Selecting a random element from an infinite set?

[danbaron]

Proposition:

It is impossible to select a random element (member) from an infinite set.

(I don't know how to rigorously prove this.)

Arguments:

1)
Say we have a box filled with 100 ping pong balls, numbered from 1 to 100. If we shake the box and select one of the balls without looking, the probability of selecting a particular ball is, 1/100.

If the box has 1000 balls, the probability is, 1/1000.

If the box has an infinity of numbered balls (assume that somehow we can write the number on the ball no matter how big the number is), then, the probability is,

the limit as n approaches infinity, of 1/n, which equals 0.

It means that if the box contains an infinite number of balls, the probability of randomly selecting any particular ball, say, number 2, is 0. (Actually, I think this argument alone comprises a proof of the proposition.)

2)
Say, that instead of a box filled with ping pong balls, we decide to simulate the experiment using a computer. We can use a random number generator to do so. Usually computers use pseudo random number generators, which are actually repeating sequences with very long periods. However, if I remember correctly, it is possible to generate real random numbers on a computer by using a device that produces electronic white noise. We could use the device to randomly set bits. It is easiest if the number of balls in the box we are simulating, is a power of 2. So, to simulate a box filled with 1024 balls, we would randomly set 10 bits, and we would get a base 2 number equivalent to the decimal range of 0 to 1023.

In the general case, if we are simulating a box filled with x ping pong balls, then, we need y bits such that, 2^y is equal to x.

But, how about if we are simulating a box filled with an infinity of ping pong balls?

Then, in order to perform the simulation we would need a computer with an infinity of bits, which is impossible.

Additionally, pretend that somehow we did have a computer with an infinity of bits. When we ran the simulation, would the computer return a random number in the range of 0 to infinity?

I don't think it would. I think the computer would always return infinity. Think of it like this. The first bit the computer returns will represent the value (0 or 1) for 2^0, the second bit it returns will represent the value for 2^1, etc. Since it has to be able to represent every possible value in the range of 0 to infinity, it will return an infinite string of bits. In order for it to return any finite number (representing the number on the ping pong ball), at some point in the string, it must never return another value of 1. All of the infinity of succeeding bits must be set to 0. And, since the probability is exactly 0.5 for each bit being 0 or 1, the probability that an infinite sequence of bits is 0, is 0. It would be the same as if you flipped a fair coin forever, and never got a tail. So, in my opinion, the result of this thought experiment is that, the computer would always return the value, infinity.

3)
You might say, "But what if I was suspended above an infinite ocean of mixed up numbered ping pong balls, couldn't I close my eyes and randomly select one?".

I think the answer is, you could not. Why? Because, I think it is impossible to mix an infinity of ping pong balls. No matter how many you mixed together, you would always have an infinity remaining that you had not mixed.