[AndrewC]

The set of all integers is provably larger than the set of all even integers, even though both sets have an infinite number of elements.

I believe this is wrong.

I'm infinity, you're infinity. Are we the same infinity?

Let us return to our question: Are there as many even integers as integers? Since we can match every integer n to a single even integer 2n, we must concede that there are the same number of each. The matching is called a one-to-one correspondence. Infinite sets can have one-to-one correspondences with "smaller-looking" subsets of themselves. Of course, this can never happen with finite sets--one will never match 14 objects one for one with any 9 of them. This difference is in fact a fundamental difference between finite sets and infinite sets. We may rest assured that our two questions:

• How many positive integers are there?
• How many even positive integers are there?

do indeed have the same answer, which we've called .

There are, however, different infinities.

[general_re]

Thus, the difference between Cantor's proofs and "common sense" ;)

Can a subset of elements be as large or larger than the set that contains it, if the set contains elements not within the subset? Anyway, the real problem here is trying to treat "infinity" as though it were a number....

[general_re]

Whoops, cut myself off. But anyway, the cardinality of some infinite sets is demonstrably greater than the cardinality of some other infinite sets....

...he said, bailing himself out at the last moment ;)

[Doctor Stochastic]

The natural density of a set of integers is defined to be the D(n)=(number in the set < n)/n where n is an arbitrary natural number, if this density exists as n gets arbitrarily large. Thus for any n, the number of integers divisible by three is about n/3 which gives a natural density of 1/3, likewise the even number have a natural density of 1/2 and the primes a density of (approximately) log(n)/n; the prime number density is a bit harder to show.

Comparison of sets may be done by a one-to-one correspondence. An infinite set is one which may be put into one-to-one correspondence with a proper subset of itself. For example, one can pair off each integer with its double, 1-2, 2-4, 3-6, 4-8, etc. This gives a one-to-one mapping of the integers to the even integers, showing that the set of integers is infinite. This is a different concept from that of natural density of integers.