Kory said... > > At 1/21/04, David Barrett-Lennard wrote: > >This allows us to say the probability that an integer is even is 0.5, or > >the probability that an integer is a perfect square is 0. > > But can't you use this same logic to show that the cardinality of the even > integers is half that of the cardinality of the total set of integers? Or > to show that there are twice as many odd integers as there are integers > evenly divisible by four? In other words, how can we talk about > probability > without implicitly talking about the cardinality of a subset relative to > the cardinality of one of its supersets?
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). Clearly there is a weak relationship between cardinality and probability measures. Why does that matter? Why do you assume infinity / infinity = 1 , when the two infinities have the same cardinality? Division is only well defined on finite numbers. > > I'm not denying that your procedure "works", in the sense of actually > generating some number that a sequence of probabilities converges to. The > question is, what does this number actually mean? I'm suspicious of the > idea that the resulting number actually represents the probability we're > looking for. Indeed, what possible sense can it make to say that the > probability that an integer is a perfect square is *zero*? > > -- Kory 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. - David
