11. Every other ordered field can be embedded in the surreals. 12. Every ordered field can be embedded into the surreal numbers. 13. Every Squares are necessarily non-negative in an ordered field . 14. Taking rational functions with rational instead of real coefficients produces a countable non-Archimedean ordered field . 15. The surreal numbers form a set, but otherwise obey the axioms of an ordered field . 16. We've gotten pretty close to an ordered field by now, but we're not quite there. 17. Formally, we say that the complex numbers cannot have the structure of an ordered field . 18. More generally, the substructures of an ordered field ( or just a group are its subgroups. 19. This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory. 20. Every ordered field is a formally real field.
