What does "Up to Isomorphism" mean? I was reading the final chapter in Spivak's Calculus and it says: There is a complete ordered field and, up to isomorphism, only one complete ordered field. I know what an isomorphism is and what it means when things are isomorphic. But I don't know what he means when it says "UP to..". I've also read things like this when talking of homotopy... Any help is appreciated!!