This is the place I go when I have a really gritty question. As usual, it's not from a book--just my curiosity:

Is there a proof that the real numbers (i.e. a complete, ordered field) are the ONLY complete ordered field (up to isomorphism)? I haven't worked much on it, but I figured it would start with... "Let A and B be two complete ordered fields..." but I'm rather at a loss for how to explain that an element of A must (through some isomorphism) be an element of B. I figured this would be how you show the equivalence of two sets, right?

Is this proof particularly complicated, or is it routine? It is possible, right?

And by the way, another thing has been bothering me. Forgive a slight lack of vocabulary. Wouldn't I be right to claim that if two sets A and B have the same cardinality, then they are the same set (up to isomorphism...)? It seems like a trivial claim, but I wanted to make sure I wasn't missing anything.

I'll appreciate any help I can get from the math community out there. Don't make fun of me--I haven't yet had the opportunity to take a set theory class.