A few thoughts:

First, if we’re going to talk about the random picking of a rational number, we need a probability *model* for that–in other words, we need to agree on things like, “What’s the probability of picking the number 1, or the number 2, or the number 3/5, or the probability of actually picking an integer from the set of rational numbers.” In other words, we need to agree on what our probability *function* is–which sets get assigned which probabilities.

I’m assuming then when you speak of “generate a truly random rational number”, you would like for the probability distribution to be *uniform*, i.e., no number is preferred over any other number. Well, right there we got a problem. Unfortunately, there is no uniform probability distribution over a countable set (such as the rational numbers). There *may* be (though I’m not sure) such a probability distribution if we involve *nonstandard* probability (where we allow infinitesimal probabilities), but that’s not really something I’m familiar with, so I can’t help you along those lines.

Anyway, getting back to the question, and making sense of it the best I can, first, there’s a significant amount of vagueness in stating that we’ll pick infinitely many members from an infinite set. How big is infinite (in either case)? Are we going to pick aleph-naught members? Aleph-one? And how big is the set? Is it aleph-naught? Aleph-one? Aleph-4543?

Since you restricted our attention to the rational numbers, the latter question is answered–this set has cardinality aleph-naught. But the former question remains. Of course, if we were to pick aleph-one (or more) rationals at random, there’s no doubt as to the answer–it is *certain* that a rational number must be repeated (pigeon hole principle, since there’s only aleph-naught rationals).

So the only interesting possibility left is if we are picking aleph-naught rationals at random.

It’s true that there are aleph-naught pairs of rationals (since the product of two countable sets is still countable). However, about your statement, “The chance of any given pair being a match would be lim m -> infinity 1/2^m”; I must admit I’m having difficulty in seeing the reasoning behind this. Additionaly, that limit would be zero, not aleph-one (and if, as I expect, you were referring to the *denominator* as approaching aleph-one, I would disagree with that as well–if we’re talking about cardinalities, and the limit as m -> aleph-naught, the denominator would approach aleph-naught, not aleph-one).

Anyway, I hate to leave the original question unanswered, but, because of the problems I mentioned earlier about the impossibility of a uniform probability distribution over a countable set, I really can’t see any way to make the question answerable under (standard, anyway) probability theory.

An alternative approach to what I’ve already mentioned would be to consider the set of all sequences of rational numbers–such a sequence would correspond to picking aleph-naught rationals, one at a time, in order (the first term in the sequence corresponds to the first rational picked, and so on). We could place a probability distribution on this set–again, however, we would first need to agree on which probability distribution to use, since there are *many*. A natural choice would be to construct our probability distribution in some way analogously to Lebesgue measure, but it’s not clear to me offhand how the details would work out in that case.

Anway, having said all of that, my gut feeling is that (using some natural approach along the lines mentioned in the previous paragraph) we would get that a rational would be repeated with probability one. The reason I say this is that it seems like the “density”, in some sense, of sequences with repeats would be much greater than the “density” of sequences with no repeats.