Infinite, but not uncountable.

Take the simple example of the dart thrown at a number line. Any measurement of its position will have a finite representation (thus part of a countable set of possibilities) and a degree of precision, thus being a specification of some interval in which the dart landed, rather than an exact point, and thus having an associated non-zero probability.
BraveHat
Posted: Sat Sep 10, 2011 11:05 pm    Post subject: -11

Thok, extro, Zag, raekuul, Nsof and others, thank you all for taking the time to answer these questions and explaining your answers the best of your abilities. I know I can seem ridiculous a lot of times with my questions, and I'm sorry if sometimes I start acting too much like I know what I'm talking about, but I really am enjoying this conversation!
BraveHat
Posted: Sat Sep 10, 2011 11:00 pm    Post subject: -12

extro wrote:
In the real world, are there possible events with probability 0?

This one is tricky semantically. I mean, theoretically, couldn't you take any single possible real world event and look at it as part of an uncountable infinite set, thus giving it a probability of 0? For example it's possible that it could rain tomorrow. "Raining tomorrow" is one element of the set of all possible future events, which is uncountable. So in that sense, we could say there is a probability of 0 that it will rain tomorrow. In that case, I would like to consider my having to go to work on Monday a member of that same set, so that I can say there is a 0 probability of me having to go to work on Monday
Thok*
Posted: Sat Sep 10, 2011 10:05 am    Post subject: -13

BraveHat wrote:

But is there a way to notate what probability 0 is greater than anther probability 0?

Yes. Instead of keeping track of the probability at a point, you keep track of the rate of change of probability at a point.

The probability of an interval in the the normal distribution is the area under the appropriate segment of the normal curve. So while any point has zero probability (the area under a point is zero), passing by points near the center of the normal curve increases the area faster than passing by points far from the center.
BraveHat
Posted: Sat Sep 10, 2011 4:34 am    Post subject: -14

Thok wrote:
No. You merely want a continuous distribution on an infinite set.

Oh, I think I made a mistake. Extro's infinite set was countable, yes? His real life situation was flipping a coin until it comes up heads, and his infinite set was the set of all possible outcomes {H, TH, TTH, TTTH,...} That's why "almost never" doesn't apply to elements of that set, because it's countable (H has a .5 chance, TH has a .25 chance, TTH a .125 chance, and TTTH a .625 chance, etc) That's why his example doesn't meet what I'm looking for.

Thok wrote:
The probability of getting any exact point on a normal distribution is 0, but it's not uniform in anyway (the interval [-1,1] in much more probable than the interval [100,102]).

But is there a way to notate what probability 0 is greater than anther probability 0?
extropalopakettle
Posted: Sat Sep 10, 2011 4:20 am    Post subject: -15

Thok wrote:
BraveHat wrote:
I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes?

No. You merely want a continuous distribution on an infinite set. The probability of getting any exact point on a normal distribution is 0, ...

I think that fails as a "real world" example though, as there's no real world way of selecting a point from a continuum. A real world dart thrown at a real world line, measured with real world methods, yields something from a countable, and perhaps finite, set.

In the real world, are there possible events with probability 0?
Thok
Posted: Sat Sep 10, 2011 1:50 am    Post subject: -16

BraveHat wrote:
I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes?

No. You merely want a continuous distribution on an infinite set. The probability of getting any exact point on a normal distribution is 0, but it's not uniform in anyway (the interval [-1,1] in much more probable than the interval [100,102]).
-----------------------------
The main reason why the normal curve is important is the central limit theorem: if you take any probability distribution that has a mean and a standard deviation and look at the probability distribution you get when you look at the average of a lot of independent experiments, it approaches the appropriate normal distribution as the number of experiments gets large.

For example, consider flipping a million coins and asking what's the odds that you get between .4 and .6 heads on average. (Here the probability distribution we start with is a coin flip, and we're looking at what happens when we do that experiment a lot.) While there is an exact answer, it requires adding up a lot of binomial coefficients and therefore is mostly hopeless to do (and runs into the problem of rounding error.)

Instead, you can use a normal approximation to the flip a million coins distribution, and you get an answer that's much easier to compute using numerical integration.

