1. Uniqueness of the Reals

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.

2. To answer your second question, it depends what you mean by an isomorphism. In general an isomorphism is a bijection which preserves all functions and relations in your language.

So an isomorphism of SETS is simply a bijection because there is no other structure.

As a less trivial example, an isomorphism of ORDERED sets is a bijection which preserves the order.

So as sets, the naturals and the integers are isomorphic. But as ordered sets (with their natural ordering) they are not.

These are just 2 examples. A structure can have many relations and/or functions that need to be preserved.

For your first question, the reals are the unique completion of the rationals. Since any countable dense linearly ordered set without endpoints is isomorphic to the rationals, then the reals are the unique completion of any countable dense linearly ordered set without endpoints (up to isomorphism). By isomorphism here, we mean an isomorphism of ordered sets.

Here is a reference: "Introduction to set theory," by Hrbacek and Jech, chapter 4.5.
This material is pretty standard however, and can be found in many places.

3. Thanks so much DrSteve. Great explanation.