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



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