[Faraday]

Got to admit that I've never studied set theory, but the discussion here got me looking into cardinality. In particular, I found this on a site:

"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."

While I understand (I think) this correspondence, what is wrong with the statement: "For every even integer, I can generate two unique integers, i.e. given n(even), return n and n+1."

[AmishDude]

The answer is: What is 2*infinity?

Well, it's infinity. The problem comes when you add them up and see what you have.

Let N be the cardinality of the integers. Then N=2*N and N=N^2. But what is different is that N != 2^N

[Aurelius]

Your statement is true - but the emphasis is on one-to-oneness (bijectivity is a somewhat less awkward technical equivalent). The fundamental notion being that two sets, infinite or not, have the same cardinality if, and only if, there is a one-to-one correspondance between them. So, the interest is in one-to-one correspondences. I hope that I interpreted your question correctly and have answered it.