(This is also why polling works: since you take the average opinion of a lot of people, you don't have to know what the original distribution of opinions was, since the distribution of the average opinion will look normal.)
BraveHat
Posted: Sat Sep 10, 2011 12:23 am    Post subject: -17

extro wrote:
Simple example: Flip a coin until it comes up heads. Count how many flips it takes. There are an infinite number of possible outcomes, each with their own non-zero probability. Is that "real world" enough?

I'm not sure that's what I'm looking for, because I'm seeking an example of an infinite set where "almost never" applies, which is a uniform probability, yes?

wikipedia on 'almost never' wrote:

If an event is almost sure, then outcomes not in this event are theoretically possible; however, the probability of such an outcome occurring is smaller than any fixed positive probability, and therefore must be 0.

Even though each possible outcome of your real world situation has a smaller probability the more flips are tossed, none of the real world outcomes have a probability which is smaller than any fixed positive probability (unless you say infinite tosses are a possible real world outcome, but that hasn't been shown to be possible in the same sense. It's only possible that it's possible) So, rather, to find a real world use of "almost never", I would need a real world use of infinite sets with uniform distribution. Unfortunately, it's taking me a while because I never took statistics and Thok's wikipedia article is a lot to take in.
extropalopakettle
Posted: Thu Sep 08, 2011 12:36 am    Post subject: -18

Thok wrote:
BraveHat wrote:
Hmm...could you give me a real world example of how probability in infinite sets is used in statistics?

The normal curve, Student's t curve, the Chi squared distribution, the gamma distribution, ....

If you don't know how the normal curve relates to probability on an infinite set, read the wikipedia article. Then reread it.

Simple example: Flip a coin until it comes up heads. Count how many flips it takes. There are an infinite number of possible outcomes, each with their own non-zero probability. Is that "real world" enough?
Thok
Posted: Wed Sep 07, 2011 9:52 pm    Post subject: -19

BraveHat wrote:
Hmm...could you give me a real world example of how probability in infinite sets is used in statistics?

The normal curve, Student's t curve, the Chi squared distribution, the gamma distribution, ....

If you don't know how the normal curve relates to probability on an infinite set, read the wikipedia article. Then reread it.
Thok
Posted: Wed Sep 07, 2011 9:49 pm    Post subject: -20

Zag wrote:
I suppose we could use the same process for choosing a positive integer uniformly across the entire number line. We generate the digits in reverse, starting at the 1's position. To actually hit any number you name, we have to generate it, then generate zeroes forever.

The probability of generating zeroes forever is 0 (well the limit as n goes to infinity of n*(9/10)^n, which is the same thing.) Which is why we don't make integers this way.*

(*Unless you are working with the p-adic integers. But that's beyond the scope of this thread and then the fractional part has finitely many digits.)
BraveHat
Posted: Wed Sep 07, 2011 9:46 pm    Post subject: -21

Thok wrote:
On infinite sets, absolutely never doesn't generalize to any useful notion of probability, while "almost never" generalizes to an incredibly useful notion that has obvious real world applications.. Unless you don't like the normal curve, in which case you've just rendered all of statistics and polling useless.

Hmm...could you give me a real world example of how probability in infinite sets is used in statistics?

(the uselessness of statistics and polling is another debate )
Thok
Posted: Wed Sep 07, 2011 9:27 pm    Post subject: -22

BraveHat wrote:
To wit: if an abstract dart is thrown at the natural number line with uniform probability

We've already explained why you can't throw a dart at the natural number line with uniform probability, or more generally why you can't have a uniform probability on any countably infinite set.

(There's also no notion of uniform probability on the real line, but that's a separate argument.)

Quote:
The seeming contradiction only points to the failure of mathematicians to notate the difference.

