Good point, and it's probably reasonable to make statements like "most integers are not prime" or "perfect numbers are very rare among the integers". However, I'd still disagree with the use of percentages. For example, "0% of integers" are prime, in the sense that primes become arbitrarily sparse as N -> infinity, but primes clearly exist, which doesn't conform with our concept of "0%".
The set of even numbers intuitively seems to be "50%" the size of the integers, but they're actually sets of the same "size", because there is a bijection between them. So I'm not fully comfortable with saying "50% of integers are even" even though it's intuitively true. A lot of intuitively true things are false with infinite sets. For example, we generally assume commutativity of addition to the point that we might believe:
because both sums contain the same terms, only in different order, and in finite sums we can rearrange terms in any way we wish. However, the sums converge to different values, and terms of such a sum (conditionally but not absolutely convergent) can be rearranged so as to converge to any real value.
However, we can both agree that the probability of choosing an even integer from {1..n} (or {-n..n}) -> 0.5 as n -> infinity.
True. Dealing with probability and infinite sets can cause some problems.
It's a bit weird how the size of positive integers is equal to the size of the positive even integers and yet the probability of choosing one from the other is not 100%.
Another more extreme example is that the size of the Z+ is equal to the size of the positive rational numbers (a/b where a,b in Z+) but what is the probability that a rational number is an integer?
Good point, and it's probably reasonable to make statements like "most integers are not prime" or "perfect numbers are very rare among the integers". However, I'd still disagree with the use of percentages. For example, "0% of integers" are prime, in the sense that primes become arbitrarily sparse as N -> infinity, but primes clearly exist, which doesn't conform with our concept of "0%".
The set of even numbers intuitively seems to be "50%" the size of the integers, but they're actually sets of the same "size", because there is a bijection between them. So I'm not fully comfortable with saying "50% of integers are even" even though it's intuitively true. A lot of intuitively true things are false with infinite sets. For example, we generally assume commutativity of addition to the point that we might believe:
1/1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = 1/1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ...,
because both sums contain the same terms, only in different order, and in finite sums we can rearrange terms in any way we wish. However, the sums converge to different values, and terms of such a sum (conditionally but not absolutely convergent) can be rearranged so as to converge to any real value.
However, we can both agree that the probability of choosing an even integer from {1..n} (or {-n..n}) -> 0.5 as n -> infinity.