Rational numbers are numbers that can be expressed as a fraction, and all prime numbers are rational numbers, actually, since they are all necessarily integers. For example, 23 is a prime number. 23 can be expressed as 23/1 or 46/2 or 69/3, etc. Perhaps you were thinking of a different set of numbers.

But when I say that nonexistence cannot be proven, you are correct in assuming that I do not speak of mathematical proofs, as those are derivations based upon previous definitions, rules, and proofs, and can only negate the existence of those things contained within its closed system. One cannot write a mathematical proof for the existence of Big Foot, for example. "Prime numbers do not exist within the set of irrational numbers" could just as easily be restated as "Prime numbers are not included within the set of irrational numbers." At no point is the actual existence of prime numbers in doubt. It is the question of existence itself with which I am concerned.

Returning to the Big Foot example, a Big Foot adherent could always claim that Big Foot was snoozing at the north end of the Forty Acre Woods while you and your team were out surveying the south end. One cannot be simultaneously everywhere and with simultaneous knowledge of everything, and for that reason, it can always be argued that those who are searching are simply looking in all the wrong places at the wrong times. Didn't see Big Foot last weekend at Yosemite? That's because he was down South with me having tea with his cousins. You didn't get any results with that telepathy experiment you ran last month? You subjects were just too preoccupied and nervous because you had them sitting in that padded white room. You haven't seen any proof of aliens visiting Earth? They're sneaky bastards with those anal probes - do you really think with their advanced technology they'd just write "Hey ya'll, we're here" in the sky?

And so it goes on. One can only prove existence, and even then there are gradations based upon the personal biases and standards of those one is trying to convince. What convinces one person will not necessarily convince another. As far as nonexistence goes, the strongest statement one can truthfully make is that there has been no conclusive evidence to suggest the existence of <insert whatever is in question>. You can't prove that it doesn't exist. You can only prove that so far, there isn't evidence that it does exist. That's actually a very important distinction.