On finite sets, "almost never" and "absolutely never" mean the same thing, for the correct definition of "almost never". On infinite sets, absolutely never doesn't generalize to any useful notion of probability, while "almost never" generalizes to an incredibly useful notion that has obvious real world applications.. Unless you don't like the normal curve, in which case you've just rendered all of statistics and polling useless.
BraveHat
Posted: Wed Sep 07, 2011 9:18 pm    Post subject: -23

If the typical precise point on the real number line could be written, the middle of it would look something like this:

...1740183470139247810293478102347.710438710934701384091734099...

if we were looking at a negative number, I suppose the hyphen would have to be written on the top or bottom since the left goes out to infinity.
BraveHat
Posted: Wed Sep 07, 2011 9:11 pm    Post subject: -24

The abstract dart has a tip dimension of 1 point, so it is going to hit an exact point on that infinite number line, no matter what kind of line it is. If we consider x to be any digit (including 0), then that exact point will have a value of ...xxx.xxx... where the number so written goes off to infinity both on it's left side and right side. (even if the number is "precisely .5" the infinite left x's would be all 0s and the infinite x's to the right of the 5 would be all 0's). So, because it will hit that precise number, that precise number cannot have an "absolute never" chance of being hit.
BraveHat
Posted: Wed Sep 07, 2011 8:52 pm    Post subject: -25

Zag wrote:
Your only termination is to fail to produce the precise number.

And it's possible it could never terminate, and just keep producing zeroes forever
BraveHat
Posted: Wed Sep 07, 2011 8:38 pm    Post subject: -26

...and no "Airplane!" jokes (RIP Leslie Nielsen)
BraveHat
Posted: Wed Sep 07, 2011 8:36 pm    Post subject: -27

Zag wrote:
If we keep up this process to infinity, we find that the chance of choosing something other than precisely 0.5 is...EXACTLY equal to 1. Not very, very, very close, but exactly equal.

But the question is, in layman's terms, does that probability mean that choosing something other than precisely .5 is "almost surely" or "absolutely surely"? If it's "almost surely", then "probability 1" has two different meanings when talking about the infinite, and those meanings should be distinguished in notation. If it's "absolutely surely" then it contradicts the plain fact that "precisely .5" can be picked.
raekuul
Posted: Wed Sep 07, 2011 8:11 pm    Post subject: -28

Which we do without even thinking about it most of the time.
Zag
Posted: Wed Sep 07, 2011 4:52 pm    Post subject: -29

I suppose we could use the same process for choosing a positive integer uniformly across the entire number line. We generate the digits in reverse, starting at the 1's position. To actually hit any number you name, we have to generate it, then generate zeroes forever.
Zag
Posted: Wed Sep 07, 2011 4:49 pm    Post subject: -30

You continue to refuse the concepts of infinity.

I won't even discuss throwing your abstract dart at the entire number line, but I'm willing to discuss throwing it at the real number line in the range [0,1] with uniform probability. What are the chances of hitting precisely 0.5? (By "precisely 0.5" I mean 0.50000...) Let me ask, instead, what are the chances of NOT hitting precisely 0.5

One way to simulate this operation is to say that we will generate the number through an infinite process of choosing digits. We start with 0. and continue to choose digits randomly.

For the first digit, there's only 1/10 chance of choosing a 5, so the chances of not hitting precisely 0.5 are, so far, 0.9.

Assuming we do choose a five, then we would need to choose a zero for the next number. There's again only 1/10 chance, so the chance of failing to stay on target for our precise 0.5 is now

0.9 + (0.1) * (0.9) = 0.99

If we keep up this process to infinity, we fine that the chance of choosing something other than precisely 0.5 is

sum (n=1 to infinity) (0.9 * (0.1) n ) = 0.9999...

Since we already established that Achilles DOES catch the tortoise, we know that this number is EXACTLY equal to 1. Not very, very, very close, but exactly equal.

Intuitively, you can see that this is also the case. If you are generating the random numbers and adding digits, you stop as soon as you generate a number other than 0 (after that initial 5), but you have to keep going if you generate a 0. Your only termination is to fail to produce the precise number.
extropalopakettle
Posted: Wed Sep 07, 2011 4:42 pm    Post subject: -31

BraveHat wrote:
Thok wrote:
You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).

That might work for my specific example, but I could easily make the same case for a countable infinite set. To wit: if an abstract dart is thrown at the natural number line with uniform probability, then the probability of 5 being hit being exactly 0 contradicts that fact that 5 has a chance to be hit. As extro points, countable versus uncountable doesn't really matter to the particular type of contradiction.

I should have took that back when I crossed out the question that followed it.

If an abstract dart is thrown at the natural number line with uniform probability, and the probability of each natural number being hit is exactly 0, then the sum of the probabilities is 0, which contradicts the definition of probability (probability of all possible outcomes must add up to 1).

