Question about random selection.

[Mano]

All this talk of infinite sets got me to thinking. Is it possible to truly make a random selection from an infinite set? It seems like this is equivalent to the axiom of choice, or something similar. But I keep thinking of examples which seem to say otherwise. Here is an example:

Suppose you are to select a positive integer completely at random. Think up as large of an integer as you can imagine, say something like 10^(10^100) and call it X. The probability that our randomly selected integer is less than X is ZERO, since there are a finite number of integers less than X and an infinite number greater than X. Therefore, can such a random selection be made?

[Candyman]

No, a random selection from an infinite set is not possible. Strictly speaking there is no such thing as 'truly' random selections, even from finite sets, but as long as we are can't understand how a selection is done, we might percieve it as random.