Yes, I agree that my definition (although well defined) doesn't have a useful interpretation given your example of perfect squares interleaved with the non perfect-squares.

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- David > -----Original Message----- > From: Kory Heath [mailto:[EMAIL PROTECTED] > Sent: Wednesday, 21 January 2004 8:30 PM > To: [EMAIL PROTECTED] > Subject: RE: Is the universe computable > > At 1/21/04, David Barrett-Lennard wrote: > >Saying that the probability that a given integer is even is 0.5 seems > >intuitively to me and can be made precise (see my last post). > > We can say with precision that a certain sequence of rational numbers > (generated by looking at larger and larger finite sets of integers from 0 > - > n) converges to 0.5. What we can't say with precision is that this result > means that "the probability that a given integer is even is 0.5". I don't > think it's even coherent to talk about "the probability of a given > integer". What could that mean? "Pick a random integer between 0 and > infinity"? As Jesse recently pointed out, it's not clear that this idea is > even coherent. > > >For me, there *is* an intuitive reason why the probability that an > >integer is a perfect square is zero. It simply relates to the fact that > >the squares become ever more sparse, and in the limit they become so > >sparse that the chance of finding a perfect square approaches zero. > > Once again, I fully agree that, given the natural ordering of the > integers, > the perfect squares become ever more sparse. What isn't clear to me is > that > this sparseness has any affect on "the probability that a given integer is > a perfect square". Your conclusion implies: "Pick a random integer between > 0 and infinity. The probability that it's a perfect square is zero." That > seems flatly paradoxical to me. If the probability of choosing "25" is > zero, then surely the probability of choosing "24", or any other specified > integer, is also zero. A more intuitive answer would be that the > probability of choosing any pre-specified integer is "infinitesimal" (also > a notoriously knotty concept), but that's not the result your method is > providing. Your method is saying that the chances of choosing *any* > perfect > square is exactly zero. Maybe there are other possible diagnoses for this > problem, but my diagnosis is that there's something wrong with the idea of > picking a random integer from the set of all possible integers. > > Here's another angle on it. Consider the following sequence of integers: > > 0, 1, 2, 4, 3, 9, 5, 16, 6, 25 ... > > Here we have the perfect squares interleaved with the non perfect-squares. > In the limit, this represents the exact same set of integers that we've > been talking about all along - every integer appears once and only once in > this sequence. Yet, following your logic, we can prove that the > probability > that a given integer from this set is a perfect square is 0.5. Can't we? > > -- Kory