Different Kinds of Infinity

Two sets are equivalent if the elements from one set can be put in one-to-one correspondence with the elements of another.  For example, if A = {a,b,c} and B = {1,2,3}, then the sets are not equal but they are equivalent, we could match a with 1, b with 2, and c with 3.  For finite sets, you just need to check the cardinalities.  If n(A)=n(B) then A and B are equivalent.

What about infinite sets?  Things get really interesting :)

The naturals {1,2,3,...} are said to be countably infinite.  We can show that this set is equivalent to the evens {2,4,6,...} (seems a bit counter-intuitive, but infinite sets are pretty cool that way).  We can also show that it is equivalent to the integers {..., -2, -1, 0, 1, 2, 3, ...}.  Any set that can be shown to be equivalent to the naturals is countably infinite.  Are all infinite sets countable?  Nope.  The reals are not, and Cantor's diagonal proof is an example of some beautiful mathematics (we will talk about it in class).

Here is a link to more math about infinite sets: http://www.trottermath.net/personal/infinity.html

And here is a link to a really cool video by Vi Hart about the different infinities:
How many kinds of infinity are there?
Proof some infinities are bigger than others