Randomly selecting a natural number In the answer to these questions:


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*Probability of picking a random natural number, 

*Given two randomly chosen natural numbers, what is the probability that the second is greater than the first? 
it is stated that one cannot pick a natural number randomly.
However, in this question:


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*What is the probability of randomly selecting $ n $ natural numbers, all pairwise coprime?
it is assumed that we can pick $n$ natural numbers randomly.
A description is given in the last question as to how these numbers are randomly selected, to which there seems to be no objection (although the accepted answer is given by one of the people explaining that one cannot pick a random number in the first question).
I know one can't pick a natural number randomly, so how come there doesn't seem to be a problem with randomly picking a number in the last question?
NB: I am happy with some sort of measure-theoretic answer, hence the probability-theory tag, but I think for accessibility to other people a more basic description would be preferable.

By partitioning the real numbers using non-measurable sets, it is possible to relate a real selected uniformly at random from an interval like [0, 1) to a natural number using a method by which all natural numbers have an "equal" chance of being mapped to (here, equal simply means that no natural is given any sort of preference as to being selected over any other natural), however, the probability of picking any given natural number is undefined as a result of having to partition the reals into non-measurable sets in order to create the mapping.  As a result, we are not able to create a cumulative distribution function that is meaningful.  To see an example of this, see the following question:
Is this fraction undefined? Infinite Probability Question.
