Probability philosophical paradox?

[yonat83]

Hi,

I am currently working on probability densities and one simple question came to my mind:
Say we lived in an "2^aleph_0 world" which contains 2^aleph_0 mathematicans (dont screem already!). Now each mathematician chooses one real number between 0 and one in a bijective way.
Now we can pick some random number between 0 and one following say a uniform repartion on [0,1].

Obviously, the probability of getting one specific number is 0. But then, one mathematician is going to get his number chosen, wish apparently contradicts the fact the probability of him to be chosen was 0 a priori.

I can think of 3 ways to interpret this apparent contradiction but none of them satisfies me completely:

1 - the problem comes from the fact our world is finite and talking about aleph_1 mathematicians is absurd. Well, it doesn't convince me because first, I wouldn't have any problem with a discrete probability on the natural numbers N for exemple: then if some mathematician picked a number associated with a null probability to happen, well, then it will never be picked! That eliminates the paradox. So the contradiction comes from the type of infinity used in the experiment. Why would there be such a "phylosophical" gap between those 2 apparently equivalent ways of describing an imaginary world?
Moreover, I don't really see why our physical would be the limiting factor. We always use mathematical concepts involving infinite quantities and I can't see why I shouldn't be able to do the same here.

2 - the problem comes from the way we are physically going to choose a number between 0 and 1: Well assuming I can randomly pick a number from a finite set, say {0,1} in some amount of time, I would also be able to choose some random sequence of them forming the binary development of some random number between 0 and 1. One can argue then, that it would take an infinite amount of time to get my final number. While I agree with that, I could then imagine that each pick takes half the time of the one before. The resulting total time would then be finite as this geometric series converges. I m assuming here that the time taken for one to randomly pick from {0,1} in not bounded, but why not? maybe because, once again, our universe is finte? Every single event takes some minimal time to occur, I must agree. But isn't that due to the specific laws of our universe? I mean, I could imagine some universe where it wouldn't be the case at all!
Makes me think that maybe it comes from the fact that each "picking" takes some energy to be done in our world, so if the time of pick wasn't bounded, it would require us unbounded "power" and energy to do it wish once again contradicts the belief that our universe has limited supply of energy...

3 - The probability function is NOT a physical concept, and consequently, the apparent absurdity of the result comes from the fact we are naturally interpreting a probability as the chance of some event to happen. But maybe this interpretation is the problem: maybe in our "2^aleph_0" world , probabilities should be interpreted in a different way. Then what would be that meaning?

Anyways, I have the feeling the answer lies somewhere between those, and I m not even sure there is only one since my question is more about philosophy than mathematics.

Yonathan

 

[yonat83]

after rereading, I saw there is a mistake in 2:

I didnt meant to say the time to pick a random number in {0,1} could be un-bounded but, that it won't take some minimal time or formally, for all e>0, I can always pick such a number in less than e seconds.

the physical diffference between a {0,1} type of random and a continuous one lies in the fact that while both of them occcure in quantum physics for exemple, only the first one is measurable. the second one is only theoretical since no-one can ever calculate a physical quantity with perfect precision.

 

[Citan Uzuki]

4: Probability zero is not the same thing as impossible. Since probabilities are only countably additive, the union of an uncountable number of probability zero events may have positive probability.

A couple of other things. First, you write: "I could then imagine that each pick takes half the time of the one before. The resulting total time would then be finite as this geometric series converges." This is true only if you have a geometric series of countable length. For a sequence of uncountable length, you cannot even form a geometric series (how much time would the ωth mathematician take to make her selection? The ω+1th?). Further, it is provable that the sum of uncountably many positive numbers is always infinite, no matter how they are selected (sketch of proof -- if the sum is finite, then for any positive integer n, only finitely many of those numbers could be greater than 1/n. So let S_n be the set of all terms in the sum greater than 1/n. The set of all terms in the sum is then the union of S_n over the positive integers, which is a countable union of finite sets and therefore countable).

Second, note that the assertion that $|\mathbb{R}|=\aleph_1$ is not known to be true, and in fact is known to unprovable in ZFC (assuming that ZFC is consistent). If you want to indicate the cardinality of the real numbers without assuming the continuum hypothesis, you should write $\beth_1$ See the Wikipedia article on http://en.wikipedia.org/wiki/Beth_number" for more information.

 

[yonat83]

You're right about aleph_1, my mistake, though it doesn't really matter here.

About the picks: I think you got me wrong. what we are picking is the next digit in the binary expension of a real number between 0 and 1. As far as I know there is a countable number of digits in the expension.

Made a couple of corrections on the original post

 

[Citan Uzuki]

Ah, you're right, I though you were talking about each mathematician picking the random real numbers in sequence. In that case, ignore my second paragraph.