LMom and Austin are right that infinities come in different sizes (different "cardinalities"). One fun fact, which corresponds to the fact that (for a given cardinality) infinity plus infinity equals infinity is this: the set of all the even numbers is exactly as large as the set of all the odd numbers. Furthermore, each of these is exactly as large as the set of all the integers. You can tell that the evens are the same size as the odds by lining them up (putting them in "one-to-one correspondence"). Across from 1 put 2, across from 3 put 4, across from 5 put 6, and so on. Since it never happens that you have an odd number that has no even pair (or vice-versa), you've proven that the two sets have the same number of members. Then you can do this with either evens or odds on one side and all the integers on the other. It's one of the weird paradoxes of infinite sets that there are as many even numbers as there are integers.
But not all infinite sets are the same size. Georg Cantor, in the late 19th c., used the notion of one-to-one correspondence to prove that there are more real numbers than integers. It's called
Cantor's diagonal argument. It's a pretty simple argument - many of the mathy kids will be able to understand it. What's neat is that it's not known whether there are any sets that are bigger than the natural numbers but smaller than the reals. Cantor hypothesized that there was no set whose size is strictly between that of the integers and that of the reals. This was called the
Continuum Hypothesis (abbreviated CH, and referring to the fact that the set of reals form a continuum). In 1940 Goedel showed that using the standard axioms of set theory CH cannot be proved. Then in 1963 Paul Cohen proved that using those same standard axioms CH cannot be disproved. So CH is said to be "independent" of the standard axioms of set theory. One of the things that set theorist do is to look for ways to augment the existing axioms of set theory that allow us to decide the truth of otherwise independent hypotheses.
Contemporary set theory is a very hoary subject, and I'm at best someone who enjoys watching from the sidelines. But the introduction to the problems is very recent - starting with Cantor in the late 19th c. - and the moves made in the first half century or so are all pretty understandable. Anyone with mathy kids might find it an interesting project to pursue.
BB
)
