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