And of course if the probabilities for all numbers are greater than 0, their sum is also not 1 (it's undefined, infinite).

For the case of reals, the continuous uniform probability distribution assigns a probability of r2 minus r1 that the chosen real is in [r1, r2]. The probability that the number chosen equals r is simply the probability that it's in [r,r], i.e. 0.
BraveHat
Posted: Wed Sep 07, 2011 3:50 pm    Post subject: -32

Thok wrote:
You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).

That might work for my specific example, but I could easily make the same case for a countable infinite set. To wit: if an abstract dart is thrown at the natural number line with uniform probability, then the probability of 5 being hit being exactly 0 contradicts that fact that 5 has a chance to be hit. As extro points, countable versus uncountable doesn't really matter to the particular type of contradiction.

No, it turns out, according to extro's link, that the real third alternative is that "probability 0" has two different meanings :"almost never" and "absolutely never". The seeming contradiction only points to the failure of mathematicians to notate the difference.
BraveHat
Posted: Wed Sep 07, 2011 1:45 pm    Post subject: -33

extro wrote:
The "throw a dart at the real number line" technique, as I said earlier, doesn't work. Darts and the real number line exist in different universes. One can't hit the other.

The unstated implication is that the dart being thrown is an abstract dart, not a real one. Both the abstract dart and the real number line co-exist in the abstract universe.
extropalopakettle
Posted: Wed Sep 07, 2011 1:39 pm    Post subject: -34

... though the notion of a randomly selected real (in [0,1], under any continuous distribution) still bugs me. Here's why:

1) I associate probability with estimates of likelihood of real world events occurring. It's the most real-world applied/motivated math, beyond simple counting and arithmetic, that there is. In the real world we measure things to compute probabilities, and the measurements, if not rational, are at least countable.

2) The "throw a dart at the real number line" technique, as I said earlier, doesn't work. Darts and the real number line exist in different universes. One can't hit the other.
BraveHat
Posted: Wed Sep 07, 2011 1:35 pm    Post subject: -35

extro wrote:
Not so. See: http://en.wikipedia.org/wiki/Almost_surely

Wow, that is a new concept for me. Utterly fascinating. But I'm very curious as to why mathematicians don't give some different or added symbol to zero when meaning "almost never". Otherwise, when talking about [0,1], "probability 0" equally applies to 1/2 being hit as 5 being hit, but clearly 1/2 being hit is "almost never" whereas 5 being hit would be "absolutely never". I'm very surprised that this difference isn't distinguished symbolically....or is it?
extropalopakettle
Posted: Wed Sep 07, 2011 12:40 pm    Post subject: -36

Thok* wrote:
BraveHat wrote:
1) To call indeterminate the chance of any particular element from an infinite set being picked under uniform distribution. 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter.

You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).

And once you reject that, there's no reason to define probability on an uncountable set point by point: you can't add up the numbers anyways to see that the entire space has probability 1. You need a generalization to do anything, and calculus is the obvious solution.

Is countable versus uncountable really what matters here?

Are you saying there is no uniform probability distribution over the rationals in [0,1]? Nevermind, there can't be ... they can be mapped one-to-one with the integers. This is ridiculously counter-intuitive.
Thok*
Posted: Wed Sep 07, 2011 11:19 am    Post subject: -37

extropalopakettle wrote:
Even with a non-uniform but continuous probability distribution, any (every) particular point on the square has a zero probability of being hit.

For example the normal curve has this property, and it's merely the most used probability distribution in statistics.
Thok*
Posted: Wed Sep 07, 2011 11:17 am    Post subject: -38

BraveHat wrote:
1) To call indeterminate the chance of any particular element from an infinite set being picked under uniform distribution. 2)To say that there is no such thing as uniform distribution of probability on an infinite set. I tend to conclude the latter.

You've missed 3: reject the idea that we can directly add up an uncountable number of numbers in a reasonable way. Which makes sense, since the most we ever add together is a countable number (in infinite series).

And once you reject that, there's no reason to define probability on an uncountable set point by point: you can't add up the numbers anyways to see that the entire space has probability 1. You need a generalization to do anything, and calculus is the obvious